Theory of Capillary Electrophoresis

 

As with other forms of electrophoresis. the electrophoretic separation results from the relative electrophoretic migration of the individual solutes that is superimposed migration resulting from electro-osmotic flow. Taking the electrophoretic mobility of the solute as () and the electro-osmotic mobility as () then the observed mobility () is given by the following equation,

where (E) is the applied voltage

 

The process is depicted diagrammatically in figure 19.

 

Figure 19. The Development of a Binary Separation by Capillary Electrophoresis

 

 

Initially the mixture of two electrolytes is placed on the capillary column and the high field potential is applied. The two migration processes immediately commence the osmotic flow starts moving both substances forward but due to Stokes law there will be a slight difference in rate due to the different respective molecular diameters of the two substances. The electrophoretic migration will also commence which will provide a significant relative differential velocity between the molecules of the two substances. The differential molecular electrophoretic velocities will be proportional to their charge to size ratios. As a consequence, as the molecules progress along the tube due to the migration driving forces, one substance will slowly move through the mixed band ahead of the other and eventually the two will be separated.  The separation is characterized in a similar manner to that used for chromatographic separation by three constants, the retention, (k), the separation  (a) and  the resolution (R).

Where.

 

 where   () is the retention time and () dead time of the solute and column respectively and,

 

where () and () are retention values of solute 1 and solute 2

 

and

 

where () and () are the base widths (in time units) of peak 1 and peak 2 respectively,

   and  () is the difference in retention time between peak 1 and 2..

 

The separation may be assessed relative to () as follows,

 

 

where () is the apparent mobility of the two peaks,

           () is the average mobility of the two peaks,

    and  (N) is the efficiency of the electrophoretic system which is given by ;-

 

 

       where () is the peak width at half height in units of time

            and () is the column length.

 

It should again be emphasized that the equation for the efficiency in theoretical plates is derived from Chromatography Plate Theory and should not be compared with values for the efficiency derived from Distillation Plate Theory.

 

The effective resolving capacity of a capillary electrophoretic columns (i.e. the column efficiency) is best described by the height of the theoretical plate () which is given by,

 

 

Where () is the variance of a peak in terms of column length and the other symbols have the meaning previously ascribed to them.

 

The value of () is made up of many contributing dispersion factors and, as it can be assumed that all the dispersive processes are random and non-interacting, the variance of the total dispersion is the sum of the individual variances arising from each dispersing effect.

 

In general terms, this can be put in the form of the following equation.

 

 

       

 

 

where () is the variance due to diffusion dispersion.

           () is the variance due to thermal dispersion,

           () is the variance resulting from dispersion in the injection system.

           () is the variance resulting from dispersion at the tube wall,

           () is the variance resulting from dispersion during electro-osmosis,

           () is the variance caused by dispersion during electrophoretic migration,

           () is the variance resulting from dispersion caused by adsorption,

   .and ()is the variance resulting from dispersion  by other undefined effects.

 

A complete equation that describes the contribution of each dispersing effect to the total band variance as a series of explicit functions in the same form as the Van Deemter equation for dispersion in liquid chromatography columns has, so far, not been reported.

 

An example of the use of a capillary electrophoretic system that has been employed to separate a series of pharmaceutical products is shown in figure 20.

 

 

 

Figure 20.  The Separation of a Series of Pharmaceutical Products by Capillary Electrophoresis.