Fluctuation effects in block copolymer melts

Self-assembly of colloids with competing interactions confined in spheres

At low temperatures, colloidal particles with short-range attractive and long-range repulsive interactions can form various periodic microphases in bulk. In this paper, we investigate the self-assembly behaviour of colloids with competing interactions under spherical confinement by conducting molecular dynamics simulations. We find that the cluster, mixture, cylindrical, perforated lamellar and lamellar structures can be obtained, but the details of the ordered structures are different from those in bulk systems. Interestingly, the system tends to form more perforated structures when confined in smaller spheres. The mechanism behind this phenomenon is driven by the relationship between the energy of the ordered structures and the bending of the confinement wall, which is different from the mechanism in copolymer systems.

Exploring Disordered Morphologies of Blends and Block Copolymers for Light-Emitting Diodes with Mesoscopic Simulations

Recently, disordered blends of semiconducting and insulating polymers have been used to prepare light-emitting diodes with increased luminous efficiency. Because the thermodynamic stability of the disordered phase in blends is limited, equivalent diblock copolymers (BCPs) could be an alternative. However, the choice between disordered blends and BCPs requires understanding structural differences and their effect on charge carrier transport. Using a hybrid mesoscopic model, we simulate blends and equivalent BCPs of two representative semiconducting and insulating polymers: poly(p-phenylene vinylene) (PPV) and polyacrylate. The immiscibility is varied to mimic annealing at different temperatures. We find stable or metastable disordered morphologies until we reach the mean-field (MF) spinodal. Disordered morphologies are heterogeneous because of thermal fluctuations and local segregation. Near the MF spinodal, segregation is stronger in BCPs than in the blends, even though the immiscibility, normalized by the MF spinodal, is the same. We link the spatial distribution of PPV with electric conductance. We predict that the immiscibility (temperature at which the layer is annealed) affects electrical percolation much stronger in BCPs than in blends. Differences in the local structure and percolation between blends and BCPs are enhanced at a high insulator content.

Calibration of a lattice model for high-molecular-weight block copolymer melts

The Morse calibration is applied to a lattice model designed for efficient simulations of two-component polymer melts of high molecular weight. The model allows multiple occupancy per site, which results in high invariant polymerization indices, and interactions are limited to monomers within the same site, which enhances the computational speed. The calibration maps the interaction parameter of the lattice model, α, onto the Flory-Huggins χ parameter of the standard Gaussian-chain model, by matching the disordered-state structure function, S(k), of symmetric diblock copolymers to renormalized one-loop predictions. The quantitative accuracy of the calibration is tested by comparing the order-disorder transition of symmetric diblock copolymer melts to the universal prediction obtained from previous simulations. The model is then used to confirm the universality of fluctuation corrections to the critical point of symmetric binary homopolymer blends.

Fluctuation correction for the critical transition of symmetric homopolymer blends

Monte Carlo simulations are performed on structurally symmetric binary homopolymer blends over a wide range of invariant polymerization indices, N¯. A finite-size scaling analysis reveals that certain critical exponents deviate from the expected 3D-Ising values as N¯ increases. However, the deviations are consistent with previous simulations and can be attributed to the fact that the system crosses over to mean-field behavior when the molecules become too large relative to the size of the simulation box. Nevertheless, the finite-size scaling techniques provide precise predictions for the position of the critical transition. Using a previous calibration of the Flory-Huggins interaction parameter, χ, we confirm that the critical point scales as (χN)_c=2+cN¯^-1∕2 for large N¯, and more importantly we are able to extract a reliable estimate, c≈1.5, for the universal constant.

Universality between Experiment and Simulation of a Diblock Copolymer Melt

The equivalent behavior among analogous block copolymer systems involving chemically distinct molecules or mathematically different models has long hinted at an underlying universality, but only recently has it been rigorously demonstrated by matching results from different simulations. The profound implication of universality is that simple coarse-grained models can be calibrated so as to provide quantitatively accurate predictions to experiment. Here, we provide the first compelling demonstration of this by simulating a polyisoprene-polylactide diblock copolymer melt using a previously calibrated lattice model. The simulation successfully predicts the peak in the disordered-state structure function, the position of the order-disorder transition, and the latent heat of the transition in excellent quantitative agreement with experiment. This could mark a new era of precision in the field of block copolymer research.

Commensurability and finite size effects in lattice simulations of diblock copolymers

Lattice Monte Carlo (MC) simulations provide an efficient method for exploring the structure and phase behavior of block polymer melts. However, the results of such simulations may differ from the equilibrium behavior of a hypothetical infinite system as a consequence of the finite size of the simulation box. Standard finite-size scaling techniques cannot be employed to remove the effects of a small system size due to incommensurability between the ordered structure domain spacing and the periodicity of the simulation box. This work describes a systematic approach to estimating the equilibrium domain spacing in lattice MC simulations of symmetric diblock copolymers, and thereby minimize the effects of incommensurability. Results for simulations with commensurate simulation boxes, which are designed to be commensurate with the preferred lattice periodicity but contain different numbers of unit cells, show that once the effects of incommensurability are removed, the effects of finite size alone are relatively small.

Universality of block copolymer melts

Simulations of five different coarse-grained models of symmetric diblock copolymers are compared to demonstrate a universal (i.e., model-independent) dependence of the free energy and order-disorder transition (ODT) on the invariant degree of polymerization N̄. The actual values of χN at the ODT approach predictions of the Fredrickson-Helfand (FH) theory for N̄ ≳ 10(4) but significantly exceed FH predictions at lower values characteristic of most experiments. The FH theory fails for modest N̄ because the competing phases become strongly segregated near the ODT, violating an underlying assumption of weak segregation.

Phase diagrams of block copolymer melts by dissipative particle dynamics simulations

Phase diagrams for monodisperse and polydisperse diblock copolymer melts and a random multiblock copolymer melt are constructed using dissipative particle dynamics simulations. A thorough visual analysis and calculation of the static structure factor in several hundreds of points at each of the diagrams prove the ability of mesoscopic molecular dynamics to predict the phase behavior of polymer systems as effectively as the self-consistent field-theory and Monte Carlo simulations do. It is demonstrated that the order-disorder transition (ODT) curve for monodisperse diblocks can be precisely located by a spike in the dependence of the mean square pressure fluctuation on χN, where χ is the Flory-Huggins parameter and N is the chain length. For two other copolymer types, the continuous ODTs are observed. Large polydispersity of both blocks obeying the Flory distribution in length does not shift the ODT curve but considerably narrows the domains of the cylindrical and lamellar phases partially replacing them with the wormlike micelle and perforated lamellar phases, respectively. Instead of the pure 3d-bicontinuous phase in monodisperse diblocks, which could be identified as the gyroid, a coexistence of the 3d phase and cylindrical micelles is detected in polydisperse diblocks. The lamellar domain spacing D in monodisperse diblocks follows the strong-segregation theory prediction, D∕N(1∕2) ~ (χN)(1∕6), whereas in polydisperse diblocks it is almost independent of χN at χN < 100. Completely random multiblock copolymers cannot form ordered microstructures other than lamellas at any composition.

Fluctuation/correlation effects in symmetric diblock copolymers: on the order-disorder transition

Using fast off-lattice Monte Carlo simulations with experimentally accessible fluctuations, we reported the first systematic study unambiguously quantifying the shift of the order-disorder transition (ODT) χ* of symmetric diblock copolymers from the mean-field prediction χ(MF)*. Our simulations are performed in a canonical ensemble with variable box lengths to eliminate the restriction of periodic boundary conditions on the lamellar period, and give the most accurate data of χ* and bulk lamellar period reported to date. Exactly the same model system (Hamiltonian) is used in both our simulations and mean-field theory; the ODT shift is therefore due to the fluctuations/correlations neglected by the latter. While χ*/χ(MF)*-1∝N(-k) is found with N denoting the invariant degree of polymerization, k decreases around the N-value corresponding to the face-centered cubic close packing of polymer segments as hard spheres, indicating the short-range correlation effects.

Effects of compositional asymmetry in phase behavior of ABA triblock copolymer melts from Monte Carlo simulation

We simulate ABA triblock copolymer melts using a lattice Monte Carlo method, known as cooperative motion algorithm, probing various degrees of compositional asymmetry. Selected order-disorder transition lines are determined in terms of the segment incompatibility, quantified by product χN , and the triblock asymmetry parameters, α and β. We correlate the results of the simulation with the self-consistent field theory and an experimental study of polyisoprene-polystyrene-polyisoprene triblock melt by Hamersky and coworkers. In particular, we confirm the mean-field prediction that for highly asymmetric triblocks the short A -block is localized in the middle of the B -domain due to an entropic advantage. This results in the middle block relaxation and is consistent with the experimental data indicating that as the relatively short A -blocks are grown into AB diblock, from the B -block side, the order-disorder transition temperature is considerably depressed.

Monte Carlo phase diagram for diblock copolymer melts

The phase diagram for diblock copolymer melts is evaluated from lattice-based Monte Carlo simulations using parallel tempering, improving upon earlier simulations that used sequential temperature scans. This new approach locates the order-disorder transition (ODT) far more accurately by the occurrence of a sharp spike in the heat capacity. The present study also performs a more thorough investigation of finite-size effects, which reveals that the gyroid (G) morphology spontaneously forms in place of the perforated-lamellar (PL) phase identified in the earlier study. Nevertheless, there still remains a small region where the PL phase appears to be stable. Interestingly, the lamellar (L) phase next to this region exhibits a small population of transient perforations, which may explain previous scattering experiments suggesting a modulated-lamellar (ML) phase.

Theoretically informed coarse grain simulations of polymeric systems

A Monte Carlo formalism for the study of polymeric melts is described. The model is particle-based, but the interaction is derived from a local density functional that appears in the field-based model. The method enables Monte Carlo simulations in the nVT, nPT, semigrandcanonical and Gibbs ensembles, and direct calculation of free energies. The approach is illustrated in the context of two examples. In the first, we consider the phase separation of a binary homopolymer blend and present results for the phase diagram and the critical point. In the second, we address the microphase separation of a symmetric diblock copolymer, examine the distribution of local stresses in lamellae, and determine the order-disorder transition temperature.

Ordering of a lamella-forming fluid near an interface

By using wedged thin films, we have measured the effect of interfaces on the ordering of an anisotropic fluid in real space. Symmetric diblock copolymers can form an ordered lamellar fluid, and the preference of the substrate for one of the blocks can induce order well into the disordered bulk phase. The induced order decays away from the substrate with a length scale that diverges at the bulk ordering transition. Ordering and disordering kinetics are found to differ: all layers relax identically upon disordering, whereas the formation of lamellae is found to vary with the distance from the substrate and can be understood from the time-dependent Ginzburg-Landau theory.

Monte carlo simulation of coarse grain polymeric systems

We introduce a particle-based Monte Carlo formalism for the study of polymeric melts, where the interaction energy is given by a local density functional, as is done in traditional field-theoretic models. The method enables Monte Carlo simulations in arbitrary ensembles and direct calculation of free energies. We present results for the phase diagram and the critical point of a binary homopolymer blend. For a symmetric diblock copolymer, we compute the distribution of local stress in lamellae and locate the order-disorder transition.

Monte Carlo simulation on symmetric ABA/AB copolymer blends in confined thin films

The effects of blend composition on morphology, order-disorder transition (ODT), and chain conformation of symmetric ABA/AB copolymer blends confined between two neutral hard walls have been investigated by lattice Monte Carlo simulation. Only lamellar structure is observed in all the simulation morphologies under thermodynamic equilibrium state, which is supported by theoretical prediction. When the composition of AB diblock copolymer (phi) increases, both lamellar spacing and the corresponding ODT temperature increase, which can be attributed to the variation of conformation distribution of the diblock and the triblock copolymer chains. In addition, both diblock and triblock copolymer, chains with bridge conformation extend dramatically in the direction parallel to the surface when the system is in ordered state. Finally, the copolymer chain conformation depends strongly on both the blend composition and the incompatibility parameter chiN.

Single chain in mean field simulations: Quasi-instantaneous field approximation and quantitative comparison with Monte Carlo simulations

The description of fluctuations by single chain in mean field (SCMF) simulations is discussed and the results of this particle-based self-consistent field technique are quantitatively compared to Monte Carlo simulations of the same discretized Edwards-Hamiltonian providing exact reference data. In SCMF simulations one studies a large ensemble of noninteracting molecules subjected to real, external fields by Monte Carlo simulations. The external fields approximate nonbonded, instantaneous interactions between molecules. In the self-consistent mean field theory the external fields are static and fluctuation effects are ignored. In SCMF simulations, the external fields fluctuate since they are frequently recalculated from the instantaneous density distribution of the ensemble of molecules. In the limit of infinitely high density or instantaneous update of the external fields, the SCMF simulation method accurately describes long-wavelength fluctuations. At high but finite updating frequency the accuracy depends on the discretization of the model. The accuracy is illustrated by studying the single chain structure and intermolecular correlations in polymer melts, and fluctuation effects on the order-disorder transition of symmetric diblock copolymers.

Morphology of multi-component polymer systems: single chain in mean field simulation studies

Recent work exploring phase separation and self-assembly in multicomponent polymer fluids using a particle-based self-consistent field simulation method is reviewed. The computational method is placed in the context of classical molecular dynamics and Monte Carlo simulations as well as field-theoretic approaches. Its potential is illustrated by applications ranging from spinodal decomposition in symmetric polymer blends and the ordering of diblock copolymers in the bulk to more complex phenomena such as solvent evaporation from thin polymer films and the fabrication of three-dimensional bicontinuous diblock copolymer morphologies reconstruction on patterned substrates.

Monte Carlo phase diagram for diblock copolymer melts

A partial phase diagram is constructed for diblock copolymer melts using lattice-based Monte Carlo simulations. This is done by locating the order-disorder transition (ODT) with the aid of a recently proposed order parameter and identifying the ordered phase over a wide range of copolymer compositions (0.2<or=f<or=0.8). Consistent with experiments, the disordered phase is found to exhibit direct first-order transitions to each of the ordered morphologies. This includes the spontaneous formation of a perforated-lamellar phase, which presumably forms in place of the gyroid morphology due to finite-size and/or nonequilibrium effects. Also included in our study is a detailed examination of disordered cylinder-forming (f=0.3) diblock copolymers, revealing a substantial degree of pretransitional chain stretching and short-range order that set in well before the ODT, as observed previously in analogous studies on lamellar-forming (f=0.5) molecules.

Liquids in equilibrium: beyond the hypernetted chain

Density functional techniques are used to derive a charging expression for the non-uniform density of a molecular liquid. In the atomic limit the equation reduces to an exact form due to Fixman. The theory is simplified greatly via a physical approximation that accounts for three-body correlations beyond those included in the hypernetted chain (HNC) closure of the Ornstein-Zernike (OZ) equation. The radial distribution function is obtained as a special case. The theory is tested by examining the phase behavior of two fundamental complex fluids: the homopolymer blend and diblock copolymer melts. For the former it is found, contrary to HNC theory and its molecular generalizations, that a critical temperature Tc is predicted from the structure route. This To scales linearly with degree of polymerization N in agreement with Flory theory. The simplest form of the theory can be considered as a way to incorporate attractive interactions within a formalism that is very similar to that of the OZ or reference interaction site model (RISM). The relevance of the theory to charged liquids is also discussed.

Formation and structure of the microemulsion phase in two-dimensional ternary AB+A+B polymeric emulsions

We present an analysis of the structure of the fluctuation-induced microemulsion phase in a ternary blend of balanced AB diblock copolymers with equal amounts of A and B homopolymers. To this end, graphical analysis methods are employed to characterize two-dimensional configuration snapshots obtained with the recently introduced field-theoretic Monte Carlo method. We find that a microemulsion forms when the mean curvature diameter of the lamellar phase coincides roughly with the periodicity of the lamellar phase. Further, we provide evidence to the effect of a subclassification of the microemulsion into a genuine and a defect-driven region.