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TODO: user-deferred biology subquestion. This part is outside the current nonbiology scope.

The transport mean free path is $\ell =v_F{\tau }_{\mathrm{tr}}$, while $\mu =e{\tau }_{\mathrm{tr}}/m^{\ast }$. Eliminating the scattering time gives $\ell =\mu \hslash k_F/e$. The usual ballistic focusing mobility criterion $\ell \gtrsim L$ therefore givesA more geometrically faithful first-peak survival criterion uses the semicircular path length $s=\pi L/2$, giving $\mu \gtrsim 17.4{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. There is no sharp universal visibility threshold: clearly resolved peaks favour $\ell \gg s$, small aperture broadening and specular reflection. The mobility inferred in part (ii) corresponds to $\ell \simeq 314\mu \mathrm{m}$, comfortably above both estimates. Higher-order peaks can still be reduced by boundary roughness even when bulk mobility is high.

Substituting $r_c=\hslash k_F/(e|B|)$ into $L=2j{r}_c$ givesUsing the density-derived Fermi wave vector gives the first three peak magnitudesThe equally spaced sequence follows from the number of reflected half-orbits, not from successive quantum Hall fillings. The field polarity selecting a given injector–collector orientation follows from the Lorentz force direction.

This is transverse magnetic focusing. The depleted gates form reflecting boundaries and two constrictions. A narrow quantum point contact collimates carriers into cyclotron trajectories with cyclotron radius $r_c=m^{\ast }{v}_F/(e|B|)=\hslash k_F/(e|B|)$. When an integer number of semicircular skipping arcs links the two openings, the collector couples particularly strongly to the injector population. A floating voltage contact then changes its electrochemical potential to satisfy zero net current, producing a peak in the measured nonlocal voltage. Sweeping the field changes the arc diameter, moving the focused population into or away from the collector.

The connection condition is $L=2j{r}_c$ for integer $j\ge 1$, with the higher orders requiring specular reflections. The effect needs ballistic electron transport, sufficiently narrow openings and a smooth boundary. Electron phase coherence can add interference structure, but the basic geometrical focusing condition is semiclassical.^[ref]

There is a polarity qualification in the printed geometry. The electron charge is negative: for a field out of the page, an electron launched upward through the left opening bends left, not toward the right opening. A direct left-opening-to-right-opening focusing path in the upper region therefore requires the field into the page; the printed outward field supports the reverse connection. Reversing conventional current reverses the occupation perturbation and voltage sign, but does not reverse the curvature of an individual electron. Thus the standard injector-to-collector explanation describes the focusing mechanism, while the asserted signed voltage increase requires a consistent field/lead convention. The calculated results below are peak-field magnitudes; taking outward field positive gives negative fields for direct left-to-right electron injection.

For the silver nanowire network, synthesize or source long, narrow wires, remove excess ligand without causing aggregation, formulate a stable coating ink and coat the flexible web by a scalable method such as slot-die coating or gravure printing. Dry below the substrate's damage temperature, then press or locally weld junctions and embed or cap the network if required. Define 100 µm stripes by masked coating, a printing-compatible patterned wetting layer, or post-coating laser ablation; provide robust end contacts and remove stray bridges between adjacent stripes. Roll processing offers throughput, but coffee-ring effect, wire alignment, random weak links and ablation damage require inline optical/electrical inspection. Verify that the local junction treatment improves resistance without simply darkening the whole film.

For the silver nanoparticle-derived grid, use a repeatable nanoimprint lithography to form a fine mesh inside the outer 25 mm by 100 µm stripe pattern. Fill the channels with a silver nanoparticle ink, remove excess from the transparent openings, and consolidate the metal by substrate-compatible chemical, photonic or low-temperature thermal sintering. Transfer or embed the grid into the polymer surface and add electrical pads. The quantitative design in part (b) sets target pitch, line width and thickness; the large-scale process must maintain those dimensions and a continuous metal cross section. A master enables repeated replication and deterministic open area, but tooling cost, incomplete filling, residual metal in windows, shrinkage cracks, imperfect sintering and adhesion are important limitations. Test the finished patterned stripes, rather than relying on a large unpatterned test coupon alone.

Three credible architectures are a welded silver nanowire network; a thin silver nanowire network with a conducting polymer overcoat; and a nanocrystalline silver grid formed from nanoparticle ink. Silver provides low intrinsic resistivity. A sparse long-wire network can be both flexible and highly transmitting if its junctions are well connected. A thin, optically weak conducting polymer overcoat can bridge junctions and smooth the surface, provided its added absorption and moisture sensitivity are controlled. A fine grid offers a more directly designed optical/electrical trade-off and does not rely on operating near random percolation.

For a quantitative grid feasibility screen, take pitch $p=5\mu \mathrm{m}$, line width $w=100\mathrm{nm}$ and metal thickness $t=200\mathrm{nm}$. An opaque-line square-grid estimate gives $T_{\mathrm{grid}}\simeq (1-w/p{)}^2=0.9604$. With a transmitting polymer substrate of 0.95, the combined estimate is 0.912. The sheet resistance is approximately $R_{\square }={\rho }_{\mathrm{eff}}p/(wt)$. Even an effective silver resistivity four times $1.6\times 10^{-8}\Omega \mathrm{m}$ gives $R_{\square }\simeq 16\Omega$, with a coating thinner than 400 nm. There are about 20 mesh periods across the stripe, making the finite-width design more credible than a pitch comparable with its whole width.

These are feasible material/process choices, not guaranteed specifications for every film carrying those names. For the two nanowire architectures, choose wire dimensions, loading and junction treatment to pass the same finished-stripe measurements; the hybrid's optical budget must include its overcoat. Measure transmission at 550 nm with its reference stated, sheet resistance, profile thickness, bend-cycle stability and pattern continuity. Bare nanotube or graphene networks should not be assumed to meet this demanding resistance–transmission combination without additional metallic conduction or other demonstrated improvement.

A transparent metallic network must span the patterned electrode with many redundant current paths while obscuring little area. For random rods, number density times squared wire length is a useful percolation measure; greater aspect ratio generally reaches a connected network at lower coverage. Conductivity also depends on intrinsic metal resistivity, wire cross section, orientation, length distribution, wire–wire contact resistance and the number of junctions along each spanning path. Insulating capping layers can make nanowire junction resistance dominate even when the individual wires are excellent conductors.

Near the connectivity threshold, small density changes or wire breaks produce large resistance changes. For a narrow stripe, finite-size effects and orientation matter: a macroscopic sheet-average percolation result does not ensure a reliable 100 µm-wide path. Optical performance depends on coverage, metal extinction, wire diameter, aggregation, optical haze, substrate transmission and any binder. Increasing wire density lowers resistance but usually lowers transmission. Welding, pressing or suitable embedding can improve contacts with less additional optical coverage than simply adding metal.^[ref]

The stripe has $L/W=250$ squares, so its end-to-end resistance is $R=250R_{\square }$; the stated sheet-resistance limit requires a stripe resistance below 5 kΩ, not 20 Ω. The sketch below separates sparse connectivity, redundant paths and junction quality.

TODO: user-deferred biology subquestion. This part is outside the current nonbiology scope.

TODO: user-deferred biology subquestion. This part is outside the current nonbiology scope.

TODO: user-deferred biology subquestion. This part is outside the current nonbiology scope.

With planar anchoring on the cylindrical side wall, uniform axial alignment $\mathbf{n}=\hat{\mathbf{z}}$ is a simple defect-free profile. The axial direction is tangent at every wall point, so it satisfies the lateral boundary everywhere without elastic distortion.

This assumes degenerate planar anchoring, or an azimuthal preference compatible with axial alignment. End constraints, selected circumferential anchoring or imposed twist can favor another state. planar anchoring alone does not require a circumferential vortex. The original axial-section sketch shows the uniform axial option.

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.

A sphere with degenerate planar anchoring can have a bipolar texture with two surface boojums at opposite poles along the far-field axis. Surface director lines remain tangent, curve between the polar cores and relax toward uniform alignment far away. The original cross-section shows both cores.

A globally regular tangent orientation field cannot cover a sphere; the associated surface winding indices sum to its Euler characteristic, two. These are surface defects, not two arbitrarily placed bulk hedgehogs. Finite anchoring allows deviations near the cores, while an azimuth-selective coating could favor a different texture.

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]

planar anchoring favors the nematic director tangent to a surface. A flat wall at $z=0$ permits uniform $\mathbf{n}=\hat{\mathbf{x}}$, as in the sketch. A rubbed surface can select one in-plane axis; degenerate planar anchoring selects only the tangent plane.

For the degenerate case, $f_s=(W/2)(\mathbf{n}\cdot \mathbf{\nu }{)}^2$ has minima at $\mathbf{n}\cdot \mathbf{\nu }=0$. A remote incompatible boundary produces elastic distortion. Tangency alone does not specify an azimuthal direction within the surface.

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.

The blue phases result when strong local chirality favors double twist that cannot fill three-dimensional space everywhere without frustration. Double-twist regions are connected through a network of nematic disclinations.

Blue phase I has body-centered cubic order and blue phase II simple-cubic order in their director/defect structures. The periodicity can cause selective optical reflection. Blue phase III is disordered or amorphous, not another ideal cubic lattice. These are structured fluid phases rather than ordinary crystals with molecules fixed to every lattice site.

A cholesteric liquid crystal is locally nematic but its nematic director rotates helically, for exampleThe sign of $q_0$ specifies handedness and $p$ is the full-turn pitch. The sketch shows rods rotating through transverse planes. This is helical orientational order, not a smectic density-layer lattice. Although an apolar director repeats physically after a half-turn, the conventional pitch denotes a full $2\pi$ turn. A pitch comparable to visible wavelengths can give selective reflection.

A liquid crystal flows while retaining partial molecular order. A nematic liquid crystal has long-range orientational order but no crystal-like positional lattice. Its local mean axis is the nematic director $\mathbf{n}$, with $\mathbf{n}\equiv -\mathbf{n}$ for an ordinary apolar nematic. Order leads to anisotropic optical and dielectric response and an elastic cost for director distortions.

Nematic order selects an axis, not a common molecular head direction or density-layer stack. The original sketch uses short rods as a coarse-grained orientation map.