Symmetric Diblock Copolymers Research Articles

THE EFFECTS OF PATTERNED SURFACES ON THE PHASE SEPARATION FOR DIBLOCK COPOLYMERS

The phase behaviors of symmetric diblock copolymer thin films confined between two hard, parallel and diversified patterned surfaces are investigated by three-dimensional dissipative particle dynamics (DPD) simulations. The induction of diversified patterned surfaces on phase separation of symmetric diblock copolymer films in snapshots, density profiles and concentration diagrams of the simulated systems are presented. The phase separations can be controlled by the patterned surfaces. In the meantime, the mean-square end-to-end distance of the confined polymer chains 〈R2〉 is also discussed. Surface-induced phase separation for diblock copolymers can help us to create novel and controlled nanostructured materials.

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.

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.

Kinetics of layer hopping in a diblock copolymer lamellar phase

In the ordered state, symmetric diblock copolymers self-assemble into an anisotropic lamellar morphology. The equilibrium thickness of the lamellae is the result of a delicate balance between enthalpic and entropic energies, which can be tuned by controlling the temperature. Here we devise a simple yet powerful method of detecting tiny changes in the lamellar thickness using optical microscopy. From such measurements we characterize the enthalpic interaction as well as the kinetics of molecules as they hop from one layer to the next in order to adjust the lamellar thickness in response to a temperature jump. The resolution of the measurements facilitate a direct comparison to predictions from self-consistent field theory.

Cooperative surface-induced self-assembly of symmetric diblock copolymers confined films with embedded nanorods

The self-assembly of symmetric diblock copolymers confined films with embedded nanorods is investigated by the self-consistent field (SCF) theory. We obtain some phase diagrams as a function of film thickness H and nanorod diameter D. The increase in preferential nanorod diameter D can promote the formation of incomplete cylindrical and spherical structures near the film surfaces, and can also induce complete lamellar, cylindrical and spherical structures in the interior. The formation of these induced self-assembled structures is due to the competition between inner surface confinement (two parallel surfaces) and outer surface confinement (nanorods). This investigation can provide some insights into the self-assembly of diblock copolymers with complex confinements.

The complex influence of the oscillatory shear on the melt of linear diblock copolymers

The phase morphologies of symmetric linear diblock copolymers subjected to the oscillatory shear are investigated with the aid of dissipative particle dynamics simulations. The frequency dependent reorientations of the lamellar phase (LAM) have been identified. We find that the parallel orientation of LAM (i.e., the lamellar normal is parallel to the velocity gradient) appears at high shear frequency, whereas the perpendicular orientation of LAM (the lamellar normal being perpendicular to the velocity gradient) takes place at low shear frequency. In both of the cases, the reorientations undergo similar processes: the original LAM phase prepared in equilibrium breaks down rapidly, and it takes a very long time for the perfectly oriented LAM being reformed. Moreover, the shear-induced isotropic to lamellar phase transitions are observed when the oscillatory shear amplitude is large enough. It indicates that the shear amplitude plays a dominant role in the order-disorder transition. The viscosity and the modulus of the melt are found to be dependent on the shear amplitude and the shear frequency in a complex way.

Study of temperature dependence of crystallisation transitions of a symmetric PEO-PCL diblock copolymer using simultaneous SAXS and WAXS measurements with synchrotron radiation

The reversible transitions of the lamellae of a crystalline-crystalline diblock copolymer from the melt to crystallites were studied using simultaneous small-angle X-ray scattering (SAXS) and wide-angle X-ray scattering (WAXS) measurements with synchrotron radiation. A symmetric poly(ethylene oxide)-poly( varepsilon -caprolactone) diblock copolymer was chosen for this study. We showed in the course of the block copolymer crystallisation that the time-resolved integrated intensity I (int) was proportional to the product of the volume fractions of the PEO and PCL phases and the scattering contrast due to the electron density difference. These results demonstrated that simultaneous SAXS/WAXS measurements could be used to monitor the crystallisation process in two domains of different sizes at the same time.

Effect of Composition of Substrate-Modifying Random Copolymers on the Orientation of Symmetric and Asymmetric Diblock Copolymer Domains

The ability of random copolymer brushes and cross-linked mats to induce the vertical orientation of domains in overlying films of lamellae- and cylinder-forming block copolymers was investigated as a function of the composition. The substrate-modifying layers consisted of styrene and methyl methacrylate random copolymers and contained either a terminal hydroxyl group or a third polar comonomer of 2-hydroxyethyl methacrylate (HEMA) for grafting brushes to silicon oxide surfaces or glycidyl methacrylate (GMA) for cross-linking the random copolymer into a mat. Polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) lamellae- and cylinder-forming block copolymers (both PS and PMMA minority block copolymers) were deposited and annealed on the modified surfaces. In all cases the vertical orientation of domains was observed for a range of random copolymer composition, but the ranges of composition were different for each combination of surface layer and block copolymer. The cylindrical domains of PS exhibited vertical structures for a very narrow range of compositions compared to cylindrical domains of PMMA or lamellae. As expected, the incorporation of polar HEMA or GMA monomers in the surface layers shifted the composition range for the perpendicular orientation of domains to higher fractions of styrene. The results are discussed in terms of the equilibration of the films in the presence of the chemically modified surfaces.

Single-chain dynamics in a homogeneous melt and a lamellar microphase: A comparison between Smart Monte Carlo dynamics, slithering-snake dynamics, and slip-link dynamics

We investigate the ability of Monte-Carlo algorithms to describe the single-chain dynamics in a dense homogeneous melt and a lamellar phase of a symmetric diblock copolymer. A minimal, coarse-grained model is employed that describes connectivity of effective segments by harmonic springs and where segments interact via soft potentials, which do not enforce noncrossability of the chain molecules. Studying the mean-square displacements, the dynamic structure factor, and the stress relaxation, we show that local, unconstraint displacements of segments via a Smart Monte Carlo algorithm give rise to Rouse dynamics for all but the first Monte Carlo steps. Using the slithering-snake algorithm, we observe a dynamics that is compatible with the predictions of the tube model of entangled melts for long times, but the dynamics inside the tube cannot be resolved. Using a slip-link model, we can describe the effect of entanglements and follow the different regimes of the single-chain dynamics over seven decades in time. Applications of this simulation scheme to spatially inhomogeneous systems are illustrated by studying the lamellar phase of a symmetric diblock copolymer. For the local, unconstraint dynamics, the single-chain motions parallel and perpendicular to the interfaces decouples; the perpendicular dynamics is slowed down but the parallel dynamics is identical to that in a homogeneous melt. Both the slithering-snake dynamics and the slip-link dynamics give rise to a coupling of parallel and perpendicular directions and a significant slowing down of the dynamics in the lamellar phase.

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.

Monte Carlo Phase Diagram of Symmetric Diblock Copolymer in Selective Solvent

With a lattice Monte Carlo method, we investigate 16−16 symmetric diblock in selective solvent, A-b-B/A, at 10 volume fractions from 1.0 to 0.1, and for each volume fraction, we perform simulations at up to 54 temperatures, using simulation boxes of different sizes. We report temperature dependencies for a number of quantities such as energy, specific heat, and mean-squared end-to-end distances and construct a phase diagram using the thermodynamic and structural quantities as well as snapshots of the selected configurations. The simulated phase diagram is compared with the experimental data of Lodge and co-workers for nearly symmetric poly(styrene-b-isoprene) mixed with dimethyl phthalate.

Flow‐Induced Morphologies of Diblock Copolymers in a Nanotube Studied by Dissipative Particle Dynamics Simulation

AbstractMorphologies of diblock copolymers confined in a cylindrical nanotube under a Poiseuille flow were studied by dissipative particle dynamics simulation. Varying with flow intensity, a symmetric diblock copolymer in a neutral tube exhibits parallel lamellae or stacked disks, stacked bowls, or cylinders; the morphologies in a non‐neutral tube are relatively less influenced by flow. An asymmetric diblock copolymer is helical in a neutral tube, but becomes hexagonal under flow; the morphologies in a non‐neutral tube are parallel to the tube axis and vary in a less pronounced manner than in a neutral tube. The simulation results reveal that confinement and external flow significantly affect the phase transitions among a variety of morphologies, particularly for copolymers with domains not parallel to the tube axis.magnified image

Variance Minimization of Free Energy Estimates from Optimized Expanded Ensembles

We explored the possibility of improving the accuracy and precision of free-energy differences estimated via expanded ensembles by manipulation of the biasing weights. Three different weighing approaches were compared: the flat histogram (FH) method, the optimized ensemble (OE) method, and a method introduced in this work, denoted MinVar, which aims to explicitly minimize the expected variance. The performance of these three methods was tested for the simulation of chemical potentials in systems of symmetric diblock copolymers with chain lengths of either 10 or 4 beads, and a system of one large hard sphere of diameter 10 d immersed in a fluid of hard spheres of diameter d. In addition, the effect of the weighing method on the observed accuracy was investigated for different choices of macrostate staging and for both optimized and nonoptimized acceptance ratio methods for calculating free-energy differences. In the diblock copolymer systems, we found that the maximum attainable accuracy can be limited by correlations between the samples, causing the "real" observed variances to be much larger than the expected "ideal" ones. Hence, if the formal minimization of the variance, as aimed by the MinVar method, occurs at the expense of increasing the correlations in the data, the accuracy may actually decrease. Although maximizing the number of round trips between initial and final macrostates (as aimed by the OE method) was found to be directly related to data decorrelation, this only translates into increased accuracy if the correlations are the major source of errors in the free energy estimates. Finally, for the hard sphere system, we found that the MinVar method performs better than both the OE and FH methods even though the MinVar method in this case never completes a round trip, illustrating that maximizing the number of round trips for fixed computational cost does not necessarily lead to increased precision.

Coarse-grained molecular dynamics simulation on the placement of nanoparticles within symmetric diblock copolymers under shear flow

We present molecular dynamics simulations coupled with a dissipative particle dynamics thermostat to model and simulate the behavior of symmetric diblock copolymer/nanoparticle systems under simple shear flow. We consider two categories of nanoparticles, one with selective interactions toward one of the blocks of a model diblock copolymer and the other with nonselective interactions with both blocks. For the selective nanoparticles, we consider additional variants by changing the particle diameter and the particle-polymer interaction potential. The aim of our present study is to understand how the nanoparticles disperse in a block copolymer system under shear flow and how the presence of nanoparticles affects the rheology, structure, and flow behavior of block copolymer systems. We keep the volume fraction of nanoparticles low (0.1) to preserve lamellar morphology in the nanocomposite. Our results show that shear can have a pronounced effect on the location of nanoparticles in block copolymers and can therefore be used as another parameter to control nanocomposite self-assembly. In addition, we investigate the effect of nanoparticles on shear-induced lamellar transition from parallel to perpendicular orientation to further elucidate nanocomposite behavior under shear, which is an important tool to induce long-range order in self-assembling materials such as block copolymers.

Simulated Annealing Study of Self‐Assembly of Symmetric ABA Triblock Copolymers Confined in Cylindrical Nanopores

AbstractWe report a simulated annealing study of the self‐assembly of symmetric lamella‐forming ABA triblock copolymers confined in cylindrical nanopores. We systematically examine the dependence of the self‐assembled morphologies and structural parameters on the degree of confinement and the strength of the surface preference. We find that the confined morphologies for the symmetric ABA triblocks with fA = 1/2 are similar to those for the symmetric or nearly symmetric AB diblock copolymers under the same confinement. We also find that different structural parameters can reflect different information. The predicted bridging fraction value for the bulk phase is in good agreement with previously established values, whereas the predicted values for the confined morphologies change with both the degree of confinement and the strength of the surface preference. We further explore the self‐assembling process by examining the morphology and various ensemble‐averaged thermodynamic quantities and structure parameters as a function of the reduced temperature.magnified image

Mesoscopic dynamics of inhomogeneous polymers based on variable cell shape dynamic self-consistent field theory

In this paper, we combine variable cell shape method with dynamic self-consistent field theory and extend to study structure and dynamics under shear for triblock copolymer melts. Due to shear, the calculation cell shape is variable and no longer orthogonal. Pseudospectral method is employed to solve the diffusion equation for chain propagator on the nonorthogonal coordinate and the shear periodical condition can be easily designed in terms of the variable cell shape method. By using this strategy, the shear induced morphology evolution is investigated for topologically complex polymeric systems such as linear and star triblock copolymers; the morphology of linear ABC triblock copolymers is more shear sensitive than that of star triblocks. In particular, once the chain propagator is obtained, the microscopic elastic stress and spatial stress distribution can be derived and thus the dynamic mechanical property can be calculated under shear. By imitating the dynamic storage modulus G' corresponding to any given morphology in the oscillatory shear measurements, we explore the relationship between the morphology and the storage modulus G' and extend to study the mechanism of phase separation dynamics as well as order-disorder transition (ODT) for linear and star triblock copolymers. The results show that the chain architecture can be easily distinguished by investigating the ODT, though the systems such as AB symmetric diblock and ABA triblock copolymers by coupling AB precursors almost exhibit similar microstructures. In addition, the storage modulus G' and loss modulus G" can be simultaneously determined in frequency sweeps of oscillatory shear measurements and the dependence of the moduli on phase separated patterns and the chain topology is investigated. The simulation findings are in qualitatively agreement with the experimental results.

Symmetric Diblock Copolymer Thin Films on Rough Substrates: Microdomain Periodicity in Pure and Blended Films

The lamellar dimension, D, for pure and blended block copolymer (BCPs) thin films of symmetric poly(styrene)-block-poly(methyl methacrylate) was measured using atomic force microscopy analysis of surface patterns of perpendicularly oriented lamellar structures. It was approximately verified, using SAXS and AFM analysis, that perpendicular structures in lamellar thin films bounded by a neutral and a roughened interface did not significantly alter the bulk block copolymer phase separation thermodynamics. In the case of blends, it was also verified that there was a uniform distribution of blend components throughout the thin film. These checks allowed a detailed comparison of D as a function of the number of statistical segments (N) and blend composition using existing bulk phase predictions and experiments. In pure BCPs we observed the two characteristic regimes; strong (D ∼ N0.66) and intermediate segregation regimes (D ∼ N0.85). In addition, we observed a more complicated, weakly segregated regime where D does not scale as N0.5 as expected for a random coil. In blended BCPs there was good agreement with existing theory when BCP components from the strongly segregated regime were used. In cases where intermediate or weakly segregating BCP components were used, an alternative semiempirical analytical function was developed. In cases where the ratio of N for the BCP blend components exceeds ∼5, macrophase separation was observed in the thin film with the smaller BCP component remaining at the air surface.

A theoretical study for nanoparticle partitioning in the lamellae of diblock copolymers

Morphology control is important for practical applications of composite materials that consist of functional polymers and nanoparticles. Toward that end, block copolymers provide useful templates to arrange nanoparticles in the scaffold of self-organized polymer microdomains. This paper reports theoretical predictions for the distribution of nanoparticles in the lamellar structures of symmetric diblock copolymers on the basis of a polymer density functional theory (DFT) and the potential distribution theorem (PDT). The DFT predicts periodic spacing of lamellar structures in good agreement with molecular dynamics simulations. With the polymer structure from DFT as the input, the PDT is used to examine the effects of particle size, surface energy, polymer chain length, and compressibility on the distribution of nanoparticles in the limit of low particle density. It is found that the nanoparticle distribution depends not only on the particle size and surface energy but also on the local structure of the microdomain interface, polymer chain length, and compressibility. The theoretical predictions are compared well with experiments and simulations.

Calculating the free energy of self-assembled structures by thermodynamic integration

We discuss a method for calculating free energy differences between disordered and ordered phases of self-assembling systems utilizing computer simulations. Applying an external, ordering field, we impose a predefined structure onto the fluid in the disordered phase. The structure in the presence of the external, ordering field closely mimics the structure of the ordered phase (in the absence of an ordering field). Self-consistent field theory or density functional theory provides an accurate estimate for choosing the strength of the ordering field. Subsequently, we gradually switch off the external, ordering field and, in turn, increase the control parameter that drives the self-assembly. The free energy difference along this reversible path connecting the disordered and the ordered state is obtained via thermodynamic integration or expanded ensemble simulation techniques. Utilizing Single-Chain-in-Mean-Field simulations of a symmetric diblock copolymer melt we illustrate the method and calculate the free energy difference between the disordered phase and the lamellar structure at an intermediate incompatibility chiN=20. Evidence for the first-order character of the order-disorder transition at fixed volume is presented. The transition is located at chi(ODT)N=13.65+/-0.10 for an invariant degree of polymerization of N=14 884. The magnitude of the shift of the transition from the mean field prediction qualitatively agrees with other simulations.

WITHDRAWN: Influences of selective solvents on phase behavior of symmetric diblock copolymers

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