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Encyclopedia of Physical Science and Technology En005H-218 June 15, 2001 20:33

368 Electrophoresis

solution but do not depend on the magnitude of the forces, molecules caused by dispersion forces. In effect, the par-
while the flow J is directly proportional to the mobility u ticles are held apart in a low potential energy well similar
[Eq. (6)]. No existing theory of electrophoretic mobility is in shape to that found for all intermolecular forces (called
consistent with Eq. (11), since most assume that u or J is the Lennard–Jones potential). Thus, it is reasonable to ask
proportional only to the electric field strength. In any anal- how a particle formed from an apparently neutral molecule
ysis it is assumed that the fluid is stationary at the boundary can become charged in suspension. (An example of such
of the particle, and fluid movement is equal but opposite a suspension is a suspension of carbon particles to give
in direction to particulate movement when viewed at dis- printer’s ink.)
tances far removed from the particle. The result of this The source of the charges is adsorbed ions from the
definition is to consider the mobility as a vector. Using solvent collecting on the exposed surface of the particle.
these mathematical boundary conditions, it was possible The energy for this adsorption comes from the interfacial
tosolveequationsformulatedusingEq.(11)andtoinclude energy found when two phases share a common bound-
the close interaction of gegenions with the charged spher- ary and is called surface excess energy (or for aqueous
ical particle (Debye–H¨uckel relationship) and viscosity η, solutions with respect to air, surface tension). A thermo-
so giving the approximate relationship dynamic argument can be used to describe the energy of
adsorption of a solute. This includes the difference be-

Q 1 + κr i
u = P(κa) (12a) tween the chemical potentials of the dissolved ions in the
6πηR 1 + κa occluded solvent around the particle and those in the bulk
2 1/2 phases. This provides the basis for the Gibbs adsorption
8π N ε 1/2
κ = I . (12b) isotherm, which shows that the surface excess concentra-
¯
1000 DkT
tion of the solute is proportional to the logarithm of the
In these equations r i is the radius of a typical gegenion, R chemical activity of the solute in the bulk solutions (this
¯
is the radius of the macroion, a = R +r i , D is the dielec- is equal to concentration for dilute solutions). The origin
tric constant of the medium, k is the Boltzman constant, of this energy is the differing interactions of the ions with
N is Avogadro’s number, ε is the charge on an electron, the solvent in the bulk of the two phases and that far re-
2
and I = 0.5 Z C i (the ionic strength of the solution, moved from the interface. These interactions lead to either
i i
where Z is the valency of the ith ion of concentration an accumulation or a deficit at the interface.
C i ); P(κa) is a dimensionless function required to allow Anaturaloutcomeofthesephenomenaisthatthecharge
for the effect I has on the effective radius of the parti- measured by electrophoresis of macroions or particles is
cle and has values that vary between 1.0 and 1.5. The not necessarily equal to that found by algebraically sum-
inclusion of a in Eq. (12) is necessary in order to include ming the ionizations of the intrinsic side groupings of
the effect that the close approximation of the neutralizing the macroion. (This can be estimated for acidic and ba-
gegenions has on the particle. The effect is to increase sic groups by chemical titration.) The term “ζ potential”
the radius of the macroion beyond that expected from the has been used to describe this total electrophoretic charge.
neutral molecule. Several workers have changed Eq. (12) It is the charge at the plane of slip between the charged
to allow for other electrical phenomena, but despite these macroion or particle and the bulk solvent. The spatial de-
efforts it is reasonable to assume that no relationship be- pendence of the charges depends on many factors, but an
tween electrophoretic mobility and molecular parameters idealized example of the distribution is shown in Fig. 3
is rigorously applicable to macroions. for a positive ζ potential. It should be mentioned that the
This conclusion is disappointing because the major rea- distribution shown in Fig. 3 is an average because there is
son for the development of electrophoresis was to relate a general randomizing movement of the ions (Brownian
the charge of colloidal suspensions and biological macro- motion). This means that the instantaneous relationship
molecules to their known molecular parameters. The use for individual ions need not equal that shown in Fig. 3,
of electrophoresis in recent times has been restricted to and it is still possible for collisions to occur between the
empirical studies where electrical forces have been used particles despite the fact that on average they all carry
to separate mixtures of charged particles or macroions into the same charge. These collisions eventually precipitate
individual components. the suspensions.
The technique has many applications and has even A ζ potential is more generally used in discussions of
been used for separating suspensions containing appar- colloidal suspensions (e.g., pigments in paints) and is not
ently neutral molecules, because all particles in stable sus- often applied to the macroions found in biology. Here, the
pensions carry a net charge. The charging of a colloidal major part of the charge arises from ionization of the po-
particle is necessary if precipitation is to be reduced. This lar groups of the condensed subunits in the polymer. For
produces a balance between the repulsive forces produced proteins (an example of polyampholytes) there are ioniz-
by like charges and the ubiquitous attraction between all able carboxylic, amino, phenolic, and mercapto groups,