Process-directed self-assembly of block copolymers refers to the processes that can reproducibly direct the kinetics of structure formation ensuing from an unstable state, generated by rapidly altering the thermodynamic parameter(s) of system, into a desired, metastable state. Compared with the strategy that focuses on the alleviation of the packing frustration of desired, metastable states by fine-tuning the chain architecture of the block copolymer or blending and therefore makes them thermodynamically stable, it is particularly suitable for the fabrication of desired, metastable states with high packing frustration, which is very hard to release. Moreover, this strategy also provides ample opportunities for the explorations on the symmetry-conserving and symmetry-altering phenomena during the kinetics of structure formation. This review mainly focuses on the theories and particle-based simulations that can be used to explore the process-directed self-assembly of block copolymers. Several representative results, which are obtained by both theories and particle-based simulations using the alchemical transformation and fast, isotropic pressure quench, respectively, to generate the unstable states as the starting points, are reviewed as well to highlight the efficiency of such a strategy as well as the symmetry-conserving and symmetry-altering phenomena during the kinetics of structure formation. The challenges encountered currently are also briefly discussed.

Protracted equilibration times and a multitude of competing periodically modulated structures are common characteristics of complex, self-assembled phases in block copolymer materials. These characteristics highlight the importance of designing processes to reproducibly direct the structure evolution of complex, spatially modulated structures. Using general symmetry considerations, we design a process that deterministically transforms the common, double-gyroid (DG) phase of linear diblock copolymers into a metastable A15 structure by irreversibly switching a conformationally symmetric ABB^{'} diblock copolymer into an ABA^{'} triblock copolymer. Key to the fabrication of the Frank-Kasper A15 structure is the nonmonotonic time evolution of the segregation that allows us to partition the large unit cell of the DG phase into 8 equivalent A15 units. Comparison between dynamic self-consistent field theory and particle-based simulations demonstrates the robustness of the designed pathway against thermal fluctuations and memory effects due to the underlying Rouse dynamics. Metastable structures that can be accessed from the DG phase are systematically explored as a function of molecular asymmetry and incompatibility.

The morphology of a multicomponent polymer melt within self-consistent field theory (SCFT) is completely characterized by the spatial density distribution of the components. SCFT therefore assumes that the molecular conformations are fast variables that adopt their equilibrium statistics with respect to a given density distribution. There are multiple situations, e.g., the early stages of structure formation, where this assumption breaks down because the densities evolve significantly on the time scale of the single-chain relaxation. Here, we develop a SCFT that uses as slow variables both densities and the variance of the first, most slowly relaxing Rouse mode and we design a numerical scheme for its solution based on single-chain propagators. Applications to diblock and multiblock copolymers are presented.

Besides dictating the equilibrium phase diagram, the rugged free-energy landscape of AB block copolymers gives rise to a multitude of non-equilibrium phenomena. Self-consistent field theory (SCFT) can be employed to calculate the mean-field free energy, F[ϕAtarget], of a non-equilibrium unstable state that is characterized by a given spatial density distribution, ϕAtarget, in the incompressible system. Such a free-energy functional is the basis of describing the structure formation by dynamic SCFT techniques or the identification of minimum free-energy paths via the string method. The crucial step consists in computing the external potential fields that generate the given density distribution in the corresponding system of non-interacting copolymers, i.e., the potential-to-density relation employed in equilibrium SCFT calculations has to be inverted (reverse SCFT calculation). We describe, generalize, and evaluate the computational efficiency of two different numerical algorithms for this reverse SCFT calculation—the Debye-function algorithm based on the structure factor and the field-theoretic umbrella-potential (FUP) algorithm. In contrast to the Debye-function algorithm, the FUP algorithm only yields the exact mean-field values of the given target densities in the limit of a strong umbrella potential, and we devise a two-step variant of the FUP algorithm that significantly mitigates this issue. For Gaussian copolymers, the Debye-function algorithm is more efficient for highly unstable states that are far away from the equilibrium, whereas the improved FUP algorithm outperforms the Debye-function algorithm closer to metastable states and is easily transferred to more complex molecular architectures.

Ellipsoids have attracted abiding attention because of their shape-dependent, anisotropic properties. In some applications, e.g., photonic crystals, both positional and orientational order of the ellipsoidal packing are required. We propose a versatile, facile, and efficient strategy to fabricate positionally and orientationally ordered crystals of soft ellipsoids in block copolymers via a step-shear deformation. Starting from the thermodynamically stable, equilibrium spherical mesophase of the copolymer material, the step-shear deformation provides an instantaneous, anisotropic stimulus to deform the spherical domains into ellipsoids and simultaneously stretches the macromolecular conformations. Subsequently, at fixed strain, the molecular stress relaxes to an equilibrium where the shape and orientation of the obtained ellipsoids are dictated by the packing frustration. Since the residual molecular stress is minuscule, the lattice relaxation via slippage in the absence of external stress is protracted, i.e., the crystal of soft ellipsoids with positional and orientational order is pseudometastable. Our strategy also allows for low volume fractions of ellipsoids (compared to colloidal systems). Both single-chain-in-mean-field (SCMF) simulations and self-consistent field theory (SCFT) calculations are employed to demonstrate the pseudometastability of the obtained ellipsoids. Varying the magnitude of the step-shear strain and the composition of the block copolymer, we can control the asphericity and orientation of the ellipsoidal domains independently. Our study provides a new concept for fabricating soft, positionally and orientationally ordered crystals of ellipsoids with potential applications in engineering functional materials.

Process-directed self-assembly of block copolymers refers to thermodynamic processes that reproducibly direct the kinetics of structure formation from a starting, unstable state into a selected, metastable mesostructure. We investigate the kinetics of self-assembly of linear ACB triblock copolymers after a rapid transformation of the middle C block from B to A. This prototypical process (e.g., photochemical transformation) converts the initial, equilibrium mesophase of the ABB copolymer into a well-defined but unstable, starting state of the AAB copolymer. The spontaneous structure formation that ensues from this unstable state becomes trapped in a metastable mesostructure, and we systematically explore which metastable mesostructures can be fabricated by varying the block copolymer composition of the initial and final states. In addition to the equilibrium mesophases of linear AB diblock copolymers, this diagram of process-accessible states includes 7 metastable periodic mesostructures, inter alia, Schoen's F-RD periodic minimal surface. Generally, we observe that the final, metastable mesostructure of the AAB copolymer possesses the same symmetry as the initial, equilibrium mesophase of the ABB copolymer.

The free-energy landscape of self-assembling block copolymer systems is characterized by a multitude of metastable minima and concomitant protracted relaxation times of the morphology. Tailoring rapid changes (quench) of thermodynamic conditions, one can reproducibly trap the ensuing kinetics of self-assembly in a specific metastable state. To this end, it is necessary to (1) control the generation of well-defined, highly unstable states and (2) design the unstable state such that the ensuing spontaneous kinetics of structure formation reaches the desired metastable morphology. This process-directed self-assembly provides an alternative to fine-tuning molecular architecture by synthesis or blending, for instance, in order to fabricate complex network structures. Comparing our simulation results to recently developed free-energy techniques, we highlight the importance of non-equilibrium molecular conformations in the starting state and motivate the significance of the local conservation of density.

Using particle-based simulation of a soft, coarse-grained model and self-consistent field theory (SCFT), we investigate the properties of dense living polymer systems with ring formation both in the bulk and in thin films. In the bulk, our results confirm that the molecular weight distribution of ring polymers exhibits a combination of an exponential decay and a power law. The exponential molecular weight distribution of linear chains is hardly affected by ring formation, only the corresponding mean molecular weight is slightly reduced. At lower segment density, the fraction of monomers of ring polymers is increased. In thin films, ring formation does not influence the width of the narrow interface, where the segment density rises to the bulk value. Since the molecular extension of ring polymers is smaller than that of linear chains, the spatial extent of the wide interphase is, however, reduced. In the narrow interface, we find that the local ring formation is affected by five aspects: (1) the mirroring of chain conformations and back-folding by the solid substrate (Silberberg argument) and other four aspects at the solid substrate, i.e., (2) the reduced segment density, (3) the enrichment of chain ends, (4) the pronounced segregation of nonbonded monomers, and (5) the reduced dimensionality of ring polymers. In the wide interphase, the local ring formation is enhanced mainly by the first aspect. By comparing the results from the particle-based simulation and SCFT, we observe good agreement in the wide interphase, but the difference in the local structure leads to differences in the narrow interface. Additionally, the SCFT results show that a decrease of the film thickness increases the global ring formation within the thin film.

Using particle-based simulations of a soft, coarse-grained model and self-consistent field theory (SCFT) we investigate the properties of dense living polymer systems in bulk and thin films. Reversible bond formation and breaking is controlled by a bonding free energy, Eb, and results in a polydisperse melt of linear flexible polymers and rings. For bulk systems we observed an exponential decay of the molecular weight distribution for all but the smallest molecular weight. The latter affect stems from the indistinguishability of the two bonding sites of a living monomeric unit. Under confinement into a film, the presence of the solid substrates gives rise to a redistribution of living monomers and chains in the wide interphase and, therefore, alters the local molecular weight distribution and local mean molecular weight. Additionally, we find that the presence of a solid substrate enhances the formation of rings with high molecular weight by reducing the dimensionality of ring living polymers in the vicinity of solid substrate and increasing the probability of two chain ends belonging to the same linear living polymer chain to meet with each other. The result of particle-based simulations and numerical self-consistent field theory are compared and, although there are differences of the two descriptions in the narrow interface, qualitative agreement is found in the wide interphase.

The free-energy landscape of self-assembling block copolymer systems is characterized by a multitude of metastable minima. Using particle-based simulations of a soft, coarse-grained model, we explore opportunities to reproducibly direct the spontaneous ordering of these self-assembling systems into a metastable complex network morphology-specifically, Schoen's I-WP periodic minimal surface-starting from a highly unstable state that is generated by a rapid expansion. This process-directed self-assembly provides an alternative to fine-tuning molecular architecture or blending for fabricating complex network structures. Comparing our particle-based simulation results to recently developed free-energy techniques, we critically assess their ability to predict spontaneous formation and highlight the importance of non-equilibrium molecular conformations in the starting state and the local conservation of density.

Cover: The morphology of simple cubic spheres connected by cylinders (SCS+C) is found stable for the H-shaped (AC)B(CA) block copolymers. In this structure, the simple-cubic-packed spheres are formed by the majority side-arm species and are connected by cylinders formed by minority side-arm species along six directions. Further details can be found in the article by D.-W. Sun, Z.-Y. Sun,* H.-F. Li, and L.-J. An* on page 100.

The microphase separation behavior of H-shaped (AC)B(CA) ternary block copolymer systems with asymmetric interaction parameters are studied in 3D space by a combinatorial screening method based on SCF theory. The focus is on systems with asymmetric interaction energies among three species, where the Flory-Huggins interaction parameters obey chi(AC) < chi(AB) = chi(BC) and chi(AC) > chi(AB) = chi(BC). It is shown that the asymmetric interaction energies enlarge the stable regions of "two-color" and "three-color" core/shell microphase morphologies near the three edges of triangle phase diagrams. The "three-color" hexagonal honeycomb packed cylinder (HEX3-PC) morphology and the simple cubic spheres connected by cylinders (SCS + C) morphology are found stable.

By using a combinatorial screening method based on the self-consistent field theory (SCFT) for polymer systems, the micro-phase morphologies of the H-shaped (AC)B(CA) ternary block copolymer system are studied in three-dimensional (3D) space. By systematically varying the volume fractions of the components A, B, and C, six triangle phase diagrams of this H-shaped (AC)B(CA) ternary block copolymer system with equal interaction energies among the three components are constructed from the weaker segregation regime to the strong segregation regime, In this study, thirteen 3D micro-phase morphologies for this H-shaped ternary block copolymer system are identified to be stable and seven 3D microphase morphologies are found to be metastable. It is found that in the weaker segregation regime (chi N-AB = chi N-AC = chi N-BC = 45), the minority component can be mixed with other two majority components to form the mixed phase regions, while in the intermediate segregation regime (chi N-AB = chi N-AC = chi N-BC = 60 and 75), the mixed phase regions phase-separate into three distinct phase regions. In the strong segregation regime (chi N-AB = chi N-AC = chi N-BC = 90, 100, and 125), the distinct blocks tend to separate with each other and the phase behavior of this H-shaped (AC)B(CA) ternary block copolymer is similar to that of the three-arm star-like ABC ternary block copolymer. Moreover, the order-disorder transitions and the order-order transitions by varying the interaction parameters are discussed. These results may help in the design of the microstructures of complex block copolymers. Crown Copyright (C) 2009 Published by Elsevier Ltd. All rights reserved.