Article Cite This: Macromolecules XXXX, XXX, XXX−XXX
A Detailed Examination of the Topological Constraints of LamellaeForming Block Copolymers Abelardo Ramírez-Hernández,*,†,§ Brandon L. Peters,§ Ludwig Schneider,∥ Marat Andreev,§ Jay D. Schieber,*,⊥ Marcus Müller,*,∥ Martin Kröger,*,# and Juan J. de Pablo*,†,§ †
Materials Science Division & Institute for Molecular Engineering, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, Illinois 60439, United States § Institute for Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, United States ∥ Institut für Theoretische Physik, Georg-August Universität, 37077 Göttingen, Germany ⊥ Center for Molecular Study of Condensed Soft Matter, Department of Chemical and Biological Engineering and Department of Physics, Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, United States # Polymer Physics, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland ABSTRACT: A microscopic molecular model of polymeric molecules that captures the effects of topological constraints is used to consider how microphase segregation can alter the distribution of entanglements both in space and along chain contours. Such topological constraints are obtained by using the Z1 algorithm, and it is found that for diblock copolymers in the lamellar morphology they are not homogeneously distributed, but instead exhibit a spatial dependence as a consequence of the self-organization of the polymer blocks. The specific shape of the inhomogeneous distribution is affected by the molecular weight of the copolymer. The microscopic information obtained by these calculations is then compared with the corresponding results generated from a coarser description of entangled block copolymers that includes soft intermolecular interactions and slip-springs, whose role is to incorporate the effects of entanglements that are lost during coarse-graining. This comparison is helpful for improving coarse-grained simulation approaches for use in multiscale studies of large-scale, self-assembled multicomponent polymer systems.
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INTRODUCTION Important efforts have been devoted to the study of complex viscoelasticity in polymeric liquids at a molecular level.1−3 Elegant theories and sophisticated simulation approaches have been developed, leading to a conceptual framework that has helped rationalize the dynamical behavior of homopolymers.4,5 For sufficiently high molecular weights, topological constraints, the so-called “entanglements”, govern the dynamics of every molecule in the melt. As a consequence of these topological interactions, molecular-scale relaxation is constrained, chain diffusivity is slow, and the viscosity of the melt is large. These processes, however, are globally isotropic. On the other hand, little is known about those topological constraints when microphase segregation comes into play. In the case of block copolymers below the order−disorder transition temperature, in particular, phase segregation breaks rotational and translational symmetry, giving place to anisotropic chain diffusion.6 Moreover, chain conformations are altered with respect to those encountered in the disordered, isotropic phase.7−9 It is therefore anticipated that topological constraints will be affected too and, perhaps, the anisotropy might be reflected on the distribution of © XXXX American Chemical Society
such constraints. Inhomogeneities of the topological constraints could in turn influence chain dynamics and the ordering kinetics of nano- or mesoscopic polymer domains. A comprehensive understanding of the relationship between molecular chain dynamics and mesoscopic self-organization is essential for molecular design of materials and processes leading to controlled self-assembly on long length scales. This is particularly relevant for lithographic applications, as block copolymers have emerged as a compelling candidates for nanoscale fabrication of nextgeneration electronic devices, where their self-assembly can be directed through the use of surface patterns to create templates for synthesis of organic and inorganic structures such as nanowires,10−12 quantum dots,13,14 magnetic storage media,15 and silicon capacitors.16 In the context of lithography, attention has focused on the ability to control the self-assembly and orientation of the nanodomains.17 However, assembling the block copolymers into defect-free structures can be difficult. It is Received: July 13, 2017 Revised: February 13, 2018
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DOI: 10.1021/acs.macromol.7b01485 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
Experiments have shown that entanglements do affect the chain diffusivity in both the parallel and perpendicular directions to the interface in lamella-forming block copolymers.35,36 Previous simulations of Lennard-Jones-type models by Murat et al. examined the effect of phase segregation on the topological constraints for short chains37 and showed that in the strong segregation limit chain stretching lowers the entanglement density at the interface. More recently, however, Sethuraman et al.38 have reported an increase of the entanglement density at the interface. Their analysis was carried out by invoking packing arguments originally introduced by Kavassalis and Noolandi39 and in the context of a self-consistent-field theory and Monte Carlo simulations of the Kremer−Grest model. These authors considered chains of length N = 100. In contrast, Murat et al.37 considered chains of up to N = 400. For the parameters used in both of these studies, the average number of polymer beads between topological constraints was approximately 48. In this work we also use a microscopic molecular model of diblock copolymers to examine melts of N = 100 and N = 400. Topological constraints are subsequently obtained for both N by using the Z1 algorithm.40−42 It is found that topological constraints are not homogeneously distributed but instead exhibit a nonuniform spatial localization as a consequence of the self-organization of the polymer blocks. The specific shape of the inhomogeneous distribution is affected by the molecular weight, N. The microscopic information obtained from these calculations is then compared with the corresponding results generated by a direct implementation of a slip-spring model that was originally developed for homopolymers. Our results indicate that the slip-spring model offers a reasonable description of the spatial organization of entanglements in ordered block copolymer systems and could serve as the basis for studies of rheology in this class of materials.
known that defective states are highly thermodynamically unfavorable;18 such defects, however, still arise in experiments,19 serving to highlight their nonequilibrium nature and the importance of kinetics and chain dynamics on the relaxation to equilibrium.20,21 In particular, it has been shown that the free energy landscape associated with defect annihilation displays complex topography with large kinetic barriers. The topography depends on the types of defects, the distance between them, and, more importantly, the presence of chemical patterns directing the assembly.22 The multiscale nature of block copolymer self-assembly processes, which encompasses the range from angstroms and picoseconds to micrometers and seconds, renders the study of such processes particularly challenging for atomistic computational models. One must therefore rely on coarse-grained approaches. On the one hand, bottom-up (or systematic) coarsegraining strategies aim to generate models with effective interactions that encode microscopic information. Typically a hard-core limit is enforced at short distances that in conjunction with a finite bond length ensures noncrossability between effective particles or molecules.23 On the other hand, in topdown coarse-grained approaches, the universality of thermodynamics is invoked to identify the effective interactions needed to capture the relevant physics at the coarse-grained level.24 The latter approaches are powerful in that they can handle longer length and time scales at a lower computational cost, while still maintaining molecular-level resolution. One drawback associated with these top-down approaches is that intermolecular potential interactions are soft, leading to violations of the noncrossability constraints. In that limit, the entangled dynamics that arise from collective topological interactions in long polymer chains are no longer captured. Given the potential relevance of such topological constraints on the ordering kinetics, it is critical to develop top-down approaches that are capable of incorporating the effects of topological interactions on coarse-grained polymer models. A promising class of such models relies on slip-springs (SS) to represent the effect of entanglements on chain dynamics.25−30 In recent work, we have examined the ability of slip-spring models to capture the main relaxation mechanisms of entangled polymers. We have shown that it is possible to reproduce transport and rheology data in quantitative agreement with experiment, both at equilibrium and far from equilibrium, in the nonlinear regime.31,32 Slip-spring-based approaches have been primarily focused on the rheology and chain dynamics of homopolymers. However, they offer the potential to describe the dynamics of entangled polymer melts subject to compositional inhomogeneities.27,33 In this regard, it is important to emphasize some of the assumptions that one would make in a first attempt to implement slip-spring dynamics in diblock copolymers. First, slip-springs would move along the chain with the same “local” dynamics, irrespective of the segments’ identity (A or B) in a diblock copolymer molecule. Second, we note that, in general, there is not a one-to-one correspondence between an entanglement and a slip-spring; the mapping between these two concepts must be elucidated through detailed comparisons between microscopic and coarse-grained representations of the same materials.34 The aim of the present work is to investigate structural information regarding topological constraints of phaseseparated block copolymers using a detailed model (LennardJones chains) and to use such information to evaluate the performance of a zeroth-order slip-spring representation of the same materials.
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MODELS AND SIMULATION METHODS
Microscopic FENE Model. As a bottom-up microscopic model to represent a block copolymer melt composed of AB polymer chains, we have chosen the “standard” model (also called Kremer−Grest multibead FENE chain model),23 which is able to capture the effects of the topological constraints naturally. Following the literature,23,43 we set the parameters that define the model as follows: energy depths ϵAA = ϵBB = ϵ and ϵAB/ϵ = 1 + ϵ̃, characteristic lengths σAA = σBB = σAB = σ, and cutoff distance rc = 21/6σij, where the values of σ = ϵ = kBT = 1.0. We study AB copolymer melts at a bead number density of ρ0σ3 = 0.85. The strength of the microphase segregation is controlled by ϵ̃. We simulate melts comprising a total of n = 200 polymer molecules, having polymerization indices N = 100 and 400. The incompatibility parameter is set to unity ϵ̃ = 1. Changing the number of beads per chain, N, simultaneously affects various properties. First, using a previous order-of-magnitude estimate χ ∼ ϵ̃ (ref 43), we find that the systems correspond to the strong segregation limit, and the incompatibility between blocks increases, χN ∼ 100 and 400, respectively. Second, the two blocks, A and B, have identical number of monomers, and the corresponding lamellar spacings, L0, obtained in our simulations are L0 ≈ 28.16σ and L0 ≈ 72.55σ for N = 100 and 400, respectively. Thus, increasing N (and therefore increasing χN), we observe that the lamellar spacing increases faster than the average end-to-end distance, Re0, in the homogeneous melt, Re02 = (N − 1)b2; i.e., the large-scale chain conformations become more stretched. Third, upon increasing N, we also increase the B
DOI: 10.1021/acs.macromol.7b01485 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
and nonbonded contributions, Hnb: H = Hb({ri}) + Hnb[ϕA,ϕB]. The intramolecular contribution is given by Hb/kBT = (3/ 2)∑i
