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 micron- scale, lateral confinement as a means of achieving defect-free configurations in thin block copolymer films. Specifically, we will focus on thin films of cylinder- forming 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).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 micron- scale, lateral confinement as a means of achieving defect-free configurations in thin block copolymer films. Specifically, we will focus on thin films of cylinder- forming 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). 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.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. ‘A’ Wetting Majority Component Figure 1: Representative density composition profiles, Φ A in yellow and corresponding Voronoi diagrams (hexagon in white, pentagon in gray, and heptagon in black) for an A-attractive wall. (a) and (b) are density profiles and Voronoi diagrams, respectively, for l = R g, (c) and (d) are for l = R g, and (e) and (f) are for l = R g. ‘A’ Wetting Majority Component Figure 1: Representative density composition profiles, Φ A in yellow and corresponding Voronoi diagrams (hexagon in white, pentagon in gray, and heptagon in black) for an A-attractive wall. (a) and (b) are density profiles and Voronoi diagrams, respectively, for l = R g, (c) and (d) are for l = R g, and (e) and (f) are for l = R g. Annealed Simuations Figure 4: Graphs of (a) vs. l and (c) vs. l after a χN anneal from random initial conditions at χN = 12 to χN = 17 for an A attractive wall (χ w N = 17). There is a region from l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Graphs of (b) vs. l and (d) vs. l after a χN anneal from random initial conditions at χN = 12 to χN = 17 for a B attractive wall (χ w N = -17). There is a region from l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Annealed Simuations Figure 4: Graphs of (a) vs. l and (c) vs. l after a χN anneal from random initial conditions at χN = 12 to χN = 17 for an A attractive wall (χ w N = 17). There is a region from l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Graphs of (b) vs. l and (d) vs. l after a χN anneal from random initial conditions at χN = 12 to χN = 17 for a B attractive wall (χ w N = -17). There is a region from l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Numerical SCFT Modified Diffusion Equation: Single Chain Partition Function: Density Equations: Saddle Point Equations: Numerical SCFT Modified Diffusion Equation: Single Chain Partition Function: Density Equations: Saddle Point Equations: Discussion and Future Plans 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 χ w N was selected to be χ w N = 17 or χ w N = -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 and the standard deviation (SD) of nearest neighbors (NN) microdomain separations inside the confining hexagon. For the quenched simulations, we observe a commensurability window of l = to l = for the case of an A-attractive wall. For the B-attractive wall, the ordered window extends from l = to l = For the annealed simulations, the ordered window extends from l = to l = for the A-attractive wall. For the B-attractive wall, we see an ordered window that extends from l = to l = Thus the χN annealing has effectively equalized the ordering effects of the A- and 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. Discussion and Future Plans 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 χ w N was selected to be χ w N = 17 or χ w N = -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 and the standard deviation (SD) of nearest neighbors (NN) microdomain separations inside the confining hexagon. For the quenched simulations, we observe a commensurability window of l = to l = for the case of an A-attractive wall. For the B-attractive wall, the ordered window extends from l = to l = For the annealed simulations, the ordered window extends from l = to l = for the A-attractive wall. For the B-attractive wall, we see an ordered window that extends from l = to l = Thus the χN annealing has effectively equalized the ordering effects of the A- and 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. 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. DMR 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. DMR 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 ComputationsReferences [1] Fredrickson, GH. The Equilibrium Theory of Inhomogeneous Polymers. Clarendon Press, Oxford, [2] Matsen, MW. Thin films block copolymer. Journal of Chemical Physics 106 (1997), [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. [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), References [1] Fredrickson, GH. The Equilibrium Theory of Inhomogeneous Polymers. Clarendon Press, Oxford, [2] Matsen, MW. Thin films block copolymer. Journal of Chemical Physics 106 (1997), [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. [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), ‘B’ Wetting Minority Component Figure 2: Representative density composition profiles, Φ A in yellow and corresponding Voronoi diagrams (hexagon in white, pentagon in gray, and heptagon in black) for a B-attractive wall. (a) and (b) are density profiles and Voronoi diagrams, respectively, for l = R g, (c) and (d) are for l = R g, and (e) and (f) are for l = R g. ‘B’ Wetting Minority Component Figure 2: Representative density composition profiles, Φ A in yellow and corresponding Voronoi diagrams (hexagon in white, pentagon in gray, and heptagon in black) for a B-attractive wall. (a) and (b) are density profiles and Voronoi diagrams, respectively, for l = R g, (c) and (d) are for l = R g, and (e) and (f) are for l = R g. Quenched Simulations Figure 3: Graphs of (a) vs. l and (c) vs. l after a quench from random initial conditions to χN = 17 for an A attractive wall (χ w N = 17). There is a region form l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Graphs of (b) vs. l and (d) vs. l after a quench from random initial conditions to χN = 17 for a B attractive wall (χ w N = -17). There is a region from l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Quenched Simulations Figure 3: Graphs of (a) vs. l and (c) vs. l after a quench from random initial conditions to χN = 17 for an A attractive wall (χ w N = 17). There is a region form l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. Graphs of (b) vs. l and (d) vs. l after a quench from random initial conditions to χN = 17 for a B attractive wall (χ w N = -17). There is a region from l = to l = over which there is a perfect array of 61 hexagonally ordered microdomains. The Model AB diblock copolymer s = contour variable in units of N (index of polymerization) f = fraction of A monomers in an AB diblock copolymer Hexagonal well χ = strength of the A and B segment repulsive interactions l = hexagonal side length We use a hexagonal wall field, φ W (r), to laterally confine the polymer melt. This field is specified as a six-fold modulated tanh function. Local incompressibility is enforced by the following constraint: φ A (r) +φ B (r) + φ W (r) =1. The Model AB diblock copolymer s = contour variable in units of N (index of polymerization) f = fraction of A monomers in an AB diblock copolymer Hexagonal well χ = strength of the A and B segment repulsive interactions l = hexagonal side length We use a hexagonal wall field, φ W (r), to laterally confine the polymer melt. This field is specified as a six-fold modulated tanh function. Local incompressibility is enforced by the following constraint: φ A (r) +φ B (r) + φ W (r) =1.