Migration in an Electric Field

 

When a charged particle is placed in an electric field, it experiences a number of different forces which, when a steady state is reached, results in the particle migrating at a constant velocity.

 

Altogether there are four effects that control the steady state migration velocity of the particle. There is the basic electrophoretic attraction () that results from the charge on the particle (Q) and the strength of the applied field (E), i.e.,

 

 

There is the Srokes friction,  (), (the viscous drag), i.e.,

 

 

    where (U) is the electrophoretic velocity of the particle.

               () is the friction coefficient which, according to Stokes Law is given by,

 

      

    where the respective symbols have the meaning previously ascribed to them.

 

Then there is the retardation effect () due to the effect of the electric field (E) on the ions of opposite charge drawn from the solution and surrounding the charged particle. This slowing effect is known as the electrophoretic retardation.

 

The fourth effect () is known as the relaxation retardation results from the movement of the particle and this movement differing from that of its surrounding ions thus distorting the ionic surroundings of the particle so that the particle itself is no longer in the center of electrical environment.

 

After the steady state is reached the sum of all the four effects must equal zero and the electrophoretic velocity (U) is given by,

 

 

One solution for an expression for () was derived by Hückel and the electrophoretic retardation force was shown to be,

 

 

Thus, by neglecting (), the electrophoretic mobility can be shown to be,

 

 

The derivation of this equation assumes that the field is not deformed by the presence of the charged particle and, thus, the spheres must be small (which in electrophoresis they will be) and the product (ka), the double-layer thickness is small compared with unity (i.e. ).