Block copolymers self-assemble to form a variety of phases with highly regular patterns, depending on the microscopic ordering of molecules. Paramount to understanding and controlling this "order" is to have good "order parameters"variables that can be used to track the changes occurring in the system as it transitions from disorder to order. In this paper, molecular dynamics are used to simulate the growth of lamella, cylinder, and gyroid phases from an isotropic liquid using as testbed a binary nanoparticle mixture model. The chosen model produces minimalistic versions of the basic repeat units of the phases of interest so that the formation of numerous such units can be efficiently simulated. Because phase domains are typically one-or two-particle-thick, local order parameters are developed based on the correlation of symmetries between a particle and its neighbors and used to identify and track the formation and growth of specific geometric motifs along the transition pathway. The proposed order parameter framework is expected to be useful in tracking the formation of other bicontinuous phases and, coupled with suitable techniques, to estimate transition barriers and rates.
In part 1 of this two-paper series, a local order parameter framework was put forth that could track the changes occurring when block copolymer-like mesophases formed from a disordered state. The framework was developed using a two-particle model and involved identifying the local symmetries and geometric motifs that were unique to a given mesophase. In this paper, this framework is suitably modified to track the mesophase formation of standard coarse-grained bead−spring simulation models of polymers and oligomers. In particular, a mesoscale chain model typically employed with dissipative-particle dynamics is used to study the ordering transition of a linear symmetric diblock copolymer into a lamellar phase, and a more detailed bead−spring model of branched bolaamphiphile molecules is used to track the formation of a single diamond phase. These applications illustrate the robustness of the method in handling molecules with intramolecular degrees of freedom (including multiple chemical blocks and branched architectures), varying levels of coarsegraining, and rare mesophases with complex 3D order (like the single diamond phase). These features are suggestive of the potential suitability of the proposed framework as a tool to map transition pathways leading to complex macromolecular morphologies.
Block polymers assemble into a variety of phase-separated morphologies based on volume fraction (φ) and interactions (χ) of the respective blocks. The arrangement of three different polymer blocks could either be a 3-arm star, with each block having one terminus attached to a common junction point or a linear A-B-C architecture. A versatile strategy is reported to synthesize a series of well-defined graft polymers that lie along the unexplored continuum between a 3-arm star and an A-B-C linear triblock polymer architecture. Using the technique of single-molecule insertion, precise control over the position of graft arm C along the B chain was achieved. A series of discrete graft polymers (PMMA-b-PS-g-PEO) with fixed φ and prescribed ω values that lie on the continuum between a 3-arm star (ω = 0) and linear triblock polymer (ω = 1) were synthesized. Morphological studies using small-angle X-ray scattering and conventional and energy-filtered transmission electron microscopy reveal the transition between lamellae, perforated lamellae, and cylindrical morphologies with systematic variation in the ω values, a trend attributed to the topological frustration and the associated χ values between the three blocks. Molecular dynamic simulations of coarse-grained models were found to predict phase diagrams that are consistent with the experimentally observed morphologies. Our results suggest that changes in ω lead to topological frustration which is an important additional new design parameter that can be used to tune the morphology of multiblock polymers in addition to φ and χ.
A previously introduced framework to identify local order parameters (OPs) distinctive of incipient complex mesophases, such as bicontinuous network phases, is used in this work to evaluate nucleation free-energy barriers. The sampling techniques considered are the mean-first-passage-time (MFPT) method and novel variants of umbrella sampling, including Hybrid Monte Carlo (HMC) and a dual-OP-method that uses a blunter global OP for the umbrella bias while keeping record of configurations for analysis with a local OP. These methods were chosen for their ability to minimize or avoid frequent calculation of the expensive local OP, which makes their continuous on-the-fly tracking computationally very inefficient. These techniques were first validated by studying phase-transition barriers of model systems, i.e., the vapor–liquid nucleation of Lennard-Jones argon and a binary nanoparticle model. The disorder-to-order free energy barrier was then traced for the double gyroid and single diamond formed by mesoscopic bead-spring macromolecular models. The dual OP method was found to be the most robust and computationally efficient, since, unlike HMC, it does not require the expensive local OP to be computed on-the-fly, and unlike the MFPT method, it can negotiate large barriers aided by the biased sampling. The dual OP method requires, however, that a cheap global OP be identified and correlated (in a post-processing step) with the local OP that describes the structure of the critical nucleus, a process that can be aided by machine learning.
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