Dynamic Self-consistent Field Research Articles

Efficient Modeling of High-Generation Dendrimers in Solution Using Dynamical Self-Consistent Field Theory

Efficient Modeling of High-Generation Dendrimers in Solution Using Dynamical Self-Consistent Field Theory

Investigating microstructure evolution in block copolymer membranes.

Block copolymer self-assembly in conjunction with nonsolvent-induced phase separation (SNIPS) has been increasingly leveraged to fabricate integral-asymmetric membranes. The large number of formulation and processing parameters associated with SNIPS, however, has prevented the reliable construction of high performance membranes. In this study, we apply dynamical self-consistent field theory to model the SNIPS process and investigate the effect of various parameters on the membrane morphology: solvent selectivity, nonsolvent selectivity, initial film composition, and glass transition composition. We examine how solvent selectivity and concentration of polymers in the film impact the structure of micelles that connect to form the membrane matrix. In particular, we find that preserving the order in the surface layer and forming a connection between the supporting and surface layer are nontrivial and sensitive to each parameter studied. The effect of each parameter is discussed, and suggestions are made for successfully fabricating viable block copolymer membranes.

The Process-Directed Self-Assembly of Block Copolymer Particles.

The kinetic paths of structural evolution and formation of block copolymer (BCP) particles are explored using dynamic self-consistent field theory (DSCFT). It is shown that the process-directed self-assembly of BCP immersed in a poor solvent leads to the formation of striped ellipsoids, onion-like particles and double-spiral lamellar particles. The theory predicts a reversible path of shape transition between onion-like particles and striped ellipsoidal ones by regulating the temperature (related to the Flory-Huggins parameter between the two components of BCP, χAB ) and the selectivity of solvent toward one of the two BCP components. Furthermore, a kinetic path of shape transition from onion-like particles to double-spiral lamellar particles, and then back to onion-like particles is demonstrated. By investigating the inner-structural evolution of a BCP particle, it is identified that changing the intermediate bi-continuous structure into a layered one is crucial for the formation of striped ellipsoidal particles. Another interesting finding is that the formation of onion-like particles is characterized by a two-stage microphase separation. The first is induced by the solvent preference, and the second is controlled by the thermodynamics. The findings lead to an effective way of tailoring nanostructure of BCP particles for various industrialapplications.

Modeling Microstructure Formation in Block Copolymer Membranes Using Dynamical Self-Consistent Field Theory.

Block copolymers have attracted recent interest as candidate materials for ultrafiltration membranes, due to their ability to form isoporous integral-asymmetric membranes by the combined processes of self-assembly and nonsolvent-induced phase separation (SNIPS). However, the dependence of surface layer and substructure morphologies on the processing variables associated with SNIPS is not well understood nor is the interplay between microphase and macrophase separation in block copolymers undergoing such coagulation. Here, we use dynamical self-consistent field theory to simulate the microstructure evolution of block copolymer films during SNIPS and find that such films form the desired sponge-like asymmetric porous substructure only if the solvent and nonsolvent have opposite block selectivities and that otherwise they form a dense nonporous microphase-separated film. Our results could have important implications for the choices of solvent and nonsolvent in the processing of block copolymer membranes.

Formation of diblock copolymer nanoparticles: Theoretical aspects

We explore the shape and internal structure of diblock copolymer (di-BCP) nanoparticles (NPs) by using the Ginzburg-Landau free-energy expansion and the dynamic self-consistent field theory (DSCFT). The self-assembly of di-BCP lamellae confined in emulsion droplets can form either ellipsoidal or onion-like NPs. The corresponding inner structure is a lamellar phase that is either perpendicular to the ellipsoid long axis (L⊥) or forms a multi-layer concentric shell (C∥), respectively. We focus on the effects of the interaction parameters between the A and B monomers χ, and polymer/solvent χPS, as well as the NP size on the nanoparticle shape and internal morphology. The aspect ratio (lAR), defined as the length ratio between the long and short axes, is used to characterize the overall NP shape. Our results show that when the solvent is neutral towards the two blocks, the ratio lAR increases as χ increases, indicating that the NP becomes more elongated. Likewise, decreasing χPS or increasing the NP size also results in a more elongated NP. Furthermore, as the solvent preference towards one of the A or B blocks increases, the NP undergoes a shape transition from a striped ellipsoid (L⊥) to onion-like sphere (C∥). Our results are in good agreement with previous experiments, and some of the predictions could be tested in future experiments.

Memory in the relaxation of a polymer density modulation.

Using analytical considerations and particle-based simulations of a coarse-grained model, we study the relaxation of a density modulation in a polymer system without nonbonded interactions. We demonstrate that shallow density modulations with identical amplitudes and wavevectors that have been prepared by different processes exhibit different nonexponential decay behaviors. Thus, in contrast to the popular assumption of dynamic self-consistent field theory, the density alone does not suffice to characterize the configuration of the polymer system. We provide an analytic description within Linear-Response Theory (LRT) and the Rouse model that quantitatively agree with the results of the particle-based simulations. LRT is equivalent to a generalized model-B dynamics with an Onsager coefficient that is nonlocal in space and time. Alternatively, the Rouse description can be cast into a dynamic density-functional theory that uses the full probability distribution of single-chain configurations as a dynamic variable and yields a memory-free description of the dynamics that quantitatively accounts for the dependence on the preparation process. An approximate scheme that only considers the joint distribution of the first two Rouse modes-the ellipsoid model-is also explored.

Dynamic Self-Consistent Field Approach for Studying Kinetic Processes in Multiblock Copolymer Melts.

The self-consistent field theory is a popular and highly successful theoretical framework for studying equilibrium (co)polymer systems at the mesoscopic level. Dynamic density functionals allow one to use this framework for studying dynamical processes in the diffusive, non-inertial regime. The central quantity in these approaches is the mobility function, which describes the effect of chain connectivity on the nonlocal response of monomers to thermodynamic driving fields. In a recent study, one of us and coworkers have developed a method to systematically construct mobility functions from reference fine-grained simulations. Here we focus on melts of linear chains in the Rouse regime and show how the mobility functions can be calculated semi-analytically for multiblock copolymers with arbitrary sequences without resorting to simulations. In this context, an accurate approximate expression for the single-chain dynamic structure factor is derived. Several limiting regimes are discussed. Then we apply the resulting density functional theory to study ordering processes in a two-length scale block copolymer system after instantaneous quenches into the ordered phase. Different dynamical regimes in the ordering process are identified: at early times, the ordering on short scales dominates; at late times, the ordering on larger scales takes over. For large quench depths, the system does not necessarily relax into the true equilibrium state. Our density functional approach could be used for the computer-assisted design of quenching protocols in order to create novel nonequilibrium materials.

Influence of Small-Scale Correlation on the Interface Evolution of Semiflexible Homopolymer Blends.

Within the framework of a dynamic self-consistent field theory, we study the effect of the correlations in a small scale on polymer dynamics, adopting the semiflexible homopolymer blends as the model system. This is accomplished by taking the pair correlation function of ideal semiflexible chains as the Onsager coefficient and the Debye function as an approximation to the Onsager coefficient. Relying on the difference of the two pair correlation functions in the small-scale region, we can identify the effect of small-scale correlations. In the equilibrium state, with the chain length growing, the interface width has a continuous transition from the contour length to radius of gyration. The investigation of interfacial evolution and chain orientation reveals that strong small-scale correlations would accelerate the small-scale dynamic process. We also expect that such a small-scale effect should be highlighted in the process where microscopic phase separation happens.

Harnessing Zone Annealing to Program Directional Motion of Nanoparticles in Diblock Copolymers: Creating Periodically Well-Ordered Nanocomposites

The integration of the dynamic self-consistent field theory for polymer molecules and molecular dynamics for nanoparticles is proposed to model the structural evolution of nanocomposites in the cou...

Dynamical self-consistent field theory captures multi-scale physics during spinodal decomposition in a symmetric binary homopolymer blend.

We study the spinodal decomposition in a symmetric, binary homopolymer blend using our recently developed dynamical self-consistent field theory. By taking the extremal solution of a dynamical functional integral, the theory reduces the interacting, multi-chain dynamics to a Smoluchowski equation describing the statistical dynamics of a single, unentangled chain in a self-consistent, time-dependent, mean force-field. We numerically solve this equation by evaluating averages over a large ensemble of replica chains, each one of which obeys single-chain Langevin dynamics, subject to the mean field. Following a quench from the disordered state, an early time spinodal instability in the blend composition develops, before even one Rouse time elapses. The dominant, unstable, growing wavelength is on the order of the coil size. The blend then enters a late-time, t, scaling regime with a growing domain size that follows the expected Lifshitz-Slyozov-Wagner t1/3 power law, a characteristic of a diffusion-driven coarsening process. These results provide a satisfying test of this new method, which correctly captures both the early and late time physics in the blend. Our simulation spans five orders-of-magnitude in time as the domains coarsen to 20 times the coil size, while remaining faithful to the dynamics of the microscopic chain model.

Ordering kinetics of lamella-forming block copolymers under the guidance of various external fields studied by dynamic self-consistent field theory.

Self-consistent field theory with a dynamic extension is exploited to investigate the kinetics of the lamellar formation of symmetric block copolymers under the direction of external fields. In particular, three types of directed self-assembly methods - a permanent field for chemo-epitaxy, a dynamic field for zone annealing and an integrated permanent/dynamic field - are examined. For the chemo-epitaxy involving sparsely prepatterned substrates or zone annealing, the block copolymers generally develop into polycrystalline nanostructures with multiple orientations due to the lack of strong driving forces for eliminating the long-lived imperfections in a limited time. As the integrated chemo-epitaxy and zone annealing method is applied to the block copolymer systems, single-crystalline nanostructures with precisely registered orientations are achieved in a short annealing time owing to the mutual acceleration of defect annihilations, which cannot be produced by the conventional techniques alone. Furthermore, the integrated method allows the rapid fabrication of well-ordered nanostructures on the extremely sparse prepatterned substrates. Our theoretical work may serve to rationalize the faster modern nanolithographic fabrication of smaller microelectronic components using lower-spatial-frequency templates.

Structure Development via Reaction-Induced Phase Separation in Polymer Mixtures: Analysis of Early- and Late-Stage Demixing and Computer Simulations at Non-Isoquench Depths

ABSTRACTThe effects of phase separation during polymerization of a monomer/polymer mixture were studied. The kinetics of the spinodal decomposition of glycidyl phenyl ether/polystyrene (95/5) during polymerization was investigated using a light-scattering technique. This phase separation during polymerization was comparatively different from the one observed at the isoquench depth such that (i) there was significantly more time before any observable fluctuation, and (ii) the periodic distance of the resulting two-phase structure in the later stages decreased with time. To further investigate the influence of a continually increasing quench depth on phase separation, a diglycidyl aminophenol/diglycidyl ether of bisphenol A (50/50) system with a simple phase diagram was investigated using ultra-small-angle X-ray scattering analysis, and a similar phase separation behavior was observed. Furthermore, computer simulations using the dynamic self-consistent field method were performed to interpret polymer morphology during the demixing driven by chemical reactions. The simulation results suggested that the periodic distance of the resulting two-phase structure formed in the later stages decreases with time. The time dependence of the concentration fluctuations observed in these simulations provided a satisfying explanation for the morphologies formed as a result of a continuously increasing quench depth.

Insights into ordered microstructures and ordering mechanisms of ABC star terpolymers by integrating dynamic self-consistent field theory and variable cell shape methods

A theoretical approach coupling dynamic self-consistent field (SCF) theory for inhomogeneous polymeric fluids and variable cell shape (VCS) method for automatically adjusting cell shape and size is developed to investigate ordered microstructures and the ordering mechanisms of block copolymer melts. Using this simulation method, we first re-examined the microphase separation of the simplest AB diblock copolymers, and tested the validity and efficiency of the novel method by comparing the results with those obtained from the dynamic SCF theory. An appropriate relaxation parameter of the VCS method effectively accelerates the system towards a stable morphology without distortions or defects. The dynamic SCF/VCS method is then applied to identify the richness morphologies of ABC star terpolymers and explore the ordering mechanisms of star terpolymer melts quenched from homogenous states. A diverse range of ordered microstructures, including two-dimensional tiling patterns, hierarchical structures and ordinary microstructures, are predicted. Three types of ordering mechanisms, namely, one-step, quick-slow and step-wise procedures, are discovered in the disorder-to-order transition of ABC star terpolymers. The procedures of microphase separation in the ABC star terpolymer melts are remarkably affected by the composition of star terpolymers and the strength of interaction parameters.

Kinetic pathways of sphere-to-cylinder transition in diblock copolymer melt under electric field

Phase transition from body-centered-cubic spheres to cylinders in a diblock copolymer melt under an external electric field is investigated by means of real-space dynamical self-consistent field theory. Different phase transition kinetic pathways and different cylindrical domains arrangements of the final phase are observed depending on the strength and direction of the applied electric field. Various transient states have been identified depending on the electric field being applied along [111], [100], and [110] directions. The electric field should be above a certain threshold value in order the transition to occur. A "dynamic critical exponent" of the transition is found to be about 3/2, consistent with other order-order transitions in diblock copolymers under electric field.

Hybrid Lattice Boltzmann/Dynamic Self-Consistent Field Simulations of Microphase Separation and Vesicle Formation in Block Copolymer Systems

We present a hybrid numerical method to introduce hydrodynamics in dynamic self-consistent field (SCF) studies of inhomogeneous polymer systems. It solves a set of coupled dynamical equations: The Navier–Stokes equations for the fluid flow, and SCF-based convection-diffusion equations for the evolution of the local monomer compositions. The Navier–Stokes equations are simulated by the lattice Boltzmann method and the dynamic self-consistent field equations are solved by a finite difference scheme. Two applications are presented: First, we study microphase separation in symmetric and asymmetric block copolymer melts with various values of shear and bulk viscosities, comparing the results to those obtained with purely diffusive dynamics. Second, we investigate the effect of hydrodynamics on vesicle formation in amphiphilic block copolymer solutions. In agreement with previous studies, hydrodynamic interactions are found to have little effect on the microphase separation at early times, but they substantially accelerate the process of structure formation at later times. Furthermore, they also contribute to selecting the pathway of vesicle formation, favoring spherical intermediates over aspherical (disklike) ones.

Minimum energy path to membrane pore formation and rupture.

We combine dynamic self-consistent field theory with the string method to calculate the minimum energy path to membrane pore formation and rupture. In the regime where nucleation can occur on experimentally relevant time scales, the structure of the critical nucleus is between a solvophilic stalk and a locally thinned membrane. Classical nucleation theory fails to capture these molecular details and significantly overestimates the free energy barrier. Our results suggest that thermally nucleated rupture may be an important factor for the low rupture strains observed in lipid membranes.

External potential dynamics simulation of the compatibility of T-shaped graft copolymer compatibilizing two immiscible homopolymers

AbstractThe compatibilizing effect of T-shaped graft copolymers between two incompatible homopolymers A and B was studied by using the external potential dynamics (EPD) method based on dynamic self-consistent field theory. We specifically focus on the role of the concentration and the junction position of the graft copolymer on the compatibility. The blend containing compatibilizers exhibits slow dynamics during the phase separation. We observed that the compatiblizers are particularly efficient when the graft point is close to the free end of the backbone chain.

Electric field-induced transitions in perforated lamella of ABA triblock copolymer thin film

Transformation of a perforated lamella of an ABA triblock copolymer melt confined in a thin film under an electric field is studied by real-space dynamical self-consistent field theory. Starting from a perforated lamella structure, we apply an electric field along [100] and [010] directions which are parallel to the surfaces of the thin film, and along [001] direction which is perpendicular to the thin film. With a sufficiently high value of the electric field, transitions to cylinders or perpendicular lamellae occur. Several kinetic pathways of the transitions have been found, some of which are very different from the ones observed for a perforated lamella in an AB diblock copolymer.

Transition in Domain Morphology of Block Copolymers Undergoing Polymerization

We have studied the dynamics of a morphology transition of domains in a reactive system composed of A−B block copolymer and B monomer, where polymerization occurs at the end of the B subchain of the block copolymer. We propose a new dynamical model for reactive phase separation, where the chain architecture, the chain conformation, and the spatial distribution of the reaction points (reactive chain ends) are taken into account by the use of random phase approximation. Our model is essentially equivalent to the dynamical self-consistent field theory if the system is close to the critical point, but it has an enormous improvement in the computational efficiency. Simulation results show that our modeling is quite useful partially because the inhomogeneity of the spatial distribution of the reactants affects the polymerization and partially because we can perform large-scale and long time simulations because of the high efficiency of our model. We found that as a result of the competition between the phase separation and the polymerization the system shows complex nonequilibrium domain structures that depend on the kinetic pathway of the phase separation and the polymerization.

Hydrodynamic effects on the disorder-to-order transitions of diblock copolymer melts

Dynamic self-consistent field theory and Navier-Stokes equation are coupled to study hydrodynamic effects on the disorder-to-order transition of diblock copolymer melts. Simulations are performed for different values of the viscosity and the results are compared with those without the hydrodynamic effects. The lower the viscosity is, the faster the free energy density decays. Starting from uniformly mixed initial states, body-centered-cubic spheres, hexagonally packed cylinders, and lamellar structures are obtained in the presence of the hydrodynamic effects. On the contrary, the bicontinuous double gyroid structure cannot be obtained even with the hydrodynamic effects. The scattering intensities of the domain structures are calculated to confirm these properties of the ordering of the structures.