Microfluidics

Microfluidics controls and studies small fluid volumes in confined channels. In cellular assays it can impose reproducible flow and chemical fields, but channel geometry, wall interactions and flow-induced stresses can themselves alter behaviour.

A nanopore is a channel or aperture with a nanometre-scale constriction. A solid-state realization can connect conducting liquid reservoirs, permitting ionic current to report particle passage through resistive-pulse sensing. A geometrical pore diameter alone does not determine its conductance: axial shape, reservoir access, surface charge, solution conductivity and interfacial effects can all matter. This definition concerns a physical aperture, without a biological application.

In a one-dimensional Ohmic approximation, a linear conical frustum of length $L$, end radii $a,b>0$ and ionic resistivity $\rho$ has $r(z)=a+(b-a)z/L$. Conservation of electric current gives $d\mathcal{R}=\rho dz/[\pi r(z{)}^2]$. Integration yields the displayed resistance. Two identical opposed frusta, each of length $l$, have resistance $2\rho l/(\pi ab)$, whether their central junction is the wider or narrower end. Their local field profiles nevertheless differ. The approximation neglects surface conduction and resistance outside the defined pore, and is not an exact three-dimensional field solution for arbitrary wall slope.

For a point-like object with constant electrophoretic mobility, deterministic traversal is the integral of inverse local drift speed. In a conical frustum of length $L$ at pore voltage $V$, the one-dimensional field is $E(z)=Vab/[Lr(z{)}^2]$. Integrating gives $t=L^2(b/a+1+a/b)/(3\mu V)$ for positive $\mu V$. For two symmetric frusta of total length $2l$ and the same total pore voltage, replace $L$ by $2l$. The inequality $b/a+a/b\ge 2$ shows that this nonuniform field cannot produce a shorter deterministic point-particle transit than a uniform field at fixed total length and voltage. Reservoir forces, diffusion, adsorption and finite-object conformation are additional physics.

Electrical detection of particles passing through a conducting-liquid aperture. A particle changes the effective pore conductance transiently; pulse amplitude and duration depend on geometry, transport and interfacial charge, so calibration is essential.

A small insulating object of excluded volume $\Omega$ perturbs an Ohmic pore most strongly where the pore cross-sectional area $A$ is small. In a local weak-blockage model, replacing a small conducting volume gives the displayed resistance scaling, with a shape-dependent coefficient. At fixed voltage, $\Delta I\simeq -I_0\Delta \mathcal{R}/{\mathcal{R}}_0$. A symmetric pore with narrow ends and a wide middle can therefore resolve two end features in a passage pulse. This is a conditional volume-exclusion model: charged-particle ion transport, surface interactions and bandwidth can change the observed sign, amplitude and shape.