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A contact line is pinned when its position remains fixed while the surrounding drop shape or contact angle changes. Surface heterogeneity or deposited material can sustain such a state over a range of angles. Evaporation from a pinned drop then requires compensating liquid motion and can generate outward solute transport associated with the coffee-ring effect. A model with a freely receding line and fixed contact angle has a different boundary constraint and need not predict the same flow direction.

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.

For a symmetric two-dimensional thin drop with uniform evaporative loss $ϵ$, a freely receding contact line and fixed leading outer contact angle $\theta$, quasistatic capillary balance makes the height quadratic. Its volume per transverse length is $2\theta s^2/3$. Since $h_t+(h\overline{u}{)}_x=-ϵ$ and the contact-line flux vanishes, volume loss is $-2ϵs$. On slow time $T=ϵt$, this gives the stated constant recession speed.

Substitute that speed into $h_t$ and integrate flux from the centre, obtaining $h\overline{u}=-ϵx(s^2-x^2)/(4s^2)$. Division by height gives the displayed inward velocity. With a pinned width instead, the same uniform-loss model gives ${\theta }_t=-3ϵ/s$ and outward velocity $\overline{u}=ϵx/(s\theta )$. Thus the contact-line assumption alone reverses solute transport. A pinned perimeter can produce a coffee-ring effect; the receding fixed-angle outer model predicts central transport instead. A strictly no-slip sharp moving contact line is singular, so the finite outer angle is a leading quasistatic approximation requiring microscopic regularization at finite speed.