Self-assembly of block copolymers confined in cylindrical nanopores is studied systematically using a simulated annealing technique. For diblock copolymers which form two-dimensional hexagonally packed cylinders with period L0 in the bulk, novel structures such as helices and stacked toroids spontaneously form inside the cylindrical pores. These confinement-induced morphologies have no counterpart in the bulk system and they depend on the pore diameter (D) and the surface-polymer interactions, reflecting the importance of structural frustration and interfacial interactions. On tightening the degree of confinement, transitions from helices to toroids to spheres are observed. Mechanisms of the morphological transitions can be understood based on the degree of structural frustration parametrized by the ratio D/L0.

Block copolymers are a class of soft matter that self-assemble to form ordered morphologies at nanometer scales, making them ideal materials for various applications. The self-assembly of block copolymers is mainly controlled by the monomer-monomer interactions, block compositions and molecular architectures. Besides these intrinsic parameters, placing block copolymers under confinement introduces a number of extrinsic factors, including the degree of structural frustration and surfacepolymer interactions, which can strongly influence the self-assembled morphologies. Therefore confinement of block copolymers provides a powerful route to manipulate their self-assembled nanostructures. In this review, we discuss the relationship between confining conditions and the resulting structures, focusing on principles governing structural formation of diblock copolymers under two-dimensional and three-dimensional confinement. In particular, the effects of commensurability condition, surface-polymer interactions, and confining geometries on the self-assembled morphologies are discussed.

Self-assembly of symmetric diblock copolymers confined in spherical nanopores is studied using simulated annealing Monte Carlo simulations. The dependence of the self-assembled morphologies and chain conformations on the degree of confinement and the strength of the surface interactions is examined systematically. A rich variety of novel structures under the three-dimensional confinement has been revealed. As the strength of the surface preference is increased gradually from neutral to weakly preferential and finally strongly preferential to one of the blocks, the observed sequence of stable structures is from perpendicular lamellae to helices and/or embedded structures and finally to concentric-spherical lamellae. As the degree of confinement decreases, the stable region of the concentric-spherical lamellae becomes larger, while that of the embedded structures becomes smaller. For the structures obtained in spherical nanopores, corresponding counterparts in two-dimensional (2D) confined systems can be identified. On the other hand, the chain conformations of the three-dimensional (3D) confined structures are different from that of their corresponding 2D counterparts. A model is proposed which gives a reasonable description for the layer thicknesses of the concentric-lamellae in both 3D and 2D confined systems. Furthermore, in the limit of large pores, the model predictions for 3D and 2D confined systems are consistent with that observed in one-dimensional confined systems.

Self-assembly of AB diblock copolymers confined in cylindrical nanopores is studied using a simulated annealing technique. The pore diameter and surface preference are systematically varied to examine their effects on the self-assembled morphologies and the chain conformations. For bulk lamella-forming and cylinder-forming diblock copolymers, novel structures such as helices and concentric (perforated) lamellae spontaneously form when the copolymers are confined in cylindrical pores. The observed equilibrium morphologies are compared with that obtained from experiments, theory, and other simulations. A simple model is proposed for symmetric diblock copolymers, which gives a reasonable description of the layer thickness for the concentric lamellae. It is found that chains near the pore surfaces are compressed relative to the bulk chains, which can be attributed to the existence of the surfaces. The dependence of the chain conformation on the degree of confinement and strength of the surface preference are reasonably explained. The energetics is discussed qualitatively and used to account for the appearance of the complex phase behavior observed for certain intermediate conditions.

A multiblock model is developed for the study of the phase behavior of gradient copolymers. The model is able to describe gradient copolymer chains with arbitrary composition profiles. The validity of the multiblock model of gradient copolymers is established by good agreement between RPA (random phase approximation) results for a continuous composition distribution and a multiblock model. The phase behavior of gradient copolymers is examined using self-consistent mean-field theory (SCMFT) for multiblock copolymers. Phase diagrams of gradient copolymer melts with different gradient profiles are constructed by solving the SCMFT equations. It is discovered that the phase behavior depends sensitively on the gradient profiles. In particular, new triple points are observed, and the stability region of phases with curved interfaces shrinks as the gradient profile becomes smooth. For linear gradient copolymers, the lamellar phase is predicted to be the only stable ordered phase.

Self-assembly of cylinder-forming diblock copolymers confined in cylindrical nanopores is studied systematically using a simulated annealing method. The diblock copolymers form hexagonally packed cylinders in the bulk with a period L 0, whereas novel structures spontaneously form when the copolymers are confined inside cylindrical pores. It is discovered that the sequence of structures is controlled by the ratio between the pore diameter (D) and L 0, as well as the selectivity of the pores. For selective small pores (D/L 0 < 2.7), the following structural sequence occurs as the pore size is increased: a string of spheres, a single cylinder, a straight band, a twisted band or stacked disks, a single helix, a set of degenerate structures (includes single helix, stacked toroids and double helices), and double helices. For larger pores (D/L 0 > 2.7), the outer ring of the minority block-domain forms helices or stacked toroids, while the inner structure repeat the sequence of structures observed in smaller pores. For neutral pores, cylinders oriented parallel and nearly perpendicular to the cylindrical pore are observed besides helical or toroidal structures. These morphologies are consistent with available experiments and theoretical studies. Mechanisms of the morphological transitions can be understood based on the degree of commensurability between the pore diameter and bulk period of the copolymer L 0, parametrized by the ratio D/L 0. The effect of the cylindrical pore length on final morphologies is investigated. The chain conformations as a function of morphologies are calculated and analyzed. A mechanism for the formation of helices is proposed based on a packing model, which gives a reasonable description of the radius and pitch of the observed helices.

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The self-assembly of diblock copolymers under soft confinement is studied systematically using a simulated annealing method applied to a lattice model of polymers. The soft confinement is realized by the formation of polymer droplets in a poor solvent environment. Multiple sequences of soft confinement-induced copolymer aggregates with different shapes and self-assembled internal morphologies are predicted as functions of solvent-polymer interaction and the monomer concentration. It is discovered that the self-assembled internal morphology of the aggregates is largely controlled by a competition between the bulk morphology of the copolymer and the solvent-polymer interaction, and the shape of the aggregates can be non-spherical when the internal morphology is anisotropic and the solvent-polymer interaction is weak. These results demonstrate that droplets of diblock copolymers formed in poor solvents can be used as a model system to study the self-assembly of copolymers under soft confinement.