Laterally Confined Diblock Copolymer Thin Films
August W. Bosse, Tanya L. Chantawansri,
Glenn H. Fredrickson, and Carlos García-Cervera
Departments of Chemical Engineering and Mathematics, UCSB
FENA Theme 3: Modeling, Simulations and Computations
‘A’ Wetting
Objective
The problem of controlling and understanding microdomain ordering in block
copolymer thin films has attracted much attention from polymer technologists. In
the context of block copolymer lithography, a challenge is to improve long-range
in-plane order of microdomains. Here we present a computational study of micronscale, lateral confinement as a means of achieving defect-free configurations in thin
block copolymer films. Specifically, we will focus on thin films of cylinderforming AB diblock copolymers confined laterally by a hexagonal well. Since the
size of the hexagonal well can be made commensurate with the optimal hexagonal
lattice formed by block copolymer cylinders in the bulk and planar steps are known
to improve order in adjacent microdomains, it is reasonable to hope that
confinement to hexagonal wells can reproducibly yield defect-free configurations.
In order to numerically simulate such a system, we apply a self-consistent field
theory (SCFT) model for an AB diblock copolymer melt combined with a masking
technique to define the well geometry[1-3]. We have chosen the size of the
hexagonal confining well such that nine cylinder rows fit across the hexagon,
which corresponds to proposed experimental confinement sizes (Kramer group –
UCSB).
‘B’ Wetting
The Model
Numerical SCFT
AB diblock copolymer
Majority Component
Minority Component
s = contour variable in units of N (index of polymerization)
f = fraction of A monomers in an AB diblock copolymer
Hexagonal well
Modified Diffusion Equation:
Single Chain Partition Function:
Density Equations:
Introduction
Block copolymer thin films represent a promising sub-optical lithographic tool.
In particular, there is considerable technological interest in using self-assembled
block copolymer microdomains to define 10 nm scale features. Thin films
consisting of a large array of microphase-separated block copolymer spheres,
cylinders, or lamellae can be used to pattern a substrate with a corresponding array
of 10 nn scale dots or lines. Such arrays are potentially useful in next generation
high-density magnetic media and semiconductor devices. However, if such devices
are to be realized, the feature arrays must exhibit high uniformity and order.
Although it is considerably difficult to generate large, 2D arrays of uniform,
well-ordered microdomains, there has been substantial work on enhancing order in
block copolymer thin films. Possible techniques of inducing order include applying
external fields, shearing the film, and graphoepitaxy (lateral confinement), among
others. Segalman et al.[4] have examined the effects of a planar wall on the ordering
of microdomains, where they observed increased lateral microdomain order within a
region extending approximately 4.75 μm from the wall. Here we study the role of
confinement geometry by investigating microdomain ordering of block copolymer
thin films laterally confined in a hexagonally-shaped well.
χ = strength of the A and B segment repulsive interactions
l = hexagonal side length
Figure 1: Representative density composition profiles, ΦA
Figure 2: Representative density composition profiles, ΦA
in yellow and corresponding Voronoi diagrams (hexagon
in yellow and corresponding Voronoi diagrams (hexagon
We use a hexagonal wall field, φW(r), to laterally confine
in white, pentagon in gray, and heptagon in black) for an
in white, pentagon in gray, and heptagon in black) for a
the polymer melt. This field is specified as a six-fold
A-attractive wall. (a) and (b) are density profiles and
B-attractive wall. (a) and (b) are density profiles and
Voronoi diagrams, respectively, for l = 15.00 Rg, (c) and
Voronoi diagrams, respectively, for l = 18.00 Rg, (c) and
(d) are for l = 16.25 Rg, and (e) and (f) are for l = 17.75
(d) are for l = 19.00 Rg, and (e) and (f) are for l = 20.00
Local incompressibility is enforced by the following
Rg.
Rg.
constraint: φA(r) +φB(r) + φW(r) =1.
modulated tanh function.
Discussion and Future Plans
Quenched Simulations
Annealed Simuations
References
[1] Fredrickson, GH. The Equilibrium Theory of Inhomogeneous Polymers. Clarendon
Press, Oxford, 2006.
In order to examine how hexagonal, lateral confinement
influences long-range order in block copolymer thin films, we conducted
2D SCFT simulations. For all simulations we set f = 0.7, thus the
majority block component is A. For the quenched simulations, χN is
held fixed at χN = 17. These values of f and χN yield SCFT solutions
corresponding to hexagonally ordered cylindrical microdomains. For the
annealing simulations χN is ramped from χN = 12 to the final value of
χN = 17. The value of χwN was selected to be χwN = 17 or χwN = -17 for
an A –attractive or B-attractive wall respectively.
To identify the width of the “commensurability window” in
hexagon size l that yields a perfect array of microdomains, we report
both the average total number of microdomains inside the confining
hexagon <N> and the standard deviation (SD) of nearest neighbors (NN)
microdomain separations inside the confining hexagon <σ>.
[2] Matsen, MW. Thin films block copolymer. Journal of Chemical Physics 106 (1997),
7781.
[3]Wu, Y, Cheng, G, Katsov, K, Sides SW, Wang J, Tang, J, Fredrickson GH, Moskovits,
M, and Stucky, GD. Chiral mesostructures by nano-confinement. Nature Materials 3
(2004), 816.
For the quenched simulations, we observe a commensurability
window of l = 15.75 to l = 17.00 for the case of an A-attractive wall. For
the B-attractive wall, the ordered window extends from l = 18.75 to l =
19.25.
[4] Segalman, RA, Hexemer, A, and Kramer EJ, Edge effects on the order and freezing
of a 2D array of block copolymer spheres. Physical Review Letters 91 (2003), 196101
Acknowledgements
We are grateful to Gila Stein, Edward Kramer, and Kirill Katsov for useful discussions.
Funding for this project was provided by the MARCO Center on Functional Engineered
Nano Architectonics (FENA).
This work made use of MRL Central Facilities supported by the MRSEC Program of the
National Science Foundation under award No. DMR05-20415.
Saddle Point Equations:
Figure 3: Graphs of (a) <N> vs. l and (c) <σ> vs. l after a quench from random initial conditions
to χN = 17 for an A attractive wall (χwN = 17). There is a region form l = 15.75 to l = 17.00 over
which there is a perfect array of 61 hexagonally ordered microdomains.
Graphs of (b) <N> vs. l and (d) <σ> vs. l after a quench from random initial conditions to χN
= 17 for a B attractive wall (χwN = -17). There is a region from l = 18.75 to l = 19.25 over which
there is a perfect array of 61 hexagonally ordered microdomains.
Figure 4: Graphs of (a) <N> vs. l and (c) <σ> vs. l after a χN anneal from random initial conditions at
χN = 12 to χN = 17 for an A attractive wall (χwN = 17). There is a region from l = 15.75 to l = 17.75 over
which there is a perfect array of 61 hexagonally ordered microdomains.
Graphs of (b) <N> vs. l and (d) <σ> vs. l after a χN anneal from random initial conditions at χN
= 12 to χN = 17 for a B attractive wall (χwN = -17). There is a region from l = 17.75 to l = 19.75 over
which there is a perfect array of 61 hexagonally ordered microdomains.
For the annealed simulations, the ordered window extends from l
= 15.75 to l = 17.75 for the A-attractive wall. For the B-attractive wall,
we see an ordered window that extends from l = 17.75 to l = 19.75. Thus
the χN annealing has effectively equalized the ordering effects of the Aand B-attractive walls. This can be explained by the formation of a
wetting layer of microdomains below (χN)ODT.
Future work includes studying defect formation in smaller and
larger hexagonally confined systems and studying other copolymer
architectures and the role of additives. In addition, we will explore other
confinement geometries such as triangular wells.