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366 Electrophoresis

circuit plus the time required for the randomly moving function; see Section I.A). If the conditions are such that
molecules to adopt a directional motion (this takes al- these factors do not distort the boundary and the starting
together <10 −6 sec). This delay is not considered here condition was similar to that described for Fig. 1, where
because the electrophoretic mobility is slow relative to the concentration of the ion is uniform on one side of the
this transition. For most purposes it can be assumed that boundary and zero on the other, then the boundary shape
the current is constant throughout the experiment and the can be described by Eq. (8):
macroions move at a uniform velocity (dx/dt) determined (π/2)(Dt) 1/2
by the field strength and total charge Q on the ion. In or- 1 2
C(x) = C 0 1 − √
der to maintain a constant velocity, Newton’s laws show 2 π 0
that the movement in one direction must be opposed by an
2
equal but opposite force. This opposing force is ascribed x
× exp − √ dx . (8)
to frictional forces f between the macroions and the sta- 2 Dt
tionary solvent. (In fact, the solvent is stationary only at
In Eq. (8), C(x) is the concentration at x after t seconds of
distances far removed from the surfaces of the particle.)
a substance having a diffusion coefficient D (the starting
The value of the frictional force is determined by the in-
concentration was C 0 ). If, however, a zone of width h con-
trinsic size and shape of the ion, as well as the viscosity
taining the macroion is introduced between the electrode
of the solvent, and increases with the velocity of the ion.
solution and the main bulk of the solvent, both sides of the
Thus, we can equate the two opposing forces to give
zone diffuse to produce a double sigmoid shape, which at
dx dx QE
QE = f or = . (6) its simplest can be described by Eq. (9):
dt dt f

If E is measured as volts per meter, then dx/dt is called C 0 h − x h + x
C(x) = erf √ + erf √ , (9)
the mobility of the ion for the chosen experimental con- 2 2 Dt 2 Dt
ditions. The coefficient f has a theoretical foundation in √ a
where [erf (z) = (2/ π) exp (−a) da] and the distri-
hydrodynamics, and a functional relationship between f 0
bution is symmetrical about a plane at x = 0. With both
and the coordinates of the particle can be derived for a
types of boundary, the centroid [¯ x, Eq. (7)] corresponds to
few regular shapes (for a sphere it is known as the Stokes’
the center of the boundary, butif the boundaryis warpedby
equation, but there are mathematical solutions for ellip-
electrical inhomogeneities or contains a mixture of unre-
soids and cylinders). It is also known that f is inversely
solved ions of slightly differing mobilities, then ¯ x will still
proportional to the randomizing effects of diffusion of a
correspond to the required centroid of the boundary but
large number of ions (Einstein–Sutherland relationship).
not necessarily its geometrical center. This explains why
Essentially, Eq. (6) describes the movement of a single
it is important to use the centroid for calculating average
ion under the influence of an electrical field. Rarely, if
mobilities of electrophoresing boundaries.
ever, can one ion be studied experimentally, because at
13
finite concentrations of ions there are >10 ions per liter The diffusion coefficient D and frictional coefficient f
(a 10 −10 M solution of 0.1 µg liter −1 for an ion of relative [Eqs. (6), (8), and (9)] of an ion are similar to that found for
3
mass 10 contains 10 13 ions per liter). Diffusion of this a neutral molecule. However, because the salt dissociates
when dissolved, but electroneutrality must be maintained
population of ions spreads the boundary about an elec-
throughout the solution despite each ion having different
trophoretically transported point called the centroid, and
diffusion rates, it is necessary to modify Eq. (6) (and the
it is the velocity of this point that is described by Eq. (6)
other equations where a diffusion coefficient is employed
for experimental situations. The centroid or first moment
to replace frictional forces). This is done by replacing the
(¯ x) can be evaluated from Eq. (7) using a set of rectangu-
lar coordinates determined experimentally over an elec- frictional coefficient in Eq. (6) by F(u + +u − )/2u + u − (u +
and u − are the mobilities of the anion and cation, respec-
trophoresing boundary,
tively) to give the Nernst equation.

x 2
xy dx In any solution of ions there always occurs electroneu-
x 1 trality, so every positive ion has a counter negative ion.
¯ x = . (7)
x 2
y 2dx For macroions these are called gegenions. They can be
small ions such as Na or Cl or larger organic ions. Each
−
+
x 1
The shape of the boundary approximates a Gaussian pro- ion transports a proportion of the current, and since small
file, but the exact description of the shape depends on the ions have greater mobilities than macroions, their trans-
starting conditions of the experiment, the mobilities of the port numbers dominate the system, which means that they
various ions, and whether sharpening of the boundary oc- carry most of the current on both sides of an interface.
curs as a result of electrical effects (Kohlrausch regulating The necessity of maintaining electroneutrality throughout