Diblock Copolymer Melts Research Articles

Investigation of structural frustration in symmetric diblock copolymers confined in polar discs through cell dynamic simulation

Nanotechnology has opened new avenues for advanced research in various fields of soft materials. Materials scientists, chemists, physicists, and computational mathematicians have begun to take a keen interest in soft materials due to their potential applications in nanopatterning, membrane separation, drug delivery, nanolithography, advanced storage media, and nanorobotics. The unique properties of soft materials, particularly self-assembly, have made them useful in fields ranging from nanotechnology to biomedicine. The discovery of new morphologies in the diblock copolymer system in curved geometries is a challenging problem for mathematicians and theoretical scientists. Structural frustration under the effects of confinement in the system helps predict new structures. This mathematical study evaluates the effects of confinement and curvature on symmetric diblock copolymer melt using a cell dynamic simulation model. New patterns in lamella morphologies are predicted. The Laplacian involved in the cell dynamic simulation model is approximated by generating a 17-point stencil discretized to a polar grid by the finite difference method. Codes are programmed in FORTRAN to run the simulation, and IBM open DX is used to visualize the results. Comparison of computational results with existing studies validates this study and identifies defects and new patterns.

Emergence of Disordered Hyperuniformity in Melts of Linear Diblock Copolymers

Emergence of Disordered Hyperuniformity in Melts of Linear Diblock Copolymers

Chain and Structural Dynamics in Melts of Sphere-Forming Diblock Copolymers

Chain and Structural Dynamics in Melts of Sphere-Forming Diblock Copolymers

Directed Self-Assembly by Sparsely Prepatterned Substrates with Self-Responsive Polymer Brushes.

The guiding pattern in the chemoepitaxially directed self-assembly (DSA) of block copolymers is often fabricated by periodically functionalizing homogeneously random copolymer brushes tethered on a substrate. The prepatterned copolymer brushes constitute a soft penetrable surface, and their two components can in principle locally segregate in response to the overlying self-assembly process of block copolymers. To reveal how the self-responsive behavior of the copolymer brushes affects the directing effect, we develop a dissipative particle dynamics model to explicitly include the prepatterned polymer brushes and implement it to simulate the DSA of a cylinder-forming diblock copolymer melt on the sparse pattern of polymer brushes. Through large-scale dynamic simulations, we identify the windows of the content of the random copolymer, the film thickness, and the diameter of the patterned spot, for the formation of perfectly ordered hexagonal patterns composed of perpendicular cylinders. Our dynamic simulations reveal that the random copolymer brushes grafted on the unpatterned area exhibit a remarkable self-responsive ability with respect to the self-assembly of the diblock copolymers overlying them, which may widen the effective window of the content of the random copolymer. Within the processing windows of these key parameters, defect-free patterns are successfully achieved both in simulations and in experiments with sizes as large as a few micrometers for 4-fold density multiplications. This work demonstrates that highly efficient computer simulations based on an effective model can provide helpful guidance for experiments to optimize the critical parameters and even may promote the application of DSA.

Photomodulated phase-separation kinetics in block copolymer melts: a DPD simulation study

ABSTRACT We study phase-separation kinetics of symmetric diblock copolymer (BCP) melt in 3 d subjected to sequential light on/off cycles. The incompatible blocks of each BCP chain are connected by a photosensitive bond, which undergoes breakage during photo-illumination (on-cycle) and recombination during the off-cycle. We set a shorter fixed time limit for each on-cycles and focus on analysing how the much longer duration of the light off-cycles affects the segregation kinetics. We explore two scenarios for each off-cycle time variation: one with equal and another with increasing durations. For both cases, we determine the rate constants for bond formation and degradation reactions and assess their influence on the dynamical scaling functions and the growth scale of evolving morphologies. The average domain size typically exhibits diffusive growth ( R ( t ) ∼ t 1 / 3 ) for a couple of cycles in the beginning. However, the domain growth nearly freezes during the rest of the cycles, resulting in bicontinuous and nearly frozen isotropic morphologies.

Frank-Kasper Phases of Diblock Copolymer Melts: Self-Consistent Field Results of Two Commonly Used Models.

We constructed phase diagrams of conformationally asymmetric diblock copolymer A-B melts using the polymer self-consistent field (SCF) calculations of both the dissipative particle dynamics chain (DPDC) model (i.e., compressible melts of discrete Gaussian chains with the DPD non-bonded potential) and the "standard" model (i.e., incompressible melts of continuous Gaussian chains with the Dirac δ-function non-bonded potential) in the χN-ε plane, where χN and ε characterize, respectively, the repulsion and conformational asymmetry between the A and B blocks, at the A-block volume fraction f = 0.2 and 0.3. Consistent with previous SCF calculations of the "standard" model, σ and A15 are the only stable Frank-Kasper (FK) phases among the five FK (i.e., σ, A15, C14, C15 and Z) phases considered. The stability of σ and A15 is due to their delicate balance between the energetic and entropic contributions to the Helmholtz free energy per chain of the system, which, within our parameter range, increases in the order of σ/A15, Z, and C14/C15. While in general the SCF phase diagrams of these two models are qualitatively consistent, A15 is not stable for the DPDC model at the copolymer chain length N = 10 and f = 0.3; any differences in the SCF phase diagrams are solely due to the differences between these two models.

Correction to “Frank–Kasper Phases of Diblock Copolymer Melts Studied with the DPD Model: SCF Results”

Correction to “Frank–Kasper Phases of Diblock Copolymer Melts Studied with the DPD Model: SCF Results”

Linear energy-stable method with correction technique for the Ohta–Kawasaki–Navier–Stokes model of incompressible diblock copolymer melt

An efficiently linear time-marching method is proposed for the incompressible fluid flows coupled Ohta–Kawasaki model of diblock copolymer melt. Although this model satisfies the energy dissipation law under appropriate boundary conditions, the existences of nonlinear terms and coupling terms lead to difficulties of designing energy-stable numerical algorithms. To overcome the difficulties resulting from nonlinear and coupling terms, we herein construct a numerical method based on a highly efficient variant of scalar auxiliary variable approach. The stabilization technique is introduced to maintain the stability with large time steps. To improve the consistency, a simple and practical relaxation strategy is used to correct the difference between time-discretized modified energy and original energy. In each time step, all variables are totally decoupled and we only need to solve linear elliptic equations in a step-by-step manner. The unique solvability and energy stability are analytically proved. Extensive numerical experiments indicate that the proposed method not only achieves desired accuracy, energy stability, and consistency, but also has good capabilities on simulating various hydrodynamically coupled pattern formations.

Confinement-induced self-assembly of a diblock copolymer within a non-uniform cylindrical nanopore.

The self-assembly of a diblock copolymer melt confined within a non-uniform cylindrical nanopore is studied using the self-consistent field theory. The non-uniformity manifests in the form of a converging-diverging cylindrical nanopore. The axial variation in pore diameter presents a range of curvatures within the same confinement pore as opposed to a single curvature in a uniform-diameter cylindrical pore. The introduction of multiple curvatures leads to the formation of novel microstructures not accessible in uniform cylindrical confinement. The well-known equilibrium structures like a single helix, double helices, and concentric lamella under cylindrical confinement transition into new morphologies such as hyperboloidal phases, microstructures containing rings with a bead, rings with spheres, and a squeezed helical phase as the pore diameter varies axially. The converging-diverging geometry of the confining pore renders the helical phases seen in the cylindrical pore less favorable. A phase diagram in the parametric space of the block fraction and the ratio of the smallest and largest pore radii has been constructed to depict the order-order transition of various microstructures. The ratio of radii, a measure of the non-uniformity of the pore, along with the pore length brings out some interesting morphologies. The mechanism of these structural transitions is understood as the interplay between the variation in pore curvature attributed to the non-uniformity, the spontaneous curvature of the block copolymer interface, and the enthalpic interaction between the segregated blocks.

Large time-stepping, delay-free, and invariant-set-preserving integrators for the viscous Cahn–Hilliard–Oono equation

Stabilization techniques are frequently used to construct efficient and stable schemes for stiff and nonlinear systems. However, the introduced errors may lead to severe delay effects when facing large time-steps or stabilization parameters. Considering a fourth-order-in-space viscous Cahn–Hilliard–Oono equation that consists of both short- and long-range interactions, and equipped with either a polynomial or logarithmic potential, we analyze its invariant sets by proposing a novel reformulation. Additionally, we devise a family of temporal integrators up to eighth-order, along with a suitable stabilization parameter, to ensure the preservation of these invariant sets for any time-step. To eliminate the lagging phenomenon, we propose rewriting the method into a parametric two-step Runge–Kutta (TSRK) formulation, quantifying the rescaled time-step, and finally performing a relaxation technique. We demonstrate that the s-stage, pth-order parametric relaxation TSRK integrators retain the order of the underlying TSRK Butcher tableau, and promote the strong-stability-preserving coefficient to at most s when p>2. To further reduce the stabilization parameter, we propose a parametric relaxation integrating factor TSRK framework that treats the stiff linear operator as an integrating factor. The proposed framework has mild restrictions on the underlying TSRK tableau, i.e., the coefficients are non-negative and the parametric abscissas are non-decreasing. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, and are used to investigate the effect of parameters on microphase separations in diblock copolymer melts.

Particle Distribution Informed by Chain Rigidity in Diblock Copolymer Melts: The Effect of Entropy

Particle Distribution Informed by Chain Rigidity in Diblock Copolymer Melts: The Effect of Entropy

Fluctuation stabilization of the Fddd network phase in diblock, triblock, and starblock copolymer melts

Fluctuation stabilization of the <i>Fddd</i> network phase in diblock, triblock, and starblock copolymer melts

Inverse design of polymer alloys using deep learning based on self-consistent field analysis and finite element analysis

In this study, a framework that deals with the inverse design of material structures based on macroscopic properties using deep learning models is proposed. For the framework, “polymer alloy’s phase separation structures” as material structures and “stresses” as macroscopic properties are considered. The results of self-consistent field analysis are used as structure data, in particular, the analysis result of two-dimensional phase separation structures of a diblock copolymer melt in the equilibrium state. For the stress data, the results of finite element analysis are used, in particular, the analysis results of average stresses over the system under the small-enforced-displacement condition. This framework consists of two deep learning models—one aimed at generating phase separation structures, and the other at predicting properties from structures. For the second model, we incorporate a convolutional neural network. By implementing a random search toward a model that combines these two deep learning models, we constructed the framework for the inverse design of phase separation structures that possess desired macroscopic properties. This framework is demonstrated to be able to suggest phase separation structures that realize desired stresses that are input to the framework through examples. The innovation of the method is the development of framework to propose the optimal structure of polymer alloys by using machine learning. The main novelties of this method are the construction of generative adversarial network that reproduces the phase-separation structure of polymer alloy specified by label information, realization of inverse analysis based on mechanical properties, and the simultaneous optimization of multiple mechanical properties.

Diblock Copolymer Melt in an Electric Field: Stability of the Homogeneous State in the Random Phase Approximation

The contribution of I.Ya. Erukhimovich (1947–2022) to the creation of the theory of microphase separation in di- and triblock copolymers has been briefly analyzed. His matrix method of calculating correlation functions for multicomponent polymer systems has been applied to find the spinodal of a diblock copolymer melt in a static electric field. It has been strictly shown that for a linear dependence of local dielectric constant of the copolymer on the order parameter the spinodal condition remains the same as in the absence of the electric field. The correction to the critical value of the Flory‒Huggins parameter has been calculated for more general case of the quadratic dependence.

Diblock Copolymer Melt in an Electric Field: Stability of the Homogeneous State in the Random Phase Approximation

Diblock Copolymer Melt in an Electric Field: Stability of the Homogeneous State in the Random Phase Approximation

Phase Behavior of Binary Blends of Diblock Copolymers: Progress and Opportunities.

The phase behavior of binary blends of diblock copolymers has been examined extensively in the past decades. Experimental and theoretical studies have demonstrated that mixing two different block copolymers provides an efficient and versatile route to regulate their self-assembled morphologies. A good understanding of the principles governing the self-assembly of block copolymer blends has been obtained from the study of A1B1/A2B2 diblock copolymer blends. The second (A2B2) diblocks could act synergistically as fillers and cosurfactants to regulate the domain size and interfacial properties, resulting in the formation of ordered phases not found in the parent (A1B1 or A2B2) diblock copolymer melts. The study of A1B1/A2B2 block copolymer blends further provides a solid foundation for future research on more complex block copolymer blends.

Liquid-Like States in Micelle-Forming Diblock Copolymer Melts.

Large cell self-consistent field theory (SCFT) solutions for a neat, micelle-forming diblock copolymer melt, initialized using the structure of a Lennard-Jones fluid, reveal the existence of a vast number of liquid-like states, with free energies of order 10-3 kBT per chain higher than the body-centered cubic (bcc) state near the order-disorder transition (ODT). Computation of the structure factor for these liquids at temperatures below the ODT indicates that their intermicellar distance is slightly swollen compared to bcc. In addition to providing a mean-field picture of the disordered micellar state, the number of liquid-like states and their near-degeneracy with the equilibrium bcc morphology suggest that self-assembly of micelle-forming diblock copolymers navigates a rugged free energy landscape with many local minima. This picture provides a basis for the anomalously slow ordering kinetics of particle-forming diblock copolymer melts observed in experiments.

Fluctuation-Corrected Phase Diagrams for Diblock Copolymer Melts.

New developments in field-theoretic simulations (FTSs) are used to evaluate fluctuation corrections to the self-consistent field theory of diblock copolymer melts. Conventional simulations have been limited to the order-disorder transition (ODT), whereas FTSs allow us to evaluate complete phase diagrams for a series of invariant polymerization indices. The fluctuations stabilize the disordered phase, which shifts the ODT to higher segregation. Furthermore, they stabilize the network phases at the expense of the lamellar phase, which accounts for the presence of the Fddd phase in experiments. We hypothesize that this is due to an undulation entropy that favors curved interfaces.

Effect of the counterion size on microphase separation in charged-neutral diblock copolymers.

In this work, the question of the influence of the counterion size on the self-assembly in melts of diblock copolymers with one charged block was studied using coarse-grained molecular dynamics simulations. It was assumed that the blocks were fully compatible, i.e., the Flory-Huggins parameter χ between them was equal to 0. Due to the presence of correlation attraction (electrostatic cohesion) between the charged species, the systems with all types of counterions underwent transitions to ordered states, forming various morphologies, including lamellae, perforated lamellae, and hexagonally packed cylinders. Phase diagrams were constructed by varying the chain composition fc and locating the order-disorder transition positions in terms of the electrostatic strength parameter λ (dimensionless Bjerrum length). Despite having a rather large ion size mismatch, the systems with smaller counterions demonstrated an even better tendency to form microphase separated states than the systems with larger ones. It was found that the differences between the phase diagrams of the systems with different counterions can be roughly rationalized by using coordinates (volume fraction of the charged block φc-modified interaction parameter λ*). The latter parameter assumes that the electrostatic energy is simply inversely proportional to the characteristic distance between the ions of different signs. Such an approach appeared to be rather effective and allowed the diagrams obtained for different counterion sizes to almost coincide. The results of this work suggest that the counterion size can be used as a tool to control the system morphology as well as the effective incompatibility between the blocks.

A computational package for measuring Topological Entanglement in Polymers, Proteins and Periodic systems (TEPPP)

Many materials, like polymer melts, solutions, biopolymers and textiles, are composed of entangled filaments. The entanglement in these systems significantly affects their mechanical properties and their function. We introduce the Topological Entanglement in Polymers, Proteins and Periodic systems (TEPPP) software, that enables to measure the topological and geometrical complexity in such systems. In particular, this software enables the computation of the Writhe, the Gauss linking integral and the Jones polynomial of each filament or pair of filaments in the system, whether they are open or closed. In particular, for systems employing Periodic Boundary Conditions (PBC), the software also allows to compute the total pairwise entanglement in PBC, using the Periodic linking number and the Periodic Writhe. For linear (open) chains, TEPPP can calculate all these topological parameters (including the Jones polynomial) without any closure scheme. In addition, TEPPP also enables measuring self and pairwise entanglement at different length-scales along a chain or a pair of chains. With appropriate preprocessing of input files, the code can also be used for star or branched architectures. We provide examples of how the code is used and we present results on the entanglement effect in polymers obtained using this package. We show how TEPPP can be used to measure the topological entanglement of linear polymer chains in a melt, revealing subtle entanglement transitions never seen before. We also used TEPPP to analyze the effect of knotting and its location in diblock copolymer melts, which reveals that knotting localization transition coincides with lamellar-disorder transition in these systems. Finally, we use TEPPP to reveal some of the topological structure of the SARS-CoV-2 Spike protein, which points to interesting structure in a region that contains the S1/S2 cleavage site. Program summaryProgram Title: Topological Entanglement in Polymers, Proteins and Periodic systems (TEPPP) softwareCPC Library link to program files:https://doi.org/10.17632/ygdbpnhpzw.1Developer's repository link:https://github.com/TEPPP-softwareLicensing provisions: BSD 3-clauseProgramming language: C++Supplementary material:Nature of problem: Measuring single and pairwise entanglement and knotting in systems of linear or ring filaments (open or closed curves) in 3-space or in systems employing Periodic Boundary Conditions (PBC) at different length scales.Solution method: TEPPP can be used to measure topological entanglement complexity in single or multi-chain filament systems in 3-space or in systems employing PBC. Given as input the coordinates of the curves, TEPPP can compute the Gauss linking integral, the Writhe, the Jones polynomial, the Periodic Linking Number and the Periodic Writhe. Also, TEPPP can measure effects of local linking and knotting using scanning methods along the chains.