= Solution
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>. \b[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 \b[escaped radial texture] can avoid it:
$$
\mathbf n(r)=\sin\beta(r)\hat{\mathbf r}+\cos\beta(r)\hat{\mathbf z},
\quad \beta(0)=0,\quad\beta(a)=\pi/2.
$$
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.
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