Microphase Separation Of Diblock Copolymers Research Articles

Numerical investigations of pattern formation in binary systems with inhibitory long-range interaction

<abstract><p>We investigate pattern formation in a two-dimensional manifold using the Otha-Kawasaki model for micro-phase separation of diblock copolymers. In this model, the total energy includes a short-range and a long-range term. The short-range term is a Landau-type free energy that is common in phase separation problems and favors large domains with minimum perimeter. The inhibitory long-range interaction term is the Otha-Kawasaki functional derived from the theory of diblock copolymers and favors small domains. The balance of these terms leads to equilibrium states that exhibit a variety of patterns, including disk-like droplets, droplet assemblies, elongated droplets, dog-bone shaped droplets, stripes, annular rings, wriggled stripes and combinations thereof. For problems where analytical results are known, we compare our numerical results and find good agreement. Where analytical results are not available, our numerical methods allow us to explore the solution space revealing new stable patterns. We focus on the triaxial ellipsoid, but our methods are general and can be applied to higher genus surfaces and surfaces with boundaries.</p></abstract>

A robust and efficient fingerprint image restoration method based on a phase-field model

In this study, we present a robust and efficient fingerprint image restoration algorithm using the nonlocal Cahn–Hilliard (CH) equation, which was proposed for modeling the microphase separation of diblock copolymers. We take a small local region embedding the damaged domain and solve the nonlocal CH equation to restore the fingerprint image. A Gauss–Seidel type iterative method, which is efficient and simple to implement, is used. The proposed method has the advantage in that the pixel values in the damaged fingerprint domain can be obtained using the image information from the outside of the damaged fingerprint region. Fingerprint restoration based on adjacent pixel information can ensure the accuracy of the fingerprint information with a low computational cost. Computational experiments demonstrated the superior performance of the proposed fingerprint restoration algorithm.

Efficient second-order unconditionally stable numerical schemes for the modified phase field crystal model with long-range interaction

In this paper, we consider numerical approximations for the modified phase field crystal model with long-range interaction, which describes the micro-phase separation in diblock copolymers. The model is a nonlinear damped wave equation with a nonlocal term that includes both diffusive dynamics and elastic interaction. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we employ the scalar auxiliary variable (SAV) approach to achieve two highly efficient and linear numerical schemes based on the second-order Crank–Nicolson and backward differentiation formula. These schemes lead to decoupled linear equations with constant coefficients at each time step and their unconditional energy stabilities are proved rigorously. The stabilization technique is adopted to further improve the stability of the numerical schemes. Various 2D and 3D numerical experiments are performed to demonstrate the accuracy, stability, and efficiency.

Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation

We present a nonlinear multigrid implementation for the two-dimensional Cahn–Hilliard (CH) equation and conduct detailed numerical tests to explore the performance of the multigrid method for the CH equation. The CH equation was originally developed by Cahn and Hilliard to model phase separation phenomena. The CH equation has been used to model many interface-related problems, such as the spinodal decomposition of a binary alloy mixture, inpainting of binary images, microphase separation of diblock copolymers, microstructures with elastic inhomogeneity, two-phase binary fluids, in silico tumor growth simulation and structural topology optimization. The CH equation is discretized by using Eyre’s unconditionally gradient stable scheme. The system of discrete equations is solved using an iterative method such as a nonlinear multigrid approach, which is one of the most efficient iterative methods for solving partial differential equations. Characteristic numerical experiments are conducted to demonstrate the efficiency and accuracy of the multigrid method for the CH equation. In the Appendix, we provide C code for implementing the nonlinear multigrid method for the two-dimensional CH equation.

Estimating the Segregation Strength of Microphase-Separated Diblock Copolymers from the Interfacial Width.

The ever-growing catalog of monomers being incorporated into block polymers affords exceptional control over phase behavior and nanoscale structure. The segregation strength, , is the fundamental link between the molecular-level detail and the thermodynamics. However, predicting phase behavior mandates at least one experimental measurement of for each pair of blocks. This typically requires access to the disordered state. We describe a method for estimating from small-angle X-ray scattering measurements of the interfacial width between lamellar microdomains, , in the microphase-separated melt. The segregation strength is determined by comparing to self-consistent field theory calculations of the intrinsic interfacial width, , as a function of the mean-field . The method is validated using a series of independent experimental measurements of and , measured via the order-disorder transition temperature, . The average absolute relative difference between calculated from and the value calculated from is a modest 11%. Corrections for nonplanarity of the interfaces are investigated but do not improve the agreement between the experiments and theory. Published 2019. This article is a U.S. Government work and is in the public domain in the USA.

Nanoscale Resolution of Electric-field Induced Motion in Ionic Diblock Copolymer Thin Films.

Understanding the responses of ionic block copolymers to applied electric fields is crucial when targeting applications in areas such as energy storage, microelectronics, and transducers. This work shows that the identity of counterions in ionic diblock copolymers substantially affects their responses to electric fields, demonstrating the importance of ionic species for materials design. In situ neutron reflectometry measurements revealed that thin films containing imidazolium based cationic diblock copolymers, tetrafluoroborate counteranions led to film contraction under applied electric fields, while bromide counteranions drove expansion under similar field strengths. Coarse-grained molecular dynamics simulations were used to develop a fundamental understanding of these responses, uncovering a nonmonotonic trend in thickness change as a function of field strength. This behavior was attributed to elastic responses of microphase separated diblock copolymer chains resulting from variations in interfacial tension of polymer-polymer interfaces due to the redistribution of counteranions in the presence of electric fields.

Stabilization of Gold Nanospheres and Nanorods in Diblock Copolymers of Styrene and Vinylpyridine

Composites based on microphase-separated diblock copolymers of styrene with vinylpyridine and gold nanoparticles, in which the block copolymers play simultaneously both the role of a stabilizer preventing the aggregation of nanoparticles and a matrix providing their selective arrangement within domains of the predefined type, are obtained for the first time. The suggested approach implies no use of additional stabilizers and is efficient for both initially organic dispersed small (~5 nm) spheres and hydrasols of larger (9 × 30 nm) rodlike nanoparticles. Nanospheres are introduced into the composites via replacement of a labile stabilizer used in synthesis with the diblock copolymer, while nanorods synthesized in aqueous media by the two-stage method are incorporated via phase transfer mediated by the diblock copolymer. Films capable of microphase separation leading to the formation of vinylpyridine cylindrical domains with a diameter of tens nanometers are spin-coated from the obtained composites. The content of nanoparticles in the composites is up to 5 wt %; they are dispersed and arranged predominantly within vinylpyridine domains, with nanospheres being localized near domain boundaries, while nanorods being mainly oriented perpendicularly to the axis of cylinders.

Ion Distribution in Microphase-Separated Copolymers with Periodic Dielectric Permittivity

We examine the distribution of ionic species in charged, microphase-separated diblock copolymers by relying on coarse-grained simulations of the underlying materials. The model adopted here has been particularly useful in understanding the behavior of neutral block polymer systems. In this work, it is extended to describe charged molecules. A simulation methodology is proposed in which dielectric inhomogeneities within the phase-separated systems are taken into account by solving on-the-fly Poisson’s equation for spatially varying dielectric permittivity and calculating the corresponding electric fields. The Green’s function appropriate for a periodic and ionic system is explicitly calculated and used in simulations to arrive at phase diagrams as a function of salt concentration and copolymer composition. The systems considered here are representative of lithium salt-doped diblock copolymers, which could be of potential use for solid-electrolyte batteries.

Metallic Nanodot Patterns with Unique Symmetries Templated from ABC Triblock Terpolymer Networks.

Nanotemplates derived from the self-assembly of AB-type block copolymers provide an elegant route to achieve well-defined metallic dot arrays, even if the variety of pattern symmetries is restricted due to the limited number of structures offered by microphase separated diblock copolymers. A strategy that relies on the use of complex network structures accessible through the self-assembly of linear ABC-type terpolymers is presented for the formation of metallic nanodots arrays with "outside-the-box" symmetries. Patterned templates formed by the cubic Q214 and orthorhombic O70 network structures are used as excellent platforms to build well-ordered gold nanodot arrays with unique p3m1 and p2 symmetries, respectively. A simple yet efficient blending strategy is used to tune the critical dimensions of the p3m1 pattern while laterally ordered gold nanodot arrays are also demonstrated through a directed self-assembly approach. Such highly ordered gold nanodots with tunable particle dimensions and array periods, enabling the control of their plasmonic responses, are attractive probes for biological imaging.

Minimality via second variation for microphase separation of diblock copolymer melts

Abstract We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta–Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the problem. Moreover, we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the L 1 {L^{1}} -topology.

Traveling pulse solutions to FitzHugh–Nagumo equations

Reaction–diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences. A localized pattern is a stable non-constant stationary solution usually located far away from neighborhoods of bifurcation induced by Turing’s instability. In the study of FitzHugh–Nagumo equations, we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial. This amounts to seeking a homoclinic orbit for a corresponding Hamiltonian system, and we utilize a variational formulation which involves a nonlocal term. Such a functional is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulse. The homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions. In contrast with the known results for positive standing pulses found in the literature, a new technique is attempted by seeking a standing pulse solution with a sign change.

Using Light To Guide the Motion of Nanorods in Photoresponsive Binary Blends: Designing Hierarchically Structured Nanocomposites

One of the challenges in creating high-performance polymer nanocomposites is establishing effective routes for tailoring the morphology of both the polymer mixture and the dispersed nanoparticles, which contribute desirable optical, electrical, and mechanical properties. Using computational modeling, we devise an effective method for simultaneously controlling the spatial regularity of the polymer phases and the distribution of the rods within this matrix. We focus on mixtures of photosensitive AB binary blends and A-coated nanorods; in the presence of light, the binary blends undergo a reversible chemical reaction and phase separation to yield a morphology resembling that of microphase-separated diblock copolymers. We simulate the effects of illuminating this sample with a uniform background light and a higher intensity, spatially localized light, which is rastered over the sample with a velocity v. The resulting material displays a periodically ordered, essentially defect-free morphology, with the A-like nanoparticles localized in lamellar A domains. The dynamic behavior of the rods within this system can be controlled by varying the velocity v and Γ2, the reaction rate coefficient produced by the higher intensity light. Specifically, the rastering light can drive the rods to be "pushed" along the lamellar domains or oriented perpendicular to these stripes. Given these attributes, we isolate scenarios where the system encompasses a complex hierarchical structure, with rods that are simultaneously ordered along two distinct directions within the periodic matrix. Namely, the rods form long nanowires that span the length of the sample and lie perpendicular to these wires in regularly spaced A lamellae. Hence, our approach points to new routes for producing self-organized rectangular grids, which can impart remarkable optoelectronic or mechanical properties to the materials.

Effect of Rod Length on the Morphology of Block Copolymer/Magnetic Nanorod Composites

The organization of magnetic nanorods in microphase-separated diblock copolymers composed of poly(styrene-b-2-vinylpyridine) (PS-PVP) as a function of rod length and rod concentration was investigated using both transmission electron microscopy and small-angle X-ray scattering. Our results reveal that the nanorods were sequestered into the PVP domains, which is attributed to the preferential interaction between pyridine-tethered nanorods and PVP. Meanwhile, the addition of nanorods in PS-PVP caused chain stretching. To minimize the energy penalty, nanorods tended to align parallel to the interface between PS and PVP to increase the conformational entropy. As the length of nanorods increased, the increasing van der Waals interaction and magnetic interaction caused extensive rod aggregation, which suppressed the domain size of PVP and amplified the local compositional fluctuations. This creates conditions to induce disorder in the polymer morphology and nanorods undergo macrophase separation.

Coassembly of Nanorods and Photosensitive Binary Blends: “Combing” with Light To Create Periodically Ordered Nanocomposites

Using computational modeling, we establish a means of controlling structure formation in nanocomposites that encompass nanorods and a photosensitive binary blend. The complex cooperative interactions in the system include a preferential wetting interaction between the rods and one of the phases in the blend, steric repulsion between the coated rods, and the response of the binary blend to light. Under uniform illumination, the binary mixture undergoes both phase separation and a reversible chemical reaction, leading to a morphology resembling that of a microphase-separated diblock copolymer. When a second, higher intensity light source is rastered over the sample, the binary blend and the nanorods coassemble into regular, periodically ordered structures. In particular, the system displays an essentially defect-free lamellar morphology, with the nanorods localized in the energetically favorable domains. By varying the speed at which the secondary light is rastered over the sample, we can control the directional alignment of the rods within the blend. Our approach yields an effective route for achieving morphological control of both the polymeric components and nanoparticles, providing a means of tailoring the properties and ultimate performance of the composites.

Standing Pulse Solutions to FitzHugh–Nagumo Equations

Reaction–diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences. A localized pattern is a stable non-constant stationary solution usually located far away from neighborhoods of bifurcation induced by Turing’s instability. In the study of FitzHugh–Nagumo equations, we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial. This amounts to seeking a homoclinic orbit for a corresponding Hamiltonian system, and we utilize a variational formulation which involves a nonlocal term. Such a functional is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulse. The homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions. In contrast with the known results for positive standing pulses found in the literature, a new technique is attempted by seeking a standing pulse solution with a sign change.

Dynamic Monte Carlo simulations of double crystallization accelerated in microdomains of diblock copolymers

We report dynamic Monte Carlo simulations of microphase separated diblock copolymers, to investigate how crystallization of one species could accelerate the subsequent crystallization of another species. Although the lattice copolymer model brings a boundary constraint to the long periods of microdomains, the single-molecular-level force balance between two blocks and its change can be revealed in this simple approach. We found two contrastable acceleration mechanisms: (1) the metastable lamellar crystals of one species become thicker at higher crystallization temperatures, sacrificing its microphase interfacial area to make a larger coil-stretching of another amorphous species and hence to accelerate subsequent crystallization of the latter with a more favorable conformation. (2) While in the case allowing chain-sliding in the crystal, the equilibrated lamellar crystals of one species become thinner at higher temperatures, sacrificing its thermal stability to gain a higher conformational entropy of another amorphous species and hence to accelerate subsequent crystallization of the latter with a stronger tension at the block junction. Parallel situations of experiments have been discussed.

Assembly of nano-particles on diblock copolymer brushes: toward laterally nano-structured composites

We have studied the self-assembly of nano-particles (NPs) at the surface of microphase-separated AB diblock copolymer brushes in a selective solvent by means of dissipative particle dynamics (DPD) simulations. The structure of such nano-composites depends both on the morphology of the underlying brush which is basically determined by chain composition fB and solvent selectivity τB, and on specific parameters of the NPs added such as polymer compatibility aBP, solvophobicity aPS, and concentration φ, where aij are the DPD interaction parameters. The immense parameter space is explored within a coarse-grained model that contains polymer A and B beads, solvent (S), nano-particle (P), and wall (W) beads. For composites with B-like nano-particles, i.e., NPs that are covered by polymer B ligands, we construct the phase diagram in the fB − φ space. For B-unlike NPs, it is found that the solvophobicity of the particles set by aPS is crucial for the spreading of nano-droplets. We show that in principle microphase-separated diblock copolymer brushes can be used to create ordered nano-dots as well as nano-wires and deduce some rules which should facilitate the design of such novel polymer nano-composites. Finally we suggest a new mechanism to move nano-objects on the surface of diblock copolymer brushes.

Small Volume-Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional

We present the first of two articles on the small volume fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. Here we focus attention on the sharp-interface version of the functional and consider a limit in which the volume fraction tends to zero but the number of minority phases (called particles) remains $O(1)$. Using the language of $\Gamma$-convergence, we focus on two levels of this convergence and derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into “point particles.” At the highest level, the effective energy is entirely local and contains information about the structure of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern ...

Investigation of Microphase Separation of PS-PPrMA Diblock Copolymer Films by Time-of-Flight Secondary Ion Mass Spectrometry

The microphase separation of diblock copolymers has been investigated by many different research groups for many years, because of the increasing use of diblock copolymers as compatibilizers, dispersants, impact modifiers, nanocarriers, and templates. In this presentation, surface characterization methods were utilized to study the surface morphology and composition produced after annealing thin polymer films. We report on the characterization of the morphology from symmetric diblock copolymers of polystyrene-b-poly(n-propyl methacrylate)(PS-PPrMA), where PS block was perdeuterated, using Time-of-Flight Secondary Ion Mass Spectrometry (TOF-SIMS). TOF-SIMS depth profiling was obtained for the lamellar morphology of deuterated PS-PPrMA which is found to orient parallel to the surface of the substrate. This preferential orientation resulted in a periodic variation in the composition of each block that continued through the entire copolymer film. Annealing studies on dPS-PPrMA thin films with different thickness on the silicon substrates were performed to investigate the lower disorder-to-order transition (LDOT) properties of diblock copolymers.

Block Copolymer Solid Battery Electrolyte with High Li-Ion Transference Number

The electrochemical properties of a solid polymer electrolyte consisting of a diblock copolymer and lithium bis(oxalato)borate, (LiBOB) salt, is reported. The spherical microphase-separated diblock copolymer is composed of a majority poly(ethylene oxide) (PEO) block and a minority random copolymer block of methyl methacrylate (MMA) and lithium salt of methacrylic acid (MAALi), . Salt-optimized electrolyte membranes of the diblock copolymer and LiBOB salt exhibited an average lithium-ion transference number value of 0.9 at room temperature . The solid-state flexible, translucent polymer electrolyte shows a wide electrochemical stability window and excellent interface properties with the lithium metal electrode. The combination of these properties makes these block copolymer membranes viable electrolyte candidates for flexible lithium-ion batteries.