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The electrophoretic mobility $\mu$ is the proportionality coefficient between drift velocity and a weak applied electric field. Its sign specifies motion relative to the field. When only travel distances are used, the mobility magnitude is often intended. For long linear DNA in a weak-field reptation regime, $|\mu |$ can scale inversely with chain length, whereas free-solution mobility is approximately length-independent. Neither law supplies a universal gel electrophoresis size calibration for arbitrary gel concentration, field strength or molecular topology.

A porous gel changes the electrophoretic mobility of charged synthetic polymers through confinement, deformation and pore-to-pore dynamics. Free-draining charge and friction can both grow proportionally to chain length, leaving free-solution mobility nearly length independent; matrix obstruction adds size sensitivity. reptation supplies one weak-field long-chain regime, not a universal calibration. Matrix concentration, effective charge, solvent conditions, conformation, topology, adsorption, electro-osmotic drift and heating can all alter migration. Size standards should match the relevant chemistry and operating regime.

In the imposed free-draining polymer friction model, the drag coefficients of the $N$ monomers add, giving ${\zeta }_1=N(6\pi \eta a)$ and $D_1=k_{B}T/(6\pi \eta aN)$. Taking the same three-dimensional centre-of-mass RMS convention as part (ii),The polymer is a one-dimensional connected contour, but that alone does not restrict its centre of mass to one spatial axis. If the intended coordinate is instead literal one-dimensional curvilinear motion along an entanglement tube, its RMS contour displacement is $\sqrt{2D_{1}t}=\sqrt{k_{B}Tt/(3\pi \eta aN)}$. That is a different coordinate and should not be compared with a three-dimensional displacement without stating the conversion/model.

Additive local drag is the free-draining Rouse model approximation. It neglects long-range solvent-mediated hydrodynamic interactions and is not a complete theory of an entangled chain. In a tube/reptation description, curvilinear diffusion can scale as $N^{-1}$ while long-time spatial centre-of-mass diffusion scales as $N^{-2}$ after the chain renews its tube. Conversely a dilute unentangled coil can exhibit Zimm model hydrodynamic coupling. Monomer motion, centre-of-mass motion and contour sliding therefore cannot all be identified with the same displacement law.

In a dilute solution, solvent-mediated hydrodynamic interactions couple distant chain segments. The Zimm model describes the resulting cooperative dynamics: an intermediate-time segment mean-square displacement scales as $t^{2/3}$, while whole-chain translation has $D_Z\sim k_{B}T/({\eta }_{s}R)$. Once internal modes relax, each tagged segment follows the chain's ordinary long-time diffusion, with mean-square displacement asymptotic to $6D_{Z}t$.

In a concentrated solution or polymer melt, hydrodynamic coupling is screened. The unentangled Rouse model has an intermediate $t^{1/2}$ segment regime and whole-chain diffusivity proportional to $N^{-1}$. Long chains with polymer entanglement are additionally confined by neighbouring chains: reptation gives a slower intermediate $t^{1/4}$ regime and an asymptotic chain diffusivity scaling approximately as $N^{-2}$. The time windows depend on segment, entanglement and terminal relaxation times, so these powers do not all describe the same time interval.

After sufficiently long times both dilute and concentrated chains undergo ordinary translational diffusion. The main long-time difference is the coefficient and its chain-length dependence, with entanglement strongly reducing mobility when the chains are long enough; concentration alone does not guarantee an entangled regime.

A polymer melt has an observation-time-dependent response because chains need time to rearrange. Short compared with their terminal relaxation time, entangled chains constrain one another and can store elastic free energy. Long compared with that time, chains escape their original constraints by reptation and the uncrosslinked melt flows. A permanently crosslinked elastomer instead retains a long-time elastic modulus.

A single-time approximation is the Maxwell fluid: a spring of shear modulus $G$ and a viscous dashpot of viscosity $\eta$ in series. Their common stress and additive shear strains give $\dot{\gamma }=\dot{\sigma }/G+\sigma /\eta$, with relaxation time $\tau =\eta /G$. For sinusoidal deformation proportional to $e^{i\omega t}$, the storage modulus and loss modulus areAt low frequency $G^{\prime \prime }\simeq \eta \omega$ dominates; at high frequency within this coarse-grained model $G^{\prime }\simeq G$ dominates. Real melts have a spectrum of relaxation times and additional segmental or glassy regimes, so the single-time model does not describe every frequency. A step-strain experiment measures the decay of stress, whereas oscillatory rheometry separates stored energy from dissipated energy through the phase of the response.

In a dilute solution, separate coils diffuse through solvent with long-ranged hydrodynamic coupling among their segments. The Zimm model gives whole-chain diffusionwhere $R_H$ is a hydrodynamic radius and ${\eta }_s$ the solvent viscosity. The coil continually changes shape while its centre of mass performs long-time Brownian motion; treating it as a rigid sphere is only a scaling analogy.

At high concentration, solvent-mediated hydrodynamic interactions are screened on sufficiently large scales and times. If chains are not entangled, additive local or blob friction gives Rouse model behaviour, $D\propto N^{-1}$ at fixed concentration. For long entangled chains, neighboring chains prevent rapid transverse escape. In the reptation tube picture the chain moves mainly along its contour until it renews the surrounding tube. Contour diffusion scales as $N^{-1}$, the tube-renewal time as $N^3$, and the random-coil size squared as $N$. ThereforeBoth cases ultimately give diffusive centre-of-mass motion, but entangled concentrated chains move much more slowly and retain constraints for longer. The inequality $\varphi \gg {\varphi }^{\ast }$ establishes strong overlap, not by itself entanglement: the latter also requires sufficiently many segments relative to an entanglement length. No numerical entanglement scale or concentrated-solution diffusion coefficient is supplied, so only the conditional regimes and scaling exponents can be fixed here.

In the weak-field reptation regime of agarose gel electrophoresis, the electrophoretic mobility of a long linear DNA chain scales as $\mu \propto M^{-1}$, where $M$ is its number of base pairs. One way to see the scaling is that the constrained-chain diffusion coefficient scales as $D\propto M^{-2}$, whereas its effective electric charge scales as $M$; the drift response is proportional to $DM$. With the same electric field and running time, the migration distance is therefore $x=\mu Et\propto M^{-1}$. The time cancels from the ratio, givingHere “kbp” means a thousand base pairs, not a rate. The second gel band has traveled farther and corresponds to the shorter chain under this migration model. The inverse-length law applies only over an appropriate migration regime.^[ref].

At very low agarose concentration the mesh becomes too open to sieve these chains effectively. In the free-solution limit the chain's electric charge and friction are both approximately proportional to its length, so both chains have nearly the same electrophoretic mobility and their gel bands cease to separate well. At very high agarose concentration the pores become small and transport is strongly hindered; over a fixed experimental time both chains migrate very little, so their absolute spatial separation again becomes small. Increasing gel concentration can improve resolution over an intermediate range, but not indefinitely.

The numerical result is conditional on the inverse-length reptation law. Actual gel electrophoresis size calibration often uses an empirical approximately linear relation between distance and $\log M$ over a specified range. One known band alone cannot fix that relation's slope. The PDF supplies neither gel concentration nor a calibration curve, so $8.9\mathrm{kbp}$ is the standard model estimate rather than a uniquely calibrated experimental size.

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, $\mu =q_{\mathrm{eff}}/\zeta$. 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.^[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 $\mu (c_g)\simeq \mu (0)e^{-K_r{c}_g}$, 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. 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.

A polymer entanglement is a constraint on a chain's motion caused by the inability of neighbouring chain contours to pass through one another. It is not a permanent chemical cross-link. In a dense polymer melt, these constraints can create long relaxation times and a transient rubbery plateau. Chains eventually release constraints through mechanisms such as reptation and motion of their neighbours. Entanglement therefore strongly affects dynamics even when the equilibrium size of a long chain remains approximately that of an ideal chain because excluded-volume interactions are screened.

Reptation is the snake-like motion of a long polymer through the constraints imposed by neighboring chains or a fixed porous matrix. A tube model restricts large transverse motion while allowing curvilinear motion along the chain contour. For a flexible chain with additive segment friction, contour diffusion scales as $N^{-1}$ and the time to renew a contour of length proportional to $N$ scales as $N^3$. With coil size squared proportional to $N$, this gives a centre-of-mass diffusion coefficient proportional to $N^{-2}$. Applying this tube picture to weak-field DNA gel electrophoresis gives an inverse-length mobility over its appropriate regime; field-induced orientation and trapping can invalidate that simple law.