Confined self-assembly of block copolymers within nanopores or emulsion droplets presents a unique platform for controlling nanoscale morphology through geometric constraints. Unlike bulk or thin film systems, confinement introduces curvature and dimensional restrictions that alter the equilibrium behavior of block copolymers, leading to novel structures and phase transitions not observed in unconfined environments. The interplay between polymer-polymer interactions, confinement geometry, and interfacial energy dictates the resulting morphologies, enabling precise engineering of nanostructures for applications ranging from drug delivery to nanopatterning.

When block copolymers are confined within cylindrical nanopores, the curvature of the pore walls and the diameter of the pore significantly influence the self-assembled structures. For instance, polystyrene-block-polybutadiene-block-polystyrene (SBS) triblock copolymers confined in cylindrical nanopores exhibit a transition from concentric lamellae to stacked disks or helical structures as the pore diameter decreases below the natural periodicity of the bulk polymer. The critical pore diameter at which these transitions occur depends on the Flory-Huggins interaction parameter and the degree of polymerization. In pores with diameters less than twice the natural lamellar spacing, the curvature energy penalty forces the system to adopt non-lamellar morphologies to minimize free energy. Similar effects are observed for diblock copolymers, where spherical, cylindrical, or bicontinuous phases emerge depending on the pore size and polymer composition.

Emulsion droplets provide another confinement geometry where the block copolymer assembly is governed by the droplet curvature and interfacial interactions. In spherical confinement, the competition between interfacial tension and chain stretching leads to morphologies such as concentric lamellae, Janus-like structures, or multi-compartmentalized spheres. The interfacial energy between the block copolymer and the surrounding medium plays a crucial role in determining the equilibrium structure. For example, in oil-in-water emulsions, hydrophilic blocks preferentially segregate to the droplet surface, while hydrophobic blocks form the interior domains. The size of the droplet relative to the polymer's natural periodicity dictates whether single or multiple domains form within the droplet. Droplets with diameters comparable to the bulk periodicity tend to form single-domain structures, whereas larger droplets accommodate multi-domain morphologies.

Dimensional restriction in nanopores or droplets also induces symmetry-breaking transitions that are absent in bulk systems. In cylindrical confinement, block copolymers with a natural tendency to form hexagonally packed cylinders in bulk may transition to stacked toroids or single helices due to the mismatch between the preferred lattice symmetry and the confining geometry. The transition between these morphologies is often first-order, with a clear coexistence region where multiple structures are energetically comparable. The confinement-induced frustration can be quantified by comparing the pore diameter to the natural periodicity of the bulk morphology. When the ratio of pore diameter to bulk periodicity is non-integer, the system adapts by introducing defects or adopting alternative symmetries to accommodate the geometric constraint.

Curvature effects are particularly pronounced in emulsion droplets, where the spherical geometry imposes a constant curvature on the assembled structures. The bending energy of the block copolymer domains becomes a dominant factor, leading to curvature-driven transitions between lamellar, cylindrical, and spherical morphologies. For example, block copolymers that form lamellae in bulk may transition to concentric spheres or interconnected networks in droplets due to the high curvature energy cost of maintaining flat interfaces. The critical curvature at which these transitions occur can be estimated using strong segregation theory, which balances the bending energy against the interfacial tension and chain stretching terms.

The confinement also affects the kinetics of self-assembly by restricting the pathways available for phase separation. In nanopores, the diffusion of polymer chains is anisotropic, with faster dynamics along the pore axis and slower dynamics perpendicular to it. This leads to preferential alignment of domains along the pore axis and slower equilibration times compared to bulk systems. In emulsion droplets, the finite volume limits the growth of domains, resulting in smaller feature sizes and faster equilibration due to the reduced diffusion distances. The interplay between confinement and kinetics can be exploited to trap metastable states that are inaccessible in bulk systems.

The choice of solvent or matrix material in emulsion-based confinement further modulates the self-assembly behavior. Selective solvents that preferentially swell one block of the copolymer can shift the effective volume fraction and induce morphological transitions. Similarly, the viscosity of the surrounding medium affects the mobility of the polymer chains and the kinetics of structure formation. In nanopores, surface functionalization can be used to tune the interfacial interactions and direct the assembly toward specific morphologies. For instance, pores with neutral wetting conditions promote the formation of concentric structures, while pores with preferential wetting for one block lead to axially aligned domains.

Applications of confined block copolymer assembly leverage the unique morphologies and size control achievable under geometric constraints. Nanoporous templates filled with block copolymers can be used to fabricate nanowires, nanotubes, or porous membranes with tailored pore sizes and arrangements. Emulsion-templated assemblies find use in drug delivery systems where the compartmentalized structures enable multi-drug loading or controlled release profiles. The ability to precisely control the size and morphology of these nanostructures through confinement parameters makes them highly versatile for functional materials design.

Future directions in this field include exploring dynamic confinement where the geometry or interfacial properties change during assembly, leading to non-equilibrium structures. Combining multiple confinement strategies, such as hierarchical pores or multi-emulsion systems, could further expand the range of accessible morphologies. Advances in in situ characterization techniques will provide deeper insights into the kinetic pathways and transient states during confined assembly. Understanding and harnessing these effects will enable the design of increasingly complex and functional nanostructures for advanced technologies.