5. Electrophoresis and the force on a charged body
5. Electrophoresis and the force on a charged body¶
| Force | Expression | Label |
|---|---|---|
| Electrophoretic | F_ep = ∫_V rho*E dV = q*E |
eq:nanopores_lorentz_force |
| Dielectrophoretic | F_dep = (p·grad)E |
eq:nanopores_dep_force |
| Electric, exact | F_ele = ∮_Gamma (T_M·n) dGamma, T_M = eps*E*E - 0.5*eps*(E·E)*Id |
eq:nanopores_electric_force |
| Drag, 1st order | F_drag = -6*pi*eta*R_h*u |
eq:nanopores_drag_force |
| Hydrodynamic, exact | F_hyd = ∮_Gamma (T_H·n) dGamma, T_H = p*Id - eta*(grad u + grad u^T) |
eq:nanopores_hydrodynamic_force |
q*E is strictly valid only in a uniform field. The stress-tensor forms assume nothing about particle shape
or field uniformity and are what to implement; T_H also captures pressure build-up ahead of a
translocating particle, which Stokes drag does not. Typical in-pore EP/EO forces: 1–100 pN (ClyA at
−50 mV: ~3 mV/nm → ~4 pN on a +10 e protein; a 5 nm sphere in 100 mm/s flow → ~4 pN). Reynolds ~5e-4, always
laminar. EP dominates for DNA; for proteins the balance is even or EO-dominated — which is how ClyA captures
proteins against the field.
Electrophoretic mobility [std]: mu_ep = v_drift/E; q/(6*pi*eta*R_h) in the thick-EDL (Hückel) limit,
eps0*eps_r*zeta/eta in the thin-EDL (Smoluchowski) limit — the exact negative of mu_eo. The thesis computes
forces directly instead. Do not use "effective charge" shortcuts without stating the EOF assumption: for dsDNA
in ClyA, EOF is estimated to halve it from −2 to −1 e/bp.