Ordered Morphologies of Confined Diblock Copolymers
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Ordered Morphologies of Confined Diblock Copolymers Yoav Tsori and David Andelman School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, 69978 Ramat Aviv, Israel ABSTRACT We investigate the ordered morphologies occurring in thin-films diblock copolymer. For temperatures above the order-disorder transition and for an arbitrary two-dimensional surface pattern, we use a Ginzburg-Landau expansion of the free energy to obtain a linear response description of the copolymer melt. The ordering in the directions perpendicular and parallel to the surface are coupled. Three dimensional structures existing when a melt is confined between two surfaces are examined. Below the order-disorder transition we find tilted lamellar phases in the presence of striped surface fields. INTRODUCTION The self-assembly of block copolymers (BCP) has been the subject of numerous studies [1]-[12]. These macromolecules are made up of chemically distinct subunits, or blocks, linked together with a covalent bond. This bonding inhibits macrophase separation, and leads to formation of mesophases with typical size ranging from nanometers up to microns. For di-block copolymers, which are made up of two partially incompatible blocks (A and B), the phase diagram is well understood [1]-[6], and exhibits disordered, lamellar, hexagonal and cubic micro-phases. The phase behavior is governed by three parameters: the chain length N = NA + NB , the fraction f = NA =N of the A block, and the Flory parameter , being inversely proportional to the temperature. The presence of a confining surface leads to various interesting phenomena [7]-[17]. The surface limits the number of accessible chain configurations and thus may lead to chain frustration. Transitions between perpendicular and parallel lamellar phases with respect to the confining surfaces have been observed in thin films [18]-[22]. In addition, the surface may be chemically active, preferring adsorption of one of the two blocks, and usually stabilizing the formation of lamellae parallel to the surface [12, 19, 21]. More complicated behavior occurs when the surface is chemically heterogeneous; namely, one surface region prefers one block while other regions prefer the second block. Compared to bulk systems, new energy and length scales enter the problem adding to its complexity. Thin-film BCP in presence of chemically patterned surfaces is of importance in many applications, such as dielectric mirrors and waveguides [23], anti-reflection coating for optical surfaces [24] and fabrication of nanolithographic templates [25].
T8.1.1
THE MODEL The copolymer order parameter is defined as (r) A (r) f , the deviation of the local A monomer concentration from its average f . Consider a symmetric (f = 1=2) BCP melt in its disordered phase (above the bulk ODT temperature) and confined by one or two flat, chemically patterned surfaces. The free energy (in units of the thermal energy kB T ) can be written as [3, 4, 5, 26, 27]:
F
Z =
1 2
2
+
1 2
h
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