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A point-defect texture that can compensate the radial boundary condition of a sphere with homeotropic anchoring, giving a dipolar particle–defect configuration in a uniform far field. It carries the opposite hedgehog charge to the radial particle boundary under a consistent convention.

homeotropic anchoring favors the nematic director normal to a surface. For a flat wall at $z=0$, uniform $\mathbf{n}=\hat{\mathbf{z}}$ is a simple compatible profile, shown as perpendicular rods. If a remote wall demands another orientation, elastic bending interpolates between the boundaries.

A common surface energy is $f_s=(W/2)[1-(\mathbf{n}\cdot \mathbf{\nu }{)}^2]$, with wall normal $\mathbf{\nu }$ and $W>0$. Its minima have $\mathbf{n}$ parallel or antiparallel to $\mathbf{\nu }$. The profile is controlled jointly by this preference and Frank elastic energy.

Strong homeotropic anchoring requires radial orientation just outside a sphere, while the remote director is uniform. A plausible dipolar texture contains a compensating hyperbolic hedgehog defect on one side. The particle's radial boundary has charge $+1$ under a consistent convention, balanced by an exterior point defect of charge $-1$ so the far field is uniform. The original cross-section depicts this configuration.

An alternative has an equatorial Saturn-ring defect. Its local disclination winding is half-integer; that local winding is not itself the net three-dimensional compensating charge. Particle size, elastic constants, confinement and anchoring strength select between configurations. The elastic theory explicitly compares dipolar, Saturn-ring and surface-ring states.^[ref]

Choose an asymmetric diblock copolymer in a cylinder-forming regime: minority cylinders lie in a continuous majority matrix. Form and anneal a film to develop microphase separation, control substrate and top-surface preferences or apply a suitable field to orient cylinders normal to the film, then selectively remove the minority phase. The retained majority matrix becomes the porous support. Infiltrate the channels with liquid crystal. The majority phase forms the channel walls; removing that continuous matrix would not generally leave a continuous nanoporous membrane.

With homeotropic anchoring, the director is radial at a cylindrical side wall. A strictly transverse radial texture has a line core, but an escaped radial texture can avoid it:The director turns axially at the center and is normal to the wall at radius $a$. The original axial-section sketch shows this three-dimensional escape. End conditions, channel dimensions and anchoring strength determine whether escape or a defect-containing profile is favored.

Homeotropic anchoring means that the nematic director is locally normal to the particle surface, hence radial on a sphere. Far away, the uniform nematic liquid crystal has a fixed director ${\mathbf{n}}_{\infty }$, with $\mathbf{n}$ and $-\mathbf{n}$ physically equivalent.

One plausible configuration is a quadrupolar Saturn-ring defect: the director is radial near the particle, bends into ${\mathbf{n}}_{\infty }$ far away, and has a disclination loop just outside the particle's equator, in the plane perpendicular to ${\mathbf{n}}_{\infty }$. A meridional section cuts the loop at two points.

The defect compensates the topological distortion introduced by the radial surface boundary condition. A dipolar configuration with a nearby hyperbolic point defect is another possible texture. Which is stable depends on particle size, anchoring strength, elastic constants and external confinement; the boundary conditions alone do not uniquely select one.^[ref]

An equatorial disclination loop surrounding a roughly spherical inclusion with homeotropic anchoring. A meridional section cuts the loop twice. It can match a radial near-surface director to a uniform far field.