= Solution
In <gel sieving of synthetic polymers>, <gel electrophoresis> combines drift under an <electric field> with size- and conformation-dependent obstruction by a porous matrix. In weak-field free solution, <force> balance gives $v=\mu E$ and, in a simple effective-charge/effective-friction model, $\mu=q_{\rm eff}/\zeta$. The <mobility> is motion relative to the medium, so fluid motion from <electro-osmosis> must be accounted for separately. This is the operational <mobility> convention, not a claim that the bare structural charge is always the effective driving charge.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry>[entry E02024, electrophoretic mobility: migration speed divided by electric field strength]{ref}
For a chain with <free-draining polymer friction> with approximately constant effective <electric charge> per segment, both charge and friction scale with $N$. The free-solution <mobility> can therefore become nearly independent of <length>, making size separation poor. A gel adds steric constraints. Coils smaller than the mesh can pass relatively easily; larger coils require deformation and pore-to-pore rearrangement. A schematic weak-sieving law is $\mu(c_g)\simeq\mu(0)e^{-K_rc_g}$, where matrix <concentration> $c_g$ and retardation coefficient $K_r$ are defined by a calibration; it is not a universal size law.
When a long flexible charged <polymer> is confined to a tube of available paths, <reptation> supplies another regime. An additive-friction model gives contour <diffusion> $D_c\propto N^{-1}$ and renewal time $\tau\sim N^2/D_c\propto N^3$. For ideal large-scale chain statistics, $R^2\propto N$, so centre-of-mass <diffusion> scales as $N^{-2}$. With total effective driving charge proportional to $N$, weak-field <mobility> can then scale as $N^{-1}$. This assumes screened/ideal large-scale conformations and <linear response>; strong-field alignment, trapping and nonideal matrices can alter the scaling.
Operationally, load narrow zones of independently characterized synthetic standards and sample into a controlled gel, apply a known field and detect migration positions after a measured time. Compare with a free-solution measurement, vary matrix <concentration> and field, and control <temperature>, <ionic strength>, <pH> and matrix charge. Larger gel pores or lower matrix <concentration> usually reduce sieving within a given calibrated material system. Excessive field causes <Joule heating> and can change conformation and <mobility>. <adsorption> or electro-osmotic drift can mimic a size effect; standards should match charge chemistry and <topology> as well as nominal mass.
Band centres move as $\mu Et$ while ordinary <diffusion> gives $\sigma_x^2(t)=\sigma_x^2(0)+2Dt$. Resolution depends on separation of centres relative to band widths, not travel distance alone. A neutral tracer helps identify bulk flow, and multiple <concentrations> test <adsorption> or overloading. \b[A <mobility> or band position does not uniquely determine size without a regime-specific calibration.] TODO: biomolecular examples, sample preparation and biological applications are user deferred; the account here concerns synthetic charged <polymers> only.
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