Optimizing topographical templates for directed self-assembly of block copolymers via inverse design simulations

Template Design for Complex Block Copolymer Patterns Using a Machine Learning Method

This study represents the first attempt to address the inverse design problem of the guiding template for directed self-assembly (DSA) patterns using solely machine learning methods. By formulating the problem as a multi-label classification task, the study shows that it is possible to predict templates without requiring any forward simulations. A series of neural network (NN) models, ranging from the basic two-layer convolutional neural network (CNN) to the large NN models (32-layer CNN with 8 residual blocks), have been trained using simulated pattern samples generated by thousands of self-consistent field theory (SCFT) calculations; a number of augmentation techniques, especially suitable for predicting morphologies, have been also proposed to enhance the performance of the NN model. The exact match accuracy of the model in predicting the template of simulated patterns was significantly improved from 59.8% for the baseline model to 97.1% for the best model of this study. The best model also demonstrates an excellent generalization ability in predicting the template for human-designed DSA patterns, while the simplest baseline model is ineffective in this task.

Non-Bulk Morphologies of Extremely Thin Block Copolymer Films Cast on Topographically Defined Substrates Featuring Deep Trenches: The Importance of Lateral Confinement

Directed self-assembly of block copolymers is evolving toward applications that are more defect-tolerant but still require high morphological control and could benefit from simple, inexpensive fabrication processes. Previously, we demonstrated that simply casting ultra-thin block copolymer films on topographically defined substrates leads to hierarchical structures with dual patterns in a controlled manner and unraveled the dependence of the local morphology on the topographic feature dimensions. In this article, we discuss the extreme of the ultraconfined thickness regime at the border of film dewetting. Additional non-bulk morphologies are observed at this extreme, which further elaborate the arsenal of dual patterns that could be obtained in coexistence with full placement control. It is shown that as the thickness confinement approaches its limit, lateral confinement imposed by the width of the plateaus becomes a critical factor influencing the local morphology.

Experimental and Computational Evaluation of Self-Assembled Morphologies in Diblock Janus Bottlebrush Copolymers

Diblock Janus-type "A-branch-B" bottlebrush copolymers (di-JBBCPs) consist of a backbone with alternating A and B side chains, in contrast to the side chain arrangement of conventional bottlebrush copolymers. As a result, A and B blocks of di-JBBCPs can microphase-separate perpendicular to the backbone, which is located at the interface between the two blocks. A reparametrized dissipative particle dynamics (DPD) model is used to theoretically investigate the self-assembly of di-JBBCPs and to compare with the experimental results of a range of polystyrene-branch-polydimethylsiloxane di-JBBCPs. The experimentally formed cylinder, gyroid, and lamellar morphologies showed good correspondence with the model phase diagram, and the effect of changing volume fraction and backbone length is revealed. The DPD model predicts a bulk-stable perforated lamella morphology together with two unconventional spherical phases, the Frank-Kasper A15 spheres and the hexagonally close-packed spheres, indicating the diversity of morphologies available from complex BCP molecular architectures.

Open-source platform for block polymer formulation design using particle swarm optimization

Facile exploration of large design spaces is critical to the development of new functional soft materials, including self-assembling block polymers, and computational inverse design methodologies are a promising route to initialize this task. We present here an open-source software package coupling particle swarm optimization (PSO) with an existing open-source self-consistent field theory (SCFT) software for the inverse design of self-assembling block polymers to target bulk morphologies. To lower the barrier to use of the software and facilitate exploration of novel design spaces, the underlying SCFT calculations are seeded with algorithmically generated initial fields for four typical morphologies: lamellae, network phases, cylindrical phases, and spherical phases. In addition to its utility within PSO, the initial guess tool also finds generic applicability for stand-alone SCFT calculations. The robustness of the software is demonstrated with two searches for classical phases in the conformationally symmetric diblock system, as well as one search for the Frank-Kasper [Formula: see text] phase in conformationally asymmetric diblocks. The source code for both the initial guess generation and the PSO wrapper is publicly available.

Inverse design of equilibrium cluster fluids applied to a physically informed model

Inverse design strategies have proven highly useful for the discovery of interaction potentials that prompt self-assembly of a variety of interesting structures. However, often the optimized particle interactions do not have a direct relationship to experimental systems. In this work, we show that Relative Entropy minimization is able to discover physically meaningful parameter sets for a model interaction built from depletion attraction and electrostatic repulsion that yield self-assembly of size-specific clusters. We then explore the sensitivity of the optimized interaction potentials with respect to deviations in the underlying physical quantities, showing that clustering behavior is largely preserved even as the optimized parameters are perturbed.

Inverse design of nonequilibrium steady states: A large-deviation approach

The design of small scale nonequilibrium steady states (NESS) is a challenging, open ended question. While similar equilibrium problems are tractable using standard thermodynamics, a generalized description for nonequilibrium systems is lacking, making the design problem particularly difficult. Here we show we can exploit the large-deviation behavior of a Brownian particle and design a variety of geometrically complex steady-state density distributions and flux field flows. We achieve this design target from direct knowledge of the joint large-deviation functional for the empirical density and flow, and a "relaxation" algorithm on the desired target states via adjustable force field parameters. We validate the method by replicating analytical results, and demonstrate its capacity to yield complex prescribed targets, such as rose-curve or polygonal shapes on the plane. We consider this dynamical fluctuation approach a first step towards the design of more complex NESS where general frameworks are otherwise still lacking.

Conductive Thin Films over Large Areas by Supramolecular Self-Assembly

We report a "one-step" method for preparing conductive thin films with cylindrical microdomains oriented normal to the surface over large areas using the supramolecular assembly of poly(styrene-block-4-vinylpyridine) (PS₁₉-b-P4VP₅) and 5,10,15,20-tetrakis(4-hydroxyphenyl)-21H,23H-porphine (HOTPP). HOTPP interacts with the P4VP block by hydrogen bonding between the hydroxyl group of HOTPP and pyridine ring of PS₁₉-b-P4VP₅, forming cylindrical P4VP(HOTPP) domains having an average diameter of ∼17 nm in a PS matrix. Dynamic light scattering, contact angle, and in situ grazing incidence small-angle X-ray scattering measurements show a morphological transition from spherical micelles in solution to cylindrical microdomains oriented normal to the substrate surface during the drying process. From the dependence of current on voltage, an average current of ∼4.0 nA is found to pass through a single microdomain, pointing to a promising route for organic semiconductor device applications.

Self-Directed Self-Assembly of 3D Tailored Block Copolymer Nanostructures

Directed self-assembly (DSA) of block copolymers (BCPs) provides a powerful tool to fabricate various 2D nanostructures. However, it still remains a challenge to extend DSA to make uniform and complex 3D nanostructures through BCP self-assembly. In this paper, we introduce a method to fabricate various nanostructures in 3D and test it using simulations. In particular, we employ dissipative particle dynamics (DPD) simulation to demonstrate that uniform multilayer nanostructures can be achieved by alternating the stacking of two "orthogonal" BCPs films, AB copolymer film and AC copolymer film, without the need to cross-link or etch any of the components. The assembly of a new layer occurs on top of the previous bottom layer, and thus the structural information from the substrate is propagated upward in the film, a process we refer to as self-directed self-assembly (SDSA). If this process is repeated many times, one can have tailored multilayer nanostructures. Furthermore, the natural (bulk) phases of the block copolymers in each layer do not need to be the same, so one can achieve complex 3D assemblies that are not possible with a single-phase 3D system. This method in conjunction with grapho (or chemo) epitaxy is able to evolve a surface pattern into a 3D nanostructure. Here we show several examples of nanostructures fabricated by this process, which include aligned cylinders, spheres on top of cylinders, and orthogonal nanomeshes. Our work should be useful for creating complex 3D nanostructures using self-assembly.

Kinetic Pathways of Block Copolymer Directed Self-Assembly: Insights from Efficient Continuum Modeling

We introduce a computationally efficient continuum technique to simulate the complex kinetic pathways of block copolymer self-assembly. Subdiffusive chain dynamics is taken into account via nonlocal Onsager coefficients. An application to directed self-assembly of thin films of diblock copolymers on patterned substrates reveals the conditions under which experimentally observed metastable structures intervene in the desired thin-film morphology. The approach generalizes easily to multiblock copolymers and more complex guiding patterns on the substrate, and its efficiency allows for the systematic optimization of guiding patterns and process conditions.

Open-source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units

Self-consistent field theory (SCFT) is a powerful approach for computing the phase behavior of block polymers. We describe a fast version of the open-source Polymer Self-Consistent Field (PSCF) code that takes advantage of the massive parallelization provided by a graphical processing unit (GPU). Benchmarking double-precision calculations indicate up to 30× reduction in time to converge SCFT calculations of various diblock copolymer phases when compared to the Fortran CPU version of PSCF using the same algorithms, with the speed-up increasing with increasing unit cell size for the diblock polymer problems examined here. Where double-precision accuracy is not needed, single-precision calculations can provide speed-up of up to 60× in convergence time. These improvements in speed within an open-source format open up new vistas for SCFT-driven block polymer materials discovery by the community at large.

Directed Self-Assembly of Patchy Microgels into Anisotropic Nanostructures

Multi-geometry nanostructures with high-order, complex, and controllable geometries have attracted extensive attention in the development of functional nanomaterials. A simple and versatile strategy is proposed to construct various anisotropic nanostructures through the directed self-assembly (DSA) of patchy microgels. A general criterion for interaction parameters is developed by the variance analysis method to achieve the formation of 1D nanorods by the single directional DSA process, and 2D or 3D polymorphs including V/T/h/cross shapes, multiple arms, multi-directional bending, single/multiple rings, nanocages, etc., by the multi-directional DSA process of binary microgel blends. At the optimum interaction parameters, the nanorods exhibit the quickest formation process and the most thermodynamically stable geometry, while the various 2D or 3D assemblies exhibit controlled jointing behaviors for versatile assembly geometries. The number of recognition sites on the patchy microgel surface guides the aggregation modes of microgels during the DSA process. These assemblies can bear large curvature variance with the increase of shear rates due to the high flexibility and the ability of adjusting orientation spontaneously. The DSA behavior of patchy microgels differs from the traditional self-assembly process of block copolymers, which may open a new route for guiding the formation of controllable nanoparticle architectures.

Customizing topographical templates for aperiodic nanostructures of block copolymers via inverse design

The limited complexity of self-assembled nanostructures of block copolymers seriously impedes their potential utility in the semiconductor industry. Therefore, the customizability of complex nanostructures has been a long-standing goal for the utilization of directed self-assembly in nanolithography. Herein, we integrated an advanced inverse design algorithm with a well-developed theoretical model to deduce inverse solutions of topographical templates to direct the self-assembly of block copolymers into reproducible target structures. The deduced templates were optimized by finely tuning the input parameters of the inverse design algorithm and through symmetric operation as well as nanopost elimination. More importantly, our developed algorithm has the capability to search inverse solutions of topographical templates for aperiodic nanostructures over exceptionally large areas. These results reveal design rules for guiding templates for the device-oriented nanostructures of block copolymers with prospective applications in nanolithography.

Directed self-assembly of a two-state block copolymer system

In this work, ladder-shaped block copolymer structures consisting of parallel bars, bends, and T-junctions are formed inside square confinement. We define binary states by the two degenerate alignment orientations, and study properties of the two-state system. We control the binary states by creating openings around the confinement, changing the confinement geometry, or placing lithographic guiding patterns inside the confinement. Self-consistent field theory simulations show templating effect from the wall openings and reproduce the experimental results. We demonstrate scaling of a single binary state into a larger binary state array with individual binary state control.

Quenching of fully symmetric mixtures of oppositely charged microgels: the role of soft stiffness

Using molecular dynamics simulations, we investigate the self-assembly of a coarse-grained binary system of oppositely charged microgels, symmetric in size and concentration. The microgel pair interactions are described by an effective pair potential which implicitly accounts for the averaged ionic contributions, in addition to a short-range elastic repulsion that accounts for the overlapping of the polymer chains, the latter being described by the Hertzian interaction. Particular emphasis is placed on the role played by the strength of the soft repulsive interaction on the resulting particle aggregation. It is found that the possibility of particle inter-penetration in oppositely charged soft particles results in a much wider variety of cluster morphologies in comparison with their hard-spheres counterparts. Specifically, the softness of the steric interactions enhances the competition between repulsive and attractive electrostatic interactions, leading to the formation of aggregates that are comprised of strongly bounded charged particles displaying a low degree of charge ordering.

Optimizing self-consistent field theory block copolymer models with X-ray metrology

A block copolymer self-consistent field theory (SCFT) model is used for direct analysis of experimental X-ray scattering data obtained from thin films of polystyrene-b-poly(methyl methacrylate) (PS-b-PMMA) made from directed self-assembly. In a departure from traditional approaches, which reconstruct the real space structure using simple geometric shapes, we build on recent work that has relied on physics-based models to determine shape profiles and extract thermodynamic processing information from the scattering data. More specifically, an SCFT model, coupled to a covariance matrix adaptation evolutionary strategy (CMAES), is used to find the set of simulation parameters for the model that best reproduces the scattering data. The SCFT model is detailed enough to capture the essential physics of the copolymer self-assembly, but sufficiently simple to rapidly produce structure profiles needed for interpreting the scattering data. The ability of the model to produce a matching scattering profile is assessed, and several improvements are proposed in order to more accurately recreate the experimental observations. The predicted parameters are compared to those extracted from model fits via additional experimental methods and with predicted parameters from direct particle-based simulations of the same model, which incorporate the effects of fluctuations. The Flory-Huggins interaction parameter for PS-b-PMMA is found to be in agreement with reported ranges for this material. These results serve to strengthen the case for relying on physics-based models for direct analysis of scattering and light signal based experiments.

A dissipative particle dynamics simulation study on phase diagrams for the self-assembly of amphiphilic hyperbranched multiarm copolymers in various solvents

Self-assembly of amphiphilic hyperbranched multiarm copolymers (HMCs) has shown great potential for preparing all kinds of delicate supramolecular structures in all scales and dimensions in solution. However, theoretical studies on the influencing factors for the self-assembly of HMCs have been greatly lagging behind. The phase diagram of HMCs in selective solvents is very necessary but has not been disclosed up to now. Here, the self-assembly of HMCs with different hydrophilic fractions in various solvents was studied systematically by using dissipative particle dynamics (DPD) simulations. Three morphological phase diagrams are constructed and a rich variety of morphologies, ranging from spherical micelles, worm-like micelles, membranes, vesicles, vesosomes, small micellar aggregates (SMAs), and aggregates of spherical and worm-like micelles to helical micelles, are obtained. In addition, both the self-assembly mechanisms and the dynamic processes for the formation of these self-assemblies have been systematically investigated. The simulation results are consistent with available experimental observations. Besides, several novel structures, like aggregates of spherical and worm-like micelles, vesosomes and helical micelles, are firstly discovered for HMC self-assembly. We believe the current work will extend the knowledge on the self-assembly of HMCs, especially on the control of supramolecular structures and on fabricating novel self-assemblies.

Numerical Simulations of Directed Self-Assembly in Diblock Copolymer Films using Zone Annealing and Pattern Templating

Bulk fabrication of surface patterns with sub-20 nm feature sizes is immensely desirable for many existing and emerging technologies. Directed self-assembly (DSA) of block copolymers (BCPs) has been a recently demonstrated approach to achieve such feature resolution over large-scale areas with minimal defect populations. However, much work remains to understand and optimize DSA methods in order to move this field forward. This paper presents large-scale numerical simulations of zone annealing and chemo-epitaxy processing of BCP films to achieve long-range orientational order. The simulations utilize a Time-Dependent Ginzburg-Landau model and parallel processing to elucidate relationships between the magnitude and velocity of a moving thermal gradient and the resulting BCP domain orientations and defect densities. Additional simulations have been conducted to study to what degree orientational order can be further improved by combining zone annealing and chemo-epitaxy techniques. It is found that these two DSA methods do synergistically enhance long-range order with a particular relationship between thermal gradient velocity and chemical template spacing.

Swarm Intelligence Platform for Multiblock Polymer Inverse Formulation Design

Multiblock polymers (MBPs) show great promise as a platform for synthesizing materials with highly customized nanostructures. In these systems, the subtle balance of chain configurational entropy and interactions between dissimilar block chemistries emerge a rich palette of self-assembled structural motifs that may be leveraged to impart novel functional, mechanical, or optical properties to the final material. Unfortunately, extensive study of MBP self-assembly has yielded few reliable heuristics for intuitively navigating the enormous molecular design space made available by modern polymer synthesis techniques. In order to make progress, new methods must be developed that allow researchers to efficiently screen molecular designs for achieving a desired state of self-assembly. Here, we introduce a platform for the automated discovery of tailored MBP formulations based on the Particle Swarm Optimization (PSO) method and a linear multiblock chain parametrization that enables continuous optimization of chain architecture. We apply the method to thin-film blends of linear ABC triblock polymers subject to lateral confinement, allowing all polymer and blend parameters to freely optimize in search of a prespecified, nontrivial target morphology. While we focus on pattern selection as a proof of principle, any computable equilibrium property can be optimized using the methods described here.

Advancing X-ray scattering metrology using inverse genetic algorithms

We compare the speed and effectiveness of two genetic optimization algorithms to the results of statistical sampling via a Markov chain Monte Carlo algorithm to find which is the most robust method for determining real space structure in periodic gratings measured using critical dimension small angle X-ray scattering. Both a covariance matrix adaptation evolutionary strategy and differential evolution algorithm are implemented and compared using various objective functions. The algorithms and objective functions are used to minimize differences between diffraction simulations and measured diffraction data. These simulations are parameterized with an electron density model known to roughly correspond to the real space structure of our nanogratings. The study shows that for X-ray scattering data, the covariance matrix adaptation coupled with a mean-absolute error log objective function is the most efficient combination of algorithm and goodness of fit criterion for finding structures with little foreknowledge about the underlying fine scale structure features of the nanograting.

Three-dimensional inverse design of nanopatterns with block copolymers and homopolymers

We propose a facile inverse design strategy to generate three-dimensional (3D) nanopatterns by using either block copolymers or a binary homopolymer blend via dissipative particle dynamics simulations. We find that the composition window of block copolymers to form a specific 3D morphology can be expanded when the self-assembly of block copolymers is directed by templates. We also find that a binary homopolymer blend can serve as a better candidate in the inverse templating design, since they have similar performances on recovering the target pattern, with much lower cost. This strategy is proved efficient for fabricating templates with desired topographical configuration, and the inverse design idea sheds lights on better control and design of materials with complex nanopatterns.

Block copolymers: controlling nanostructure to generate functional materials - synthesis, characterization, and engineering

In this perspective, we survey recent advances in the synthesis and characterization of block copolymers, discuss several key materials opportunities enabled by block copolymers, and highlight some of the challenges that currently limit further realization of block copolymers in promising nanoscale applications. One significant challenge, especially as the complexity and functionality of designer macromolecules increases, is the requirement of multiple complementary techniques to fully characterize the resultant polymers and nanoscale materials. Thus, we highlight select characterization and theoretical methods and discuss how future advances can improve understanding of block copolymer systems. In particular, we consider the application of theoretical/simulation methods to the rationalization, and prediction, of observed experimental self-assembly phenomena. Finally, we explore several next steps for the field and emphasize some general areas of emerging research that could unlock additional opportunities for nanostructure-forming block copolymers in functional materials.

Turning statistical physics models into materials design engines

Despite the success statistical physics has enjoyed at predicting the properties of materials for given parameters, the inverse problem, identifying which material parameters produce given, desired properties, is only beginning to be addressed. Recently, several methods have emerged across disciplines that draw upon optimization and simulation to create computer programs that tailor material responses to specified behaviors. However, so far the methods developed either involve black-box techniques, in which the optimizer operates without explicit knowledge of the material's configuration space, or require carefully tuned algorithms with applicability limited to a narrow subclass of materials. Here we introduce a formalism that can generate optimizers automatically by extending statistical mechanics into the realm of design. The strength of this approach lies in its capability to transform statistical models that describe materials into optimizers to tailor them. By comparing against standard black-box optimization methods, we demonstrate how optimizers generated by this formalism can be faster and more effective, while remaining straightforward to implement. The scope of our approach includes possibilities for solving a variety of complex optimization and design problems concerning materials both in and out of equilibrium.

Celebrating Soft Matter's 10th Anniversary: toward jamming by design

In materials science, high performance is typically associated with regularity and order, while disorder and the presence of defects are assumed to lead to sub-optimal outcomes. This holds for traditional solids such as crystals as well as for many types of nanoscale devices. However, there are circumstances where disorder can be harnessed to achieve performance not possible with approaches based on regularity. Recent research has shown opportunities specifically for soft matter. There, the phenomenon of jamming leads to unique emergent behavior that enables disordered, amorphous systems to switch reversibly between solid-like rigidity and fluid-like plasticity. This makes it possible to envision materials that can change stiffness or even shape adaptively. We review some of the progress in this direction, discussing examples where jamming has been explored from micro to macro scales in colloidal systems, suspensions, granular-materials-enabled soft robotics, and architecture. We focus in particular on how the jammed aggregate state can be tailored by controlling particle level properties and discuss very recent ideas that provide an important first step toward actual design of specifically targeted jamming behavior.

Evolutionary Optimization of Directed Self-Assembly of Triblock Copolymers on Chemically Patterned Substrates

Directed self-assembly of block copolymers on chemical patterns is of considerable interest for sublithographic patterning. The concept of pattern interpolation, in which a subset of features patterned on a substrate is multiplied through the inherent morphology of an ordered block copolymer, has enabled fabrication of extremely small, defect-free features over large areas. One of the central challenges in design of pattern interpolation strategies is that of identifying system characteristics leading to ideal, defect-free directed assembly. In this work we demonstrate how a coarse-grained many-body model of block copolymers, coupled to an evolutionary computation (EC) strategy, can be used to design and optimize substrate-copolymer combinations for use in lithographic patterning. The proposed approach is shown to be significantly more effective than traditional algorithms based on random searches, and its results are validated in the context of recent experimental observations. The coupled simulation-evolution method introduced here provides a general and efficient method for potential design of complex device-oriented structures.

Harnessing Anisotropic Nanoposts to Enhance Long-Range Orientation Order of Directed Self-Assembly Nanostructures via Large Cell Simulations

Self-assembly behaviors of cylinder-forming diblock copolymers directed by an array of anisotropic nanoposts with elliptical shape are explored by large cell simulations of self-consistent field theory. The strategy using elliptical nanoposts allows us to generate long-range order cylinders with single orientation by suppressing other selections of cylinder alignment. The anisotropy of nanoposts plays a significant role in improving the tolerance of commensurability conditions between the dimensions of nanopost lattices and the period of cylinders. Moreover, the local defect structures could be regulated through varying the spacing and orientation of elliptical nanoposts. This work may provide useful guidelines for designing the topographical templates and lay the groundwork for fabricating well-ordered nanostructures of block copolymer lithography.

Determining the resolution limits of electron-beam lithography: direct measurement of the point-spread function

One challenge existing since the invention of electron-beam lithography (EBL) is understanding the exposure mechanisms that limit the resolution of EBL. To overcome this challenge, we need to understand the spatial distribution of energy density deposited in the resist, that is, the point-spread function (PSF). During EBL exposure, the processes of electron scattering, phonon, photon, plasmon, and electron emission in the resist are combined, which complicates the analysis of the EBL PSF. Here, we show the measurement of delocalized energy transfer in EBL exposure by using chromatic aberration-corrected energy-filtered transmission electron microscopy (EFTEM) at the sub-10 nm scale. We have defined the role of spot size, electron scattering, secondary electrons, and volume plasmons in the lithographic PSF by performing EFTEM, momentum-resolved electron energy loss spectroscopy (EELS), sub-10 nm EBL, and Monte Carlo simulations. We expect that these results will enable alternative ways to improve the resolution limit of EBL. Furthermore, our approach to study the resolution limits of EBL may be applied to other lithographic techniques where electrons also play a key role in resist exposure, such as ion-beam-, X-ray-, and extreme-ultraviolet lithography.

Three-dimensional nanofabrication by block copolymer self-assembly

Thin films of block copolymers are widely seen as enablers for nanoscale fabrication of semiconductor devices, membranes, and other structures, taking advantage of microphase separation to produce well-organized nanostructures with periods of a few nm and above. However, the inherently three-dimensional structure of block copolymer microdomains could enable them to make 3D devices and structures directly, which could lead to efficient fabrication of complex heterogeneous structures. This article reviews recent progress in developing 3D nanofabrication processes based on block copolymers.