Electrophoresis and Capillary Electrophoresis

A fused-silica tube thinner than a strand of spaghetti sits between two platinum electrodes. It is filled with buffer and held at 30,000 volts. This is the entire separation engine of capillary electrophoresis. Inject a nanoliter of sample and apply the voltage. Within minutes, ions differing by just one charge state glide apart. They form sharp, symmetric peaks. No pump, no packed bed, no van Deemter C-term broadening from stagnant mobile phase is involved. Yet efficiencies routinely exceed 200,000 theoretical plates. This far surpasses what high-performance liquid chromatography achieves on the same analytes. This matters because electrophoresis separates on a completely different physical axis than Principles of Chromatographic Separation does. Chromatography exploits differential partitioning between two phases. Electrophoresis exploits differential migration of charged species through a continuous electric field. No phase equilibrium is required at all. That single distinction explains why capillary electrophoresis (CE) can resolve DNA fragments differing by one base. It can separate protein charge variants that co-elute in reversed-phase HPLC. It does this with peak efficiencies that make the Golay equation from Rate Theory and Band Broadening look almost pessimistic. In ideal CE, the only broadening mechanism left standing is longitudinal diffusion. The big picture is a competition between two independent transport processes. Both run in the same tube at the same time. These are electrophoretic migration of the analyte ion relative to the buffer. The other is electroosmotic flow (EOF) of the entire bulk liquid relative to the capillary wall. Electrophoretic mobility depends on charge, size, and shape of the analyte. Electroosmotic mobility depends on the surface chemistry of the silica wall. In the classic case, it is identical for every species in the tube, neutral or charged. EOF in bare fused silica is unusually fast and directed toward the cathode. It can carry even negatively charged anions past the detector. So cations, neutrals, and anions all emerge in a single run. They are ordered by their net mobility. No chromatographic mechanism can match this with one column and one mobile phase. This article builds the theory from first principles. It covers the Debye–Hückel–Henry treatment of electrophoretic mobility. It covers the electrical double layer and Smoluchowski's equation for EOF. It also covers the combined migration model. This model predicts observed velocity, resolution, and plate count in CE. Three worked examples carry real numbers through to final answers. These include a small inorganic ion, a peptide separation with resolution optimization. They also include an isoelectric-focusing-adjacent pI calculation. Each example concludes with correct units and significant figures.

Key takeaways

Electrophoretic mobility $\mu_{ep}$ is an intrinsic property: charge-to-friction ratio, corrected for ionic-atmosphere relaxation and retardation (Debye–Hückel–Henry theory).
Electroosmotic flow arises from the electrical double layer at the charged silica–buffer interface; under thin-double-layer conditions it obeys the Smoluchowski equation $\mu_{eo} = \varepsilon\zeta/\eta$ with a flat (plug) velocity profile.
Observed velocity is the vector sum $v_{obs} = (\mu_{ep} + \mu_{eo})E$ ; species migrate in an order set by net mobility, letting cations, neutrals, and anions all be seen in one run.
Because the EOF profile is flat, not parabolic, CE largely escapes the Van Deemter C-term that limits chromatographic columns, leaving longitudinal diffusion as the dominant broadening term.
Resolution in CE depends on $\Delta\mu$ , average mobility, applied voltage, and diffusion coefficient — not on a distribution coefficient or phase ratio as in chromatography.
Buffer pH, ionic strength, and additives (surfactants, chiral selectors, coated capillaries) tune $\mu_{ep}$ , $\mu_{eo}$ , or both, giving enormous method-development flexibility.

Key idea: Every analyte in a capillary moves at its own electrophoretic velocity superimposed on one common bulk electroosmotic velocity, and it is this vector sum — not a partition equilibrium — that generates the separation.

Where this lives today

Capillary electrophoresis is the workhorse confirmatory technology behind essentially every DNA sequencer used in clinical and forensic laboratories: capillary array electrophoresis, run in parallel across dozens to hundreds of thin capillaries filled with a sieving polymer matrix, separates fluorescently labeled DNA fragments differing by a single nucleotide, underlying both Sanger sequencing instruments and short-tandem-repeat (STR) profiling used in forensic identity testing and paternity determination. The sieving matrix converts the separation from pure free-solution electrophoresis (which cannot resolve DNA fragments above a few hundred bases, since mobility becomes size-independent for long, uniformly charged rods) into a size-dependent process resembling gel electrophoresis, while retaining the capillary format's speed and automation. In the biopharmaceutical industry, capillary electrophoresis with sodium dodecyl sulfate (CE-SDS) is a mandated regulatory method for confirming the purity and molecular-weight profile of therapeutic monoclonal antibodies, replacing slab-gel SDS-PAGE with a faster, quantitatively reproducible, automatable instrumental analog. Capillary isoelectric focusing (cIEF) resolves antibody charge variants — differences arising from deamidation, C-terminal lysine clipping, or glycosylation — that are essential quality attributes tracked at every stage of biologic drug manufacturing, often run in tandem with the same charge-variant analysis performed by ion-exchange chromatography under Principles of Chromatographic Separation as an orthogonal confirmation. Micellar electrokinetic chromatography (MEKC), a hybrid technique that adds charged surfactant micelles to the running buffer, extends CE to neutral analytes by giving them a differential partitioning interaction with the micellar pseudo-stationary phase — effectively borrowing a chromatographic retention mechanism and running it inside an electrophoretic engine, and it inherits band-broadening considerations directly analogous to those from Rate Theory and Band Broadening. Chiral CE, using cyclodextrin or crown-ether additives as chiral selectors dissolved directly in the buffer, resolves enantiomers of pharmaceutical intermediates and is used industrially wherever a required enantiomeric excess must be verified without derivatization. In clinical chemistry, capillary electrophoresis quantifies hemoglobin variants (including HbA1c for diabetes monitoring and HbS for sickle-cell screening) by resolving them on tiny charge differences, and in metabolomics and glycomics, CE coupled to mass spectrometry (CE-MS) separates highly polar, often zwitterionic metabolites and released glycans that are poorly retained on reversed-phase HPLC columns.

Ask the tutor: How would the resolution equation change if I ran the same separation with a partially suppressed EOF (via a neutral polymer coating), and is there an optimal EOF magnitude that maximizes resolution per unit analysis time?

The physical picture

An ion of charge $q$ sitting in a uniform electric field $E$ feels an electrical force $F_{el} = qE$ pushing it toward the electrode of opposite sign. As it accelerates, viscous drag builds up, and for a spherical particle of radius $r$ moving through a fluid of viscosity $\eta$ , Stokes' law gives a retarding force $F_{drag} = 6\pi\eta r v$ . These forces balance almost instantly — the terminal velocity is reached in nanoseconds — giving a constant migration velocity proportional to field strength. The proportionality constant is the electrophoretic mobility, $\mu_{ep} = v/E$ , with SI units of m² V⁻¹ s⁻¹ (commonly reported in cm² V⁻¹ s⁻¹ or 10⁻⁹ m² V⁻¹ s⁻¹). But a bare ion in vacuum is not the real picture. In solution, the ion carries a cloud of counter-ions — its ionic atmosphere — which is distorted by the applied field. Two corrections apply. The relaxation effect: as the ion moves, the ionic atmosphere lags behind and must re-form, creating an asymmetric charge cloud that pulls backward on the ion. The electrophoretic retardation effect: counter-ions moving in the opposite direction drag solvent with them, so the ion effectively swims against a countercurrent. The full Debye–Hückel–Henry treatment folds both into a mobility that depends on ionic strength through the Debye screening length $\kappa^{-1}$ . The second engine in the capillary is electroosmotic flow. Fused-silica walls, at pH values above roughly 3, bear ionized silanol groups (Si–O⁻), giving the wall a fixed negative surface charge. This attracts a compensating layer of cations from the buffer, forming an electrical double layer: a rigid Stern layer of tightly bound counter-ions immediately at the wall, and a diffuse layer extending into the bulk solution over a distance of order $\kappa^{-1}$ (a few nanometers at typical CE ionic strengths). When the axial field is applied, the mobile diffuse-layer cations are dragged toward the cathode, and because they are solvated, they drag the bulk solvent with them. Because the driving force is applied at the wall over the entire tube surface rather than at a single frit, and because the diffuse layer is only nanometers thick compared to a 25–75 µm bore, the resulting velocity profile is flat across almost the entire cross-section — a plug profile, not the parabolic profile of pressure-driven (Poiseuille) flow.

Figure

Electrical double layer at the silica–buffer interface and the resulting flat EOF profile

Once EOF is established, every species in the capillary — cations, neutrals, anions — is carried along at the same bulk velocity $v_{eo}$ , on top of which each charged analyte adds its own electrophoretic velocity $v_{ep}$ (positive toward the cathode, negative toward the anode, by convention). The observed velocity of any analyte, measured as it passes a fixed detection window, is

v_{obs} = v_{ep} + v_{eo} = (\mu_{ep} + \mu_{eo})E

This single superposition is the entire separation mechanism of CE, and it is why elution order in a bare-silica capillary with cathodic detection is typically: small fast cations first, then slower cations, then all neutrals together (unresolved, moving exactly at $v_{eo}$ ), then anions — with the fastest, least mobile (most weakly retarded) anions arriving last, or not at all if $|\mu_{ep}| > \mu_{eo}$ pushes them backward out through the injection end.

Figure

Migration order in bare fused silica: net mobility determines arrival time at the cathodic detector

The core ideas

Electrophoretic mobility of a small ion (Debye–Hückel–Henry). For a spherical ion of radius $r$ and charge $ze$ in a medium of viscosity $\eta$ and permittivity $\varepsilon$ , with Debye length $\kappa^{-1}$ :

\mu_{ep} = \frac{ze}{6\pi\eta r}\, f(\kappa r)

where:

$z$ = charge number of the ion (signed),
$e$ = elementary charge, $1.602\times10^{-19}$ C,
$\eta$ = solvent viscosity (Pa·s),
$r$ = hydrodynamic (Stokes) radius of the solvated ion (m),
$\kappa^{-1}$ = Debye length, decreasing with ionic strength,
$f(\kappa r)$ = Henry's function, ranging from 1 (Hückel limit, $\kappa r \ll 1$ , small ions/low ionic strength) to 1.5 (Smoluchowski limit, $\kappa r \gg 1$ , large particles/high ionic strength).

Reading: mobility rewards small, highly charged, compact ions and penalizes large, weakly charged, or heavily solvated ones — exactly the intuition that a bare proton (small, $z=+1$ , minimally solvated relative to its charge density) has an enormous mobility compared with a bulky, singly charged organic anion.

Electroosmotic mobility (Smoluchowski equation). When the double layer is thin compared with the capillary radius ( $\kappa a \gg 1$ , always true for CE bores of tens of micrometers):

\mu_{eo} = \frac{\varepsilon \zeta}{\eta}

where:

$\varepsilon$ = permittivity of the buffer (F/m),
$\zeta$ = zeta potential at the shear plane of the double layer (V), set by wall charge density and ionic strength,
$\eta$ = buffer viscosity (Pa·s).

Reading: EOF is a bulk-fluid property controlled entirely by the wall's surface chemistry and the buffer's ionic composition, completely independent of what the analyte is — which is exactly why it acts as a common "conveyor belt" under every species simultaneously.

Observed velocity and migration time.

v_{obs} = (\mu_{ep} + \mu_{eo})\,E, \qquad E = \frac{V}{L_{tot}}

t_m = \frac{L_{det}}{v_{obs}} = \frac{L_{det}\,L_{tot}}{(\mu_{ep}+\mu_{eo})\,V}

where:

$L_{det}$ = length from injection to the detection window (m),
$L_{tot}$ = total capillary length (m),
$V$ = applied voltage (V).

Reading: migration time is inversely proportional to net mobility and to applied voltage, so raising the voltage speeds every peak up proportionally — a direct, simple lever unavailable in chromatography, where flow rate and plate height trade off in a non-monotonic van Deemter curve.

Plate number and resolution in CE. Neglecting injection-plug and thermal broadening, longitudinal diffusion alone sets the peak variance, $\sigma^2 = 2Dt_m$ , giving

N = \frac{\mu_{net}\,V}{2D}\cdot\frac{L_{det}}{L_{tot}}

R_s = \frac{1}{4}\sqrt{N}\,\frac{\Delta\mu}{\bar\mu}

where:

$D$ = analyte diffusion coefficient (m²/s),
$\mu_{net} = \mu_{ep}+\mu_{eo}$ ,
$\Delta\mu$ = difference in net mobility between two analytes,
$\bar\mu$ = their mean net mobility.

Reading: plate count grows linearly with voltage and is intrinsically higher for small, fast-diffusing-slowly... actually the opposite: larger, more slowly diffusing analytes (small $D$ ) give sharper peaks and higher $N$ , which is why CE is spectacularly efficient for large biomolecules like proteins and DNA, whose diffusion coefficients are tiny.

Derivation ADebye–Hückel–Henry mobility from a force balance with double-layer corrections

Goal. Derive the electrophoretic mobility of a spherical ion, starting from the simple Stokes force balance and adding the two ionic-atmosphere corrections that convert it into the full Henry equation.

Step 1 — Bare force balance. An ion of charge $q = ze$ in field $E$ feels electrical force $F_{el} = zeE$ . Viscous drag on a sphere of Stokes radius $r$ moving at velocity $v$ is $F_{drag} = 6\pi\eta r v$ (Stokes' law, valid at low Reynolds number, always true for ions). Terminal velocity requires $F_{el} = F_{drag}$ :

zeE = 6\pi\eta r v

\mu_{ep}^{(0)} = \frac{v}{E} = \frac{ze}{6\pi\eta r}

This is the Hückel mobility — correct in the limit of an infinitely thin or absent ionic atmosphere, i.e. very low ionic strength.

Step 2 — Electrophoretic retardation. In real solution the ion is surrounded by a counter-ion atmosphere of opposite sign. Under the applied field, these counter-ions migrate the opposite way, and because they are solvated, they entrain solvent with them, creating a local backflow that the central ion must swim against. Solving the linearized Poisson–Boltzmann/Navier–Stokes problem around a sphere embedded in its ionic atmosphere (Henry, 1931) shows that this backflow does not simply subtract a constant; it rescales the mobility by a factor $f_1(\kappa r)$ that depends on how many Debye lengths fit inside the ion's radius.

Step 3 — Relaxation effect. As the ion moves, its surrounding atmosphere — built for the equilibrium (stationary) situation — must continuously re-form ahead of the ion and decay behind it. This finite relaxation time distorts the atmosphere into an asymmetric, elongated shape with excess opposite charge trailing behind the ion, generating a small retarding electric field on the ion itself (the "relaxation field"). For univalent ions at CE ionic strengths (tens of mM), this correction is typically only a few percent and is often folded into the same Henry function to leading order, or treated as a separate multiplicative activity-coefficient-like term in more rigorous treatments (Onsager–Fuoss extensions).

Step 4 — Combine into Henry's equation. Collecting both double-layer effects into a single correction factor $f(\kappa r)$ gives the working equation

\mu_{ep} = \frac{ze}{6\pi\eta r} f(\kappa r)

with limiting forms $f\to 1$ as $\kappa r \to 0$ (dilute solution, small ion — the Hückel limit used above) and $f\to 3/2$ as $\kappa r\to\infty$ (large particle or concentrated solution — the Smoluchowski limit, which is the same mathematical limit that gives the electroosmotic-flow equation its factor of unity when written as $\varepsilon\zeta/\eta$ rather than $2\varepsilon\zeta/3\eta$ ). Real small buffer ions in typical CE run buffers (10–100 mM, $\kappa^{-1} \approx 1$ –3 nm, $r \approx 0.2$ –0.4 nm) sit in an intermediate regime, so tabulated limiting mobilities (from conductivity measurements) are used directly in practice rather than recomputing $f(\kappa r)$ from scratch for every species.

Go deeper · Derivation B · 2 minSmoluchowski electroosmotic velocity from the double-layer force balance

Derivation BSmoluchowski electroosmotic velocity from the double-layer force balance

Goal. Derive $\mu_{eo} = \varepsilon\zeta/\eta$ by balancing electrical body force on the diffuse-layer charge against viscous drag, in the thin-double-layer limit relevant to capillaries tens of micrometers wide.

Step 1 — Set up the local charge density. Near the negatively charged wall, the diffuse layer carries a net positive volumetric charge density $\rho(x)$ that decays with distance $x$ from the wall over the Debye length $\kappa^{-1}$ . The Poisson equation in one dimension (flat-wall approximation, valid because $\kappa^{-1} \ll$ capillary radius $a$ ) relates this charge density to the local electrostatic potential $\psi(x)$ :

\frac{d^2\psi}{dx^2} = -\frac{\rho(x)}{\varepsilon}

Step 2 — Write the steady-state force balance. Under the applied tangential field $E$ (along the capillary axis, $z$ -direction), each volume element of diffuse-layer fluid experiences an electrical body force per unit volume $\rho(x)E$ , balanced in steady, low-Reynolds-number flow by the viscous shear stress gradient:

\eta \frac{d^2 v_z(x)}{dx^2} = -\rho(x) E

Step 3 — Substitute the Poisson equation. Replacing $\rho(x)$ using Step 1,

\eta \frac{d^2 v_z}{dx^2} = \varepsilon E \frac{d^2\psi}{dx^2}

Both sides are now exact second derivatives of $x$ , so the equation integrates directly:

\eta \frac{dv_z}{dx} = \varepsilon E \frac{d\psi}{dx} + C_1

Far from the wall, both the shear rate $dv_z/dx$ and the field gradient $d\psi/dx$ vanish (bulk fluid, no field distortion, uniform velocity), forcing $C_1 = 0$ .

Step 4 — Integrate again with the correct boundary conditions. Integrating once more from the shear plane (where $\psi = \zeta$ and, by the no-slip condition, $v_z = 0$ ) out to the bulk (where $\psi \to 0$ and $v_z \to v_{eo}$ , the bulk EOF velocity):

\eta\, v_z\Big|_{shear}^{bulk} = \varepsilon E\, \psi\Big|_{shear}^{bulk}

\eta(v_{eo} - 0) = \varepsilon E(0 - \zeta)

v_{eo} = -\frac{\varepsilon\zeta E}{\eta}

Step 5 — Fix sign and read off mobility. For a negatively charged silica wall, $\zeta < 0$ , so $v_{eo} > 0$ when $E$ points from anode to cathode — meaning the bulk flow moves toward the cathode, matching the physical picture of cations in the diffuse layer being dragged cathodically and entraining the whole fluid column. Defining electroosmotic mobility as $\mu_{eo} = v_{eo}/E$ ,

\mu_{eo} = \frac{\varepsilon \zeta}{\eta}

(sign convention absorbed so that $\mu_{eo} > 0$ for the normal cathodic-EOF case in bare silica). Because the derivation never referenced the capillary radius $a$ except through the requirement $\kappa a \gg 1$ , the velocity profile outside the thin double layer is uniform across the entire bore — the flat plug profile that is CE's chief efficiency advantage over pressure-driven flow, whose parabolic profile is the direct origin of the Golay C-term in open-tubular chromatography.

Worked examples

Example 1 — Foundational: mobility and migration time for a small inorganic ion. Potassium ion, $\mathrm{K^+}$ , has a limiting (infinite-dilution) electrophoretic mobility of $\mu_{ep} = 7.62\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}}$ (a standard tabulated ionic conductance value). A fused-silica capillary is $L_{tot} = 60.0$ cm total length, $L_{det} = 50.0$ cm to the detector, run at $V = 25.0$ kV, with an electroosmotic mobility measured (via a neutral marker, mesityl oxide) of $\mu_{eo} = 5.90\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}}$ toward the cathode. Detection is at the cathodic end. Find the migration time. Step 1 — field strength.

E = \frac{V}{L_{tot}} = \frac{25{,}000\ \mathrm{V}}{0.600\ \mathrm{m}} = 4.167\times10^{4}\ \mathrm{V/m}

Step 2 — net mobility. K⁺ is a cation, so its electrophoretic migration is also toward the cathode, same direction as EOF:

\mu_{net} = \mu_{ep} + \mu_{eo} = 7.62\times10^{-8} + 5.90\times10^{-8} = 1.352\times10^{-7}\ \mathrm{m^2\,V^{-1}\,s^{-1}}

Step 3 — observed velocity.

v_{obs} = \mu_{net}E = (1.352\times10^{-7})(4.167\times10^{4}) = 5.634\times10^{-3}\ \mathrm{m/s}

Step 4 — migration time.

t_m = \frac{L_{det}}{v_{obs}} = \frac{0.500\ \mathrm{m}}{5.634\times10^{-3}\ \mathrm{m/s}} = 88.8\ \mathrm{s}

So K⁺ reaches the detector in 88.8 s (1.48 min). Sanity check on limiting behavior: if EOF were switched off entirely ( $\mu_{eo}=0$ , e.g. a heavily coated neutral capillary), $\mu_{net}$ drops to $7.62\times10^{-8}$ , $v_{obs}$ falls to $3.175\times10^{-3}$ m/s, and $t_m$ stretches to 157 s — slower, as expected, since the cation now relies solely on its own electrophoretic push with no bulk conveyor assisting it.

Example 2 — Applied: resolving two peptides by adjusting voltage and buffer. Two peptides have net mobilities in a pH 2.5 phosphate buffer of $\mu_1 = 2.10\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}}$ and $\mu_2 = 2.38\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}}$ (both net, i.e. already including whatever small EOF survives at this low pH, taken here as $\mu_{eo} \approx 0.15\times10^{-8}$ , negligible but included for realism). Diffusion coefficients are $D_1 = 3.2\times10^{-10}\ \mathrm{m^2/s}$ and $D_2 = 3.0\times10^{-10}\ \mathrm{m^2/s}$ (peptide 2 slightly larger, diffuses slightly slower — consistent with its larger $|\mu|$ arising partly from higher charge rather than smaller size, so treat $D_2$ as given rather than back-calculated). Capillary: $L_{tot}=40.0$ cm, $L_{det}=30.0$ cm, $V=20.0$ kV. Find $N$ for peptide 2 and the resolution $R_s$ . Step 1 — field.

E = \frac{20{,}000}{0.400} = 5.00\times10^{4}\ \mathrm{V/m}

Step 2 — plate number for peptide 2. Using $N = \mu_{net}V(L_{det}/L_{tot})/(2D)$ — this form comes directly from combining $t_m = L_{det}/(\mu_{net}E)$ with $\sigma^2 = 2Dt_m$ and $N=(L_{det}/\sigma)^2$ :

N_2 = \frac{\mu_2 V}{2D_2}\cdot\frac{L_{det}}{L_{tot}} = \frac{(2.38\times10^{-8})(20{,}000)}{2(3.0\times10^{-10})}\times\frac{0.300}{0.400}

N_2 = \frac{4.76\times10^{-4}}{6.0\times10^{-10}}\times 0.750 = 7.933\times10^{5}\times 0.750 = 5.95\times10^{5}

Nearly 600,000 theoretical plates — a value routinely reported in real CE peptide separations and roughly an order of magnitude beyond typical HPLC efficiency on the same solutes. Step 3 — resolution. Mean mobility $\bar\mu = (2.10+2.38)/2 \times10^{-8} = 2.24\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}}$ ; $\Delta\mu = 0.28\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}}$ . Using $\sqrt{N} \approx \sqrt{N_2} = 771$ (a good approximation since $N_1$ and $N_2$ differ by only a few percent here):

R_s = \frac{1}{4}\sqrt{N}\,\frac{\Delta\mu}{\bar\mu} = \frac{1}{4}(771)\left(\frac{0.28\times10^{-8}}{2.24\times10^{-8}}\right)

R_s = 192.75\times 0.1250 = 24.1

An $R_s$ of 24 is far beyond baseline (baseline resolution is conventionally $R_s \geq 1.5$ ), so in practice the analyst would lower the voltage substantially to shorten run time while retaining adequate resolution — a direct, simple trade unavailable in gradient HPLC method development. Halving the voltage to 10.0 kV halves $E$ , which quarters $N$ (since $N \propto V$ ) to about $1.49\times10^{5}$ , and $\sqrt{N}$ drops by the factor $1/\sqrt{2}$ , giving $R_s \approx 24.1/\sqrt{2} = 17.0$ — still comfortably resolved, but the run finishes in half the time.

Example 3 — Exam-level: locating an isoelectric point from mobility versus pH, and predicting focusing behavior. A weak-acid/weak-base peptide's electrophoretic mobility (EOF-subtracted, i.e. true $\mu_{ep}$ ) is measured at three pH values in free-solution CE:

pH	$\mu_{ep}$ (10⁻⁸ m² V⁻¹ s⁻¹)
4.00	+3.10
6.50	+0.40
9.00	−2.90

Assume, as a first approximation valid near the isoelectric point, that mobility varies linearly with pH over this narrow window (a reasonable local approximation even though the true titration-curve dependence is sigmoidal, since only two ionizable groups dominate net charge here). Find the isoelectric point pI, and state what happens to this peptide in an isoelectric focusing (IEF) experiment run in the same capillary with an immobilized pH gradient spanning pH 3–10. Step 1 — use the two points bracketing the sign change (pH 6.50, $\mu=+0.40$ ) and (pH 9.00, $\mu=-2.90$ ), since pI must lie between them (mobility changes sign there):

\text{slope} = \frac{-2.90 - 0.40}{9.00 - 6.50} = \frac{-3.30}{2.50} = -1.32\ (10^{-8}\ \mathrm{m^2V^{-1}s^{-1}\,pH^{-1}})

Step 2 — solve for pH where $\mu_{ep}=0$ , starting from the point (6.50, +0.40):

0 = 0.40 + (-1.32)(\mathrm{pI} - 6.50)

\mathrm{pI} - 6.50 = \frac{0.40}{1.32} = 0.303

\mathrm{pI} = 6.50 + 0.303 = 6.80

Step 3 — check with the other segment (pH 4.00 to 6.50) to test linearity assumption:

\text{slope}_{4-6.5} = \frac{0.40-3.10}{6.50-4.00} = \frac{-2.70}{2.50} = -1.08

The slope is somewhat shallower between pH 4 and 6.5 (−1.08) than between 6.5 and 9.0 (−1.32), confirming the expected sigmoidal (not perfectly linear) titration behavior — the local linear estimate of pI = 6.80 from the bracketing segment is the more reliable one since it uses data closest to the zero crossing. Step 4 — physical consequence in IEF. In an isoelectric focusing experiment, the peptide will migrate electrophoretically through the pH gradient until it reaches the local pH equal to its pI (6.80), at which point $\mu_{ep}=0$ and net migration stops — any small diffusive excursion into a region of different pH immediately reinstates a mobility pushing it back toward pH 6.80 (positive mobility, hence cathode-directed push, at pH below pI; negative, anode-directed, above it). This self-correcting restoring force is precisely why IEF bands are so sharp: it is a stable equilibrium focusing mechanism, fundamentally different from the kinetic separation of free-solution CE described in the rest of this article, though it shares the same underlying mobility formalism.

Limiting cases, scope & common traps

Situation	Behavior	Why
$\kappa r \to 0$ (dilute buffer, small ion)	Henry function $f\to 1$ (Hückel limit)	Ionic atmosphere is diffuse compared to ion size; retardation/relaxation corrections vanish
$\kappa r \to \infty$ (concentrated buffer, large particle)	Henry function $f\to 3/2$ (Smoluchowski limit)	Double layer collapses onto particle surface, matching the same physics used in the EOF derivation
$\mu_{eo} = 0$ (coated, neutral capillary)	Only cations reach a cathodic detector; anions and neutrals never elute (or need reversed field)	No bulk conveyor; only self-mobility carries analytes past the window
$\|\mu_{ep}\| = \mu_{eo}$ , opposite sign	Analyte appears stationary; migration time formally infinite	Vector sum of electrophoretic and electroosmotic velocity is exactly zero
High ionic strength buffer	EOF drops ( $\zeta$ magnitude shrinks as double layer compresses), Joule heating rises	Both hurt efficiency: less EOF is not free — it also risks band broadening from resistive heating
Very high voltage / small-bore capillary	$N$ rises linearly with $V$ but Joule heating rises with $V^2/R$	A practical ceiling exists; thermostatting and narrow bores manage heat, not the theory itself
pH near analyte's pI	$\mu_{ep} \to 0$	Net charge vanishes; species becomes indistinguishable from a neutral marker in free-solution CE

The chemist behind it

Arne Tiselius built the first rigorous, quantitative moving-boundary electrophoresis apparatus in the early 1930s at Uppsala University, turning a qualitative curiosity — that charged colloids and proteins migrate in an electric field — into a precise analytical technique capable of resolving serum proteins into distinct albumin and globulin fractions by their differing mobilities. His apparatus used a U-tube with a sharp boundary between protein solution and buffer, tracked optically via refractive-index (Schlieren) changes as the boundary moved under an applied field. This was the direct ancestor of every modern electrophoretic method, from slab-gel protein electrophoresis to the free-solution capillary format used today. Tiselius received the 1948 Nobel Prize in Chemistry for this work and for his related studies of adsorption analysis, and his moving-boundary method directly established the conceptual and mathematical framework — mobility as charge-to-friction ratio, boundary sharpening, and the role of buffer composition — on which capillary electrophoresis theory still rests. The technique remained largely a slab- and gel-based method for decades, limited by Joule heating in the bulky formats Tiselius and his successors used, until James Jorgenson and Krynn Lukacs, working at the University of North Carolina in the early 1980s, showed that shrinking the separation channel to a fused-silica capillary of 75 µm internal diameter solved the heating problem by dramatically improving the surface-to-volume ratio for heat dissipation. This let them apply far higher fields than any slab format tolerated, and they demonstrated separations with plate counts exceeding 400,000 — a result that launched capillary electrophoresis as a modern instrumental technique in its own right, complementary to and in some respects surpassing HPLC for charged, polar, and biomacromolecular analytes.

Origins · 1 minArne Wilhelm Kaurin Tiselius and how this was discovered

Arne Wilhelm Kaurin Tiselius

1902–1971 · Uppsala University, Sweden, 1930s–1940s

Tiselius trained under Theodor Svedberg, the inventor of the analytical ultracentrifuge, and inherited from him an insistence on rigorous, quantitative physical measurement of biomolecules rather than purely qualitative biochemical characterization. His moving-boundary electrophoresis apparatus required him to solve, for the first time with real experimental precision, the coupled problem of ionic mobility, buffer conductivity, and boundary-sharpening effects in multicomponent mixtures — mathematics that is the direct conceptual ancestor of the mobility and resolution equations used in capillary electrophoresis today.

History

- 1807 — Ferdinand Frederic Reuss observes electrophoretic migration of clay particles in water under an applied field, the earliest documented electrophoretic effect.

- 1930–1937 — Arne Tiselius develops quantitative moving-boundary electrophoresis, resolving serum proteins into albumin and α-, β-, γ-globulin fractions.

- 1948 — Tiselius awarded the Nobel Prize in Chemistry for electrophoresis and adsorption analysis.

- 1967 — Stellan Hjertén demonstrates free-solution electrophoresis in narrow-bore rotating tubes, an early precursor to capillary formats.

- 1981 — James Jorgenson and Krynn Lukacs publish high-efficiency separations in 75 µm fused-silica capillaries, establishing modern capillary electrophoresis.

Why does capillary electrophoresis achieve higher plate counts than HPLC for the same analyte?

The dominant broadening mechanism differs. In packed-column HPLC, the Van Deemter equation's C-term (resistance to mass transfer, including the parabolic flow profile inside particle pores and between particles) contributes substantially to plate height, especially at higher flow rates. In capillary electrophoresis, the electroosmotic flow profile is flat (plug-shaped) because the driving force is confined to the nanometers-thick double layer at the wall, not distributed as a pressure gradient through the bulk. With that source of broadening essentially eliminated, the main remaining contributor is longitudinal molecular diffusion, $\sigma^2=2Dt_m$ , which scales down as $t_m$ shortens (raise voltage) or as $D$ shrinks (larger analytes). That is why CE routinely reports 200,000–600,000 plates for peptides and proteins, an order of magnitude above typical HPLC values for comparable analytes and run times.

What determines the direction and magnitude of electroosmotic flow, and can it be reversed?

EOF direction and magnitude are set by the sign and size of the zeta potential at the capillary wall, which in bare fused silica is negative above about pH 3 due to ionized silanol groups, producing cathode-directed EOF via the Smoluchowski relation $\mu_{eo}=\varepsilon\zeta/\eta$ . Lowering pH protonates silanols and reduces $|\zeta|$ , shrinking EOF; raising ionic strength compresses the double layer and also reduces $|\zeta|$ (and hence EOF) somewhat, though the ionic-strength dependence is weaker than the pH dependence. EOF can be reversed entirely by adsorbing a cationic surfactant (such as cetyltrimethylammonium bromide) onto the wall, which overcoats the native negative charge with a positive bilayer, flipping $\zeta$ positive and reversing bulk flow toward the anode — a standard trick used to analyze anions with cathodic-position detectors by making the anodic-directed EOF dominate.

How is capillary electrophoresis able to separate cations, neutrals, and anions all in a single run?

Because electroosmotic flow moves the entire bulk fluid at one common velocity regardless of charge, while each charged analyte additionally migrates electrophoretically at its own mobility superimposed on that common flow. In normal-polarity bare-silica CE, EOF toward the cathode is typically faster than most analyte electrophoretic mobilities, so it dominates net transport for everything: cations arrive earliest since their electrophoretic push adds to EOF, neutrals all co-migrate exactly at the EOF velocity (unresolved from each other, but distinguishable in time from ions), and anions arrive last because their electrophoretic mobility partially subtracts from, but does not usually overcome, the EOF. Chromatography cannot replicate this in a single run with one stationary phase because retention there depends on partitioning, which is not systematically ordered by simple charge sign the way electrophoretic mobility is.

Why does Joule heating matter so much in capillary electrophoresis, and how is it managed?

Applying tens of kilovolts across a buffer-filled capillary drives substantial current, and the power dissipated as heat is $P=I^2R$ (equivalently $V^2/R$ ), all of which must be removed through the capillary wall or the buffer temperature will rise, lowering viscosity, altering mobilities non-uniformly across the capillary cross-section (since the center is hotter than the wall-adjacent fluid), and broadening peaks through this radial temperature-viscosity gradient. Because heat generation scales with the capillary's cross-sectional volume but heat removal scales with its surface area, narrower capillaries (typically 25–75 µm i.d.) have a far more favorable surface-to-volume ratio and can tolerate much higher fields before thermal effects degrade efficiency — this scaling argument is exactly what let Jorgenson and Lukacs push to the high fields that made modern CE efficiencies possible. Active thermostatting (forced-air or liquid cooling of the capillary cassette) further controls this in commercial instruments.

What is micellar electrokinetic chromatography and why is it called a hybrid technique?

MEKC adds an ionic surfactant, commonly sodium dodecyl sulfate, above its critical micelle concentration to the CE running buffer, forming charged micelles that act as a mobile pseudo-stationary phase. Neutral analytes, which would otherwise all co-migrate at exactly the EOF velocity and be completely unresolved in ordinary CE, now partition differentially between the bulk aqueous buffer and the micelle interior based on their hydrophobicity, exactly analogous to the partition equilibrium underlying Principles of Chromatographic Separation in reversed-phase HPLC. The micelles themselves have their own electrophoretic mobility (since they're charged), which combines with EOF to carry them through the capillary at a velocity different from the free buffer, creating an effective retention-time window in which neutral analytes elute according to their hydrophobicity — hence "chromatography" performed inside an electrophoretic instrument.

How does one measure electroosmotic mobility experimentally, separate from an analyte's own electrophoretic mobility?

The standard method injects a small, easily detected neutral marker (mesityl oxide, dimethylformamide, or acetone are common choices) that carries no charge, so its electrophoretic mobility $\mu_{ep}$ is exactly zero and its observed migration velocity equals $v_{eo}$ alone. Measuring its migration time $t_{eo}$ to the detector at known field strength $E$ and detector distance $L_{det}$ gives $\mu_{eo} = L_{det}L_{tot}/(V t_{eo})$ directly. Once $\mu_{eo}$ is known for a given buffer and capillary, any analyte's electrophoretic mobility can be recovered from its own observed migration time by subtracting the neutral marker's contribution: $\mu_{ep} = \mu_{net} - \mu_{eo}$ , where $\mu_{net}$ is calculated from the analyte's own migration time by the same formula.

Does electrophoretic mobility depend on molecular weight the way retention in size-exclusion chromatography does?

Not directly, and this is a common point of confusion. Free-solution electrophoretic mobility depends on the charge-to-friction ratio, roughly $ze/(6\pi\eta r)$ , and for many biomolecules like proteins and short nucleic acids, both charge and hydrodynamic radius scale together with size in ways that partially cancel, meaning mobility can be nearly size-independent over a considerable range — this is exactly why long, uniformly charged DNA cannot be size-separated by free-solution CE alone. Only when a sieving matrix (an entangled polymer solution or crosslinked gel) is added does frictional drag become strongly size-dependent, because larger fragments must reptate through a smaller effective pore network more slowly, restoring a size-based separation mechanism analogous in spirit, though not mechanism, to size-exclusion chromatography.

Electrophoresis and Capillary Electrophoresis

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Limiting cases, scope & common traps

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