Theory of capillary electrophoresis

Here is a detailed breakdown of the theory and fundamental principles of Capillary Electrophoresis (CE).

1. Core Principle

Capillary electrophoresis (CE) is an analytical separation technique in which charged particles migrate through a narrow capillary filled with an electrolyte solution under the influence of an electric field. The separation occurs because different analytes possess unique charge-to-size ratios and electrophoretic mobilities, causing them to migrate at different rates.

When a high voltage is applied across the capillary, the overall migration of an analyte is governed by two simultaneous types of movement: Electrophoresis and Electroosmotic Flow (EOF).

2. The Two Types of Movement

A. Electrophoresis (Electrophoretic Migration) This is the movement of individual charged species toward the electrode of the opposite charge. The velocity of this movement is directly proportional to the ion's charge-to-size ratio.

• Electrophoretic Mobility ($\mu_{ep}$): This depends on the ion's charge and hydrodynamic radius, described by the equation: $\mu_{ep} = \frac{q}{6\pi\eta r}$ Where $q$ is the ionic charge, $\eta$ is the viscosity of the buffer, and $r$ is the ionic radius.
• Conclusion: Small, highly charged ions will move significantly faster than large or weakly charged ions.

B. Electroosmotic Flow (EOF) EOF is the bulk, "plug-like" flow of the liquid inside the capillary.

• Mechanism: In an uncoated fused silica capillary, the inner wall contains silanol groups that ionize and carry a negative charge. These negative charges attract positively charged buffer cations, forming an electrical double layer.
• When the electric field is applied, these cations migrate toward the cathode, dragging the bulk solvent molecules along with them.
• Significance: EOF is typically strong enough that it carries all species—anions, cations, and even neutral molecules—in the same direction toward the detector. The strength of the EOF depends heavily on the buffer's pH, its ionic strength, and the surface properties of the capillary wall.

3. Mathematical Equations of Migration

• Overall Migration Velocity ($v$): The total velocity of an analyte is the sum of its individual electrophoretic mobility and the bulk electroosmotic mobility, multiplied by the electric field. $v = (\mu_{ep} + \mu_{eo})E$ (where $\mu_{ep}$ is electrophoretic mobility, $\mu_{eo}$ is electroosmotic mobility, and $E$ is the applied electric field).
• Migration Time ($t_m$): The time it takes for an analyte to reach the detector is calculated by dividing the effective capillary length ($L$) by the migration velocity. $t_m = \frac{L}{(\mu_{ep} + \mu_{eo})E}$

4. Efficiency and Resolution

One of the defining theoretical advantages of CE is its extremely high efficiency, capable of generating plate numbers up to $10^6$.

• Why is it so efficient? This high efficiency is theoretically due to the complete absence of a stationary phase (eliminating mass transfer resistance) and the "plug-like" flow profile of the EOF, which minimizes the band broadening that typically occurs in pressure-driven systems.