Microphase Separation Of Diblock Copolymers Research Articles

On the Convergence of the Ohta–Kawasaki Equation to Motion by Nonlocal Mullins–Sekerka Law

In this paper, we establish the convergence of the Ohta–Kawasaki equation to motion by nonlocal Mullins–Sekerka law on any smooth domain in space dimensions $N\leq3$. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are (i) well-preparedness of the initial data and (ii) smoothness of the limiting interface. Our method makes use of the “Gamma-convergence” of a gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta–Kawasaki equation characterizing their transport mechanism.

Ordered Microphase Separation in Thin Films of PMMA−PBA Synthesized by RAFT: Effect of Block Polydispersity

The microphase separation of diblock copolymers synthesized by reversible addition−fragmentation chain transfer (RAFT) polymerization, containing one monodisperse block (poly(methyl methacrylate), PMMA) and one polydisperse block (poly(butyl acrylate), PBA), was investigated in thin films (<100 nm). The formation of ordered microphase-separated domains was observed by atomic force microscopy (AFM) and resulted in four morphologies, depending on composition and film thickness: parallel lamellae, hexagonally packed perforated lamellae (PL), parallel cylinders (C∥), and hexagonally packed spheres, and in C∥-to-PL-to-C∥ transitions. Polydispersity of the PBA block shifts the phase boundaries toward higher PBA volume fraction values with respect to those expected for monodisperse block copolymers and stabilizes the perforated lamella morphology. Neutron reflectivity data confirmed that lamellae parallel to the substrate form at a very low PBA volume fraction, fPBA = 0.23. Polydispersity of the PBA block also h...

On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn–Hilliard Functional

We consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers. Our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal Cahn–Hilliard functional. Based upon two parameters which characterize the phase diagram, we give a preliminary analysis of the phase plane. That is, we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers. In particular we identify a regime wherein the uniform (disordered state) is the unique global minimizer; a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry; and a regime of small volume fraction wherein we conjecture that small well-separated approximately spherical objects are the unique global minimizer. For this last regime, we present an asymptotic analysis from the point of view of the energetics which will be complemented by rigorous $\Gamma$-convergence results to appear in a subsequent article. For all regimes, we present numerical simulations to support and expand on our findings.

Microphase separation of diblock copolymers with amphiphilic segment

We present a statistical mechanical approach for predicting the self-assembled morphologies of amphiphilic diblock copolymers in the melt. We introduce two conformationally asymmetric linear copolymer models with a local structural asymmetry, one of a “comb-tail” type and another that we call “continuous jackknife model.” The copolymers consist of amphiphilic and “monophilic” (non-amphiphilic) blocks, which have different segmental volume and tend to segregate into subphases. Using a self-consistent field theory (SCFT) framework, we explore the phase diagrams for these copolymers and compare them with that known for conventional, conformationally symmetric diblock copolymers. To determine the impact of structural effects on the self-assembly of copolymer melts, copolymers with a variation in both molecular architecture and chemical composition, f, are studied for different values of the Flory–Huggins parameter, χ. The composition dependence of the phase diagrams is shown to be basically determined by the conformational asymmetry. Remarkably, the stable lamellar structures exist even in the very compositionally asymmetric case, f < ¼. An interesting geometric distinction of the “direct” and “inverse” morphologies is introduced. The presence of an internal structure is found to influence the high χ behavior, where a stable two-scale (structure-in-structure) hexagonal morphology is found to be formed for some compositions. Therefore, the local chemical structure of monomer units can dictate the global morphology of copolymer melts.

E6 Periodic cylindrical nanostructure localization by means of affinity control in diblock copolymer microphase separation(Other manufacturing-related technologies)

E6 Periodic cylindrical nanostructure localization by means of affinity control in diblock copolymer microphase separation(Other manufacturing-related technologies)

Coarse-Grained Brownian Dynamics Simulations for Symmetric Diblock Copolymer Melts Based on the Soft Dumbbell Model

We propose a highly coarse-grained dynamics simulation model for symmetric diblock copolymers based on the soft dumbbell model. We perform the simulations of microphase separation dynamics of diblock copolymer melts with and without shear flow and show that Brownian dynamics simulations for soft dumbbells can reproduce dynamics of microphase separated diblock copolymers qualitatively well with small computational costs. We also calculate the shear relaxation moduli for unoriented and oriented lamellar structures. It is shown that lamellars can be oriented to the perpendicular direction by shear flow, and the shear thinning behaviour is observed.

Uniform energy distribution for an isoperimetric problem with long-range interactions

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are sufficiently large with respect to the intrinsic length scale. Moreover, we also prove an L ∞ L^\infty bound on the optimal potential associated with the long-range interactions. This bound allows for an interesting interpretation: Note that the average volume fraction of the optimal pattern in a subsystem of size R R fluctuates around the system average m m . The bound on the potential yields a rate of decay of these fluctuations as R R tends to + ∞ +\infty . This rate of decay is stronger than the one for a random checkerboard pattern. In this sense, the optimal pattern has less large-scale variations of the average volume fraction than a pattern with a finite correlation length.

Effects of interaction range and compressibility on the microphase separation of diblock copolymers: Mean-field analysis

Using the random-phase approximation and self-consistent field calculations, we have investigated the effects of finite interaction range and compressibility on the order-disorder transition (ODT) and the lamellar structure of symmetric diblock copolymers. While the compressibility does not affect the ODT, both the values of chiN and bulk lamellar period at the ODT increase with increasing interaction range. On the other hand, both the free-energy density and bulk period of the lamellae increase with either increasing interaction range or decreasing compressibility. Even with a finite compressibility, the mean-field ODT is still a second-order phase transition. The scaling exponent of bulk lamellar period with chiN, however, decreases with increasing compressibility. Our mean-field analysis provides a well understood reference for the study of fluctuation effects in diblock copolymers with finite interaction range and compressibility.

Microphase separation in diblock copolymers with amphiphilic block: Local chemical structure can dictate global morphology

Using dissipative particle dynamics, we study microphase separation in the melt of amphiphilic/nonpolar diblock copolymers in the strong segregation regime. We show that the phase diagram for the copolymers with an amphiphilic block can be significantly different from that known for the conventional diblocks. In the limit of significant amphiphilicity (surface activity), the resulting morphology corresponds to thin channels and slits of amphiphilic units penetrating through the matrix of a majority nonpolar component. The physical reason behind this is connected with the surface activity of amphiphilic monomers, which forces them to be located in the regions of maximum concentration gradient.

Molecular dynamics simulation of phase separating binary liquids in cylindrical Couette flow

We use molecular dynamics simulations to study phase separation of a 50:50 (by volume) fluid mixture in a confined and curved (Taylor-Couette) geometry, consisting of two concentric cylinders. The inner cylinder may be rotated to achieve a shear flow. In nonsheared systems we observe that, for all cases under consideration, the final equilibrium state has a stacked structure. Depending on the lowest free energy in the geometry the stack may be either flat, with its normal in the z direction, or curved, with its normal in the r or theta direction. In sheared systems we make several observations. First, when starting from a prearranged stacked structure, we find that sheared gradient and vorticity stacks retain their character for the durations of the simulation, even when another configuration is preferred (as found when starting from a randomly mixed configuration). This slow transition to another configuration is attributed to a large free energy barrier between the two states. In case of stacks with a normal in the gradient direction, we find interesting interfacial waves moving with a prescribed angular velocity in the flow direction. Because such a wave is not observed in simulations with a flat geometry at similar shear rates, the curvature of the wall is an essential ingredient of this phenomenon. Second, when starting from a randomly mixed configuration, stacks are also observed, with an orientation that depends on the applied shear rate. Such transitions to other orientations are similar to observations in microphase separated diblock copolymer melts. At higher shear rates complex patterns emerge, accompanied by deviations from a homogeneous flow profile. The transition from steady stacks to complex patterns takes place around a shear rate 1/tau(dv), where tau(dv) is the crossover time from diffusive to viscous dominated growth of phase-separated domains, as measured in equilibrium simulations.

The role of entanglements on the stability of microphase separated diblock copolymers in shear flow

Various possible orientations of lamellar structures of diblock copolymers under shear are investigated with respect to their stability. A Brownian dynamics model is put forward in which each diblock is modeled as a dumbbell. The blobs in each dumbbell are held together by a finite extendable nonlinear elastic (FENE) potential and interact with all surrounding blobs by soft dissipative particle dynamics (DPD) potentials. In addition to this, the blobs have the possibility to entangle with each other. The corresponding interactions lead to large viscosities, which, however, quickly diminish with increasing shear rate. This fact turns out to be crucial for the stabilization of structures with the lamellae parallel to the velocity-vorticity plane. As a second result it is found that asymmetry in the entanglement interactions stimulates the actual reorientation into this state.

Structural Rheology of Microphase Separated Diblock Copolymers

We formulate a viscoelastic theory for interconnected microphase separated structures such as a double gyroid in diblock copolymer melts. We consider the energy increase and the energy dissipation due to the deformation of interconnected periodic domains. The dominant elastic part agrees with previous analytical results obtained by a phase formulation of deformations. It is shown that a viscous response also appears which is mainly due to two relaxation modes of the gyroid.

Highly Ordered Arrays of Mesoporous Silica Nanorods with Tunable Aspect Ratios from Block Copolymer Thin Films

Mesoporous silica thin films have received much attention in applications as diverse as separation devices, sensors, and optoelectronic devices. The evaporation-induced self-assembly method (EISA) has been established as an efficient process for the rapid preparation of mesoporous silica thin films. Usually, beginning with a highly dilute homogenous solution of a soluble silica species and a surfactant in an ethanol/water mixture, preferential evaporation of ethanol concentrates the nonvolatile surfactant and silica species in water, thereby inducing the self-assembly of silica-surfactant micelles and their further organization into liquid-crystalline mesophases. Recently, this efficient EISA method has been applied to prepare ordered mesoporous materials within the confined channels of anodic alumina membranes (AAM), silicon membranes, and other resist molds. Compared to mesoporous powders and thin films, a hierarchically ordered mesoporous material is advantageous for self-assembly on a macroscopic scale and allows easy control over the morphology. A reduction in diameter below 50 nm to obtain mesoporous nanowires and -fibers is believed to offer even more benefits. However, studies of this type of materials have been limited so far due to the size limitation of the templates available. This shortcoming can be overcome by using the block copolymer lithography technique, which provides a powerful way to fabricate highly ordered arrays of inorganic nanoparticles with diameters of 5 – 50 nm. Both the diameter and interparticle distance of the resulting nanoparticle arrays can be tuned by adjusting the volume fractions of the block copolymer templates. Unfortunately, control of the aspect ratio (or the height) still remains a challenge. Herein, we report for the first time preparation of a highly ordered array of mesoporous silica nanorods with tunable aspect ratios, through an integrated strategy of block copolymer lithography and EISA. Block copolymer thin films with normal, cylindrical domains are used as external scaffolds to define the morphology of SiO2 nanorods. The surfactant cetyltrimethylammonium bromide (CTAB) is used as an internal template to form mesoporous structures inside the SiO2 nanorods. The control of the diameter, the center-to-center distance, and the height of SiO2 nanorods were studied systematically. Microphase-separated diblock copolymer films of amphiphilic PEOm-b-PMA(Az)n (m and n denote repeated units of the individual segments), consisting of hydrophilic poly(ethylene oxide) (PEO) and hydrophobic poly(methacrylate) with azobenzene mesogens (PMA(Az)) in the side chain, can be prepared by spin-coating 1∼ 3 wt % toluene or chloroform solutions thereof on silicon wafers substrates followed by annealing at 140 °C for 24 h in vacuum. Figure 1A–C show typical atomic force microscopy (AFM) height images of PEO114-b-PMA(Az)45, PEO272-b-PMA(Az)116, and PEO454b-PMA(Az)184 thin films, respectively, on a silicon wafer after annealing. Dark dots and bright surroundings in the images can be assigned to cylindrical PEO microdomains and PMA(Az) matrix, respectively. The insets showing fast Fourier transformed (FFT) images indicate a hexagonal or quasihexagonal arrangement of the PEO domains. The average diameter of PEO cylinders and their average center-to-center distance are (11.2± 1.6) nm and (22.1± 1.4) nm for PEO114-bPMA(Az)45, (18.6± 1.8) nm and (33.4± 1.9) nm for PEO272-bPMA(Az)116, and (23.7± 4.0) nm and (51.7± 3.1) nm for PEO454-b-PMA(Az)184, respectively. When the silicate sol was introduced into the PEOm-bPMA(Az)n thin film, the swollen PEO domains had been selectively doped, which was confirmed by TEM (See Supporting Information Figure S1). Through the sol-gel process driven by the solvent evaporation, the silicate sol will penetrate into the swollen PEO cylindrical domains to form one dimensional SiO2 nanoparticles. Figure 2A shows an AFM height image of the silicate sol hybridized with the PEO domains in a PEO272-b-PMA(Az)116 thin film (thickness: ∼ 150 nm) after immersion in the silicate sol for 3 h. A regular nanoparticle array protruded from the surface of the PEO272-bPMA(Az)116 thin film (height: ∼ 2 nm), which was not observed in cases with thicker PEOm-b-PMA(Az)n film templates or shorter immersion times. As shown by cross-sectional AFM image (Figure 2B) the cylindrical structure was still present after immersion, but most of the SiO2 nanorods appeared to lie flat on the substrate after calcination (Fig. S2). C O M M U N IC A IO N

Effect of Molecular Properties of Block Copolymers and Nanoparticles on the Morphology of Self-Assembled Bulk Nanocomposites

Self-assembly of thiol-terminated polystyrene-tethered Au nanoparticles in microphase-separated diblock copolymers composed of poly(styrene-b-2vinylpyridine) (PS−PVP) as a function of particle concentration and composition of block copolymers was investigated using anomalous small-angle X-ray scattering (ASAXS) and transmission electron microscopy (TEM). Results reveal that the self-assembly of nanoparticles in the PS domain causes swelling and increases the interfacial curvature that, in turn, induces order−order transitions. At intermediate loading, the presence of nanoparticles amplifies the local compositional fluctuations, hence the roughness at the PS and PVP interface, which creates conditions to induce disorder in the polymer morphology. The system thus undergoes an order−disorder transition. At high particle loading, packing constraints prevent all particles from assembling in the PS domain, and the excess nanoparticles undergo macrophase separation. The present systematic study augments experimental data to the scarce literature on the phase behavior of bulk nanocomposites. We present a generalized phase map for the bulk composites as a function of effective volume fraction of PS (FPS) for a given nanoparticle size. We believe that the results from this study will enable comparison of the phase maps from various studies and will serve to validate the simulation studies of inorganic particle/block copolymer composites.

Micro-phase separation of diblock copolymer in a nanosphere: Dissipative particle dynamics approach

Micro-phase separation and morphologies of diblock copolymer in a nanosphere with different size and interfacial energy were studied by dissipative particle dynamics approach. The surface of a nanosphere is modeled by a layer of frozen soft particles. For symmetric diblock copolymer A 5B 5, if the surface is neutral, i.e., with the same interactions toward different blocks, the morphology consists of bended distorted lamellae connected by channels. They might be mushroom-like, helical or bi-continuous depending on the sphere size. When the surface is non-neutral, i.e., with the different interactions toward different blocks, the meso-structure is onion like. The outmost layer is the block with weaker repulsion toward the surface. The number of the alternative inner layers is determined by the size. For asymmetric diblock, the morphology is strong associated with the composition. For A 3B 7, owing to the strong restriction of the nanosphere, the hexagonal phase originally in the bulk does not occur. When the surface is neutral, the minority block forms bended sticks touched the surface. With the increase of the sphere size, more and more cylindrical or star like joints link these sticks. When the surface is non-neutral and less repulsive toward the minority block, more bended sticks near the nanosphere surface than those in the neutral nanosphere. Increasing the sphere size causes spherical and cylindrical domain formed in the center of the sphere. For A 2B 8, the minority A-block forms spheres or peanut like domains consisting concentric spheres; the number of them is related to the sphere size. However, for A 1B 9, no micro-phase separation was found. The polymer surfactant in aqueous solution was also studied. The results indicate that the polymer in a nanosphere can be used to mimic the polymer surfactant in some cases.

The Effect of Pressure on the Microphase Separation of Diblock Copolymer Melts Studied by Dynamic Density Functional Theory Based on Equation of State

AbstractThe effect of pressure on the microphase separation of diblock copolymer melts was investigated by dynamic density functional theory based on equation of state. The results correspond well with experiment data. With the application of high pressure, all of the phase regions corresponding to the different ordered morphologies become narrower. However, the pressure dependence of the order‐disorder transition temperature (TODT) relies on the symmetry of the diblock copolymer. In the very non‐symmetrical case when f is small, TODT decreases with increasing pressure, while in the symmetrical case when f = 0.5, TODT increases with increasing pressure. For the latter case, the increase in the total bead number of the system at the ODT is found, which is in good accordance with the experimental phenomenon that there is an increase in volume accompanying with the transition from ordered to disordered state. In contrast to the temperature, the pressure does not influence the starting time and the duration of microphase separation.magnified image

Fluctuation Effects in the Theory of Microphase Separation of Diblock Copolymers in the Presence of an Electric Field

We generalize the Fredrickson−Helfand theory of the microphase separation in symmetric diblock copolymer melts by taking into account the influence of a time-independent homogeneous electric field on the composition fluctuations within the self-consistent Hartree approximation. We predict that electric fields suppress composition fluctuations, and consequently weaken the first-order transition. In the presence of an electric field the critical temperature of the order−disorder transition is shifted toward its mean-field value. The collective structure factor in the disordered phase becomes anisotropic in the presence of the electric field. Fluctuational modulations of the order parameter along the field direction are strongest suppressed. The latter is in accordance with the parallel orientation of the lamellae in the ordered state.

Density of polymers in a layered structure: An exactly solvable model

We have developed an exactly solvable model that makes it possible to calculate the density profile of polymers filled inside a chemically nonuniform structure that consists of alternating layers that have different affinity for polymers. It is shown that the mean density of polymers in this layered structure is in excess relative to that in the uniform system that has the same average affinity for polymers. The average excess density of polymers in the layered structure is calculated as a function of their degree of polymerization, layer thickness, and affinities of the layers for polymers. The developed theoretical model is shown to have relevance to several experimentally important chemically nonuniform layered systems like microphase separated diblock copolymers and selective mixed brushes.

On the first and second variations of a nonlocal isoperimetric problem

We consider a nonlocal perturbation of an isoperimetric variational problem. The problem may be viewed as a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation associated with competing short and long-range interactions. In particular, it arises as a Γ-limit of a model for microphase separation of diblock copolymers. In this article, we establish precise conditions for criticality and stability (i.e. we explicitly compute the first and second variations). We also present some applications.

Exploiting Photoinduced Reactions in Polymer Blends to Create Hierarchically Ordered, Defect-Free Materials

Computer simulations reveal how photoinduced chemical reactions can be exploited to create long-range order in binary and ternary polymeric materials. The process is initiated by shining a spatially uniform light over a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and phase separation. We then introduce a well-collimated, higher intensity light source. Rastering this secondary light over the sample locally increases the reaction rate and causes formation of defect-free, spatially periodic structures. These binary structures resemble either the lamellar or hexagonal phases of microphase-separated diblock copolymers. We measure the regularity of the ordered structures as a function of the relative reaction rates for different values of the rastering speed and determine the optimal conditions for creating defect-free structures in the binary systems. We then add a nonreactive homopolymer C, which is immiscible with both A and B. We show that this component migrates to regions that are illuminated by the secondary, higher intensity light, allowing us to effectively write a pattern of C onto the AB film. Rastering over the ternary blend with this collimated light now leads to hierarchically ordered patterns of A, B, and C. The findings point to a facile, nonintrusive process for manufacturing high quality polymeric devices in a low-cost, efficient manner.