Impact of Cyclic Block Copolymer Chain Architecture and Degree of Polymerization on Nanoscale Domain Spacing: A Simulation and Scaling Theory Analysis

Abstract

Cyclic block copolymers are predicted to assemble into nanostructured domains up to 40% smaller than their linear analogues, making them promising alternatives for nanoscale patterning applications. The limited cyclic block copolymer structures observed experimentally, however, have not met the domain reductions predicted by scaling theory. Through a systematic dissipative particle dynamics simulation study of linear and cyclic block copolymer assembly into lamellar and cylindrical nanostructures, we explore these discrepancies. We find standard implementations of scaling theory, which assume a cyclic BCP behaves as a linear molecule of half the contour length, fail to account for finite chain size effects, resulting in overpredictions of the extent of domain shrinkage upon cyclization. We propose a revised scaling law that clarifies the interplay of chain length, segregation strength, and chain architecture in determining domain spacing reduction attainable by molecular cyclization and offers an explanation for the discrepancies between prior theoretical predictions and experimental results.

Blending Linear and Cyclic Block Copolymers to Manipulate Nanolithographic Feature Dimensions

“…Because the values of χ N in these simulations are well above the order–disorder transition and in a regime where SST scaling behavior has been established previously, 40 we can rationalize our simulation observations by extending SST 44 − 46 to polymer blends to describe the dependence of the domain spacing on blend composition. Our application of SST principles differs from prior SST- and mean-field theory work related to linear–linear BCP blends 47 − 49 because our framework also draws on prior work related to the scaling behavior of cyclic BCPs compared to their linear counterparts.…”

“…Parameter Λ consists of an architecture-dependent term that grows with N modified by a correction to capture the fact that the bonds between unlike monomers seated at a lamellar interface, b AB , are ∼35% longer than those between like monomers, b AA = b BB . 40 This approach gives the following definitions for cyclic and linear BCPs, respectively: As the polymer chains become longer, the extents asymptotically approach Λ lin → N lin and Λ cyc → N cyc /2 as expected based on random walk scaling. For the short, coarse-grained polymers simulated here, however, the impacts of finite polymer length and localized stretching of the bonds at the interface ( b AB > b AA ) must be taken into account.…”

“…Figure 2 a shows the domain spacings of cyclic and linear BCP blends as a function of φ cyc at N = 8, 12, and 16 (corresponding to χ N = 98, 147, and 196, respectively; thus, all blend components exhibit SST scaling behavior as established previously by Goodson et al . 40 ). In all three blends, the cyclic and linear polymers have the same number of monomers, so previous experimental 12 and simulation 19 , 40 work suggest that, for the pure components, d cyc should be approximately 30% smaller than d lin ( d lin ≈ for ideal random walk scaling).…”

“… 40 ). In all three blends, the cyclic and linear polymers have the same number of monomers, so previous experimental 12 and simulation 19 , 40 work suggest that, for the pure components, d cyc should be approximately 30% smaller than d lin ( d lin ≈ for ideal random walk scaling). Indeed, we see that d (φ cyc = 1)/ d (φ cyc = 0) (the ratio of the pure cyclic to pure linear BCP lamellar spacings, d cyc / d lin ) ranges from 0.68 ( N = 16) to 0.71 ( N = 8).…”

“…Our application of SST principles differs from prior SST- and mean-field theory work related to linear–linear BCP blends 47 − 49 because our framework also draws on prior work related to the scaling behavior of cyclic BCPs compared to their linear counterparts. 40 In the original development of SST, the free energy of a pure BCP melt is expressed as a linear combination of contributions due to unlike monomer contact at the interface between domains, which favors domain swelling to minimize inter-block contact and chain stretching, which opposes domain growth that drives chains away from their ideal Gaussian confirmations. 45 , 50 This free energy of a single chain in a pure BCP lamella can subsequently be expressed as where the first term in this expression represents the stretching penalty as a Hookean spring, the second represents the contact penalty that acts to minimize A and B interfacial growth, and l is the characteristic segment length of the polymer.…”