Electrostatics in Macromolecular Solutions - Food Colloids: Self-Assembly and Material Science - page 139

crucial and the same qualitative behaviour is found for both interacting planes
and interacting spheres. The screened Coulomb potential captures the change in
free energy when the two macromolecules approach each other. It is, however,
questionable to partition the screened Coulomb interaction into energy and
entropy terms. More elaborate forms of the screened Coulomb potential can be
derived, 15 where the macromolecular size is taken into account. The comparison
in this section has been limited to a 1:1 electrolyte and to situations where k 1 is
of the same order (or greater) than the macromolecular dimension. To extend
the use of the screened Coulomb potential to multivalent electrolytes usually
leads to qualitatively incorrect results, as will be discussed below.
9.4.2 Effect of Multivalent Ions
We have shown above how a simple theory, the screened Coulomb potential, is
capable of an almost quantitative description of the interaction between two
charged proteins. This good agreement is limited to systems containing only
monovalent counter-ions. There is a qualitative difference between the interac-
tion of two charged macromolecules in the presence of monovalent counter-ions
and in the presence of multivalent counter-ions. In the latter case the mean-field
approximation behind the DH equation breaks down and one has to rely on
simulations or more accurate theories like the hypernetted chain equation. 16,17
The deviation from the mean-field description due to ion-ion correlations has
such a physical origin that the effect should be independent of the particular
geometry of the charged aggregates. Clearly there are quantitative differences
between cylindrical, spherical, irregularly shaped or flexible charged colloidal
species, but the basic mechanism operates in the same way. The importance of
ion-ion correlations can be seen in Figure 7, where the free energy of interaction
for two charged spherical aggregates has been calculated by MC simulation. For
monovalent counter-ions there is a monotonic repulsion in accordance with the
screened Coulomb equation [Equation (10)]; but with multivalent counter-ions,
or with a solvent of low dielectric permittivity, the entropic double-layer
repulsion decreases and eventually the correlation term starts to dominate.
This phenomenon can be seen as a balance between entropy and energy. For
two weakly or moderately charged macromolecules with monovalent counter-
ions, the dominant contribution to the free energy of interaction comes, as we
have seen in Figure 5, from a reduction in entropy when the two counter-ion
clouds start to overlap. The energy of interaction is always attractive and is
only weakly dependent on the counter-ion valency. The important difference
between a system with monovalent or divalent counter-ions is the reduced
entropy of the latter due to a lower number density of counter-ions. Thus, any
change that reduces the entropy and/or increases the electrostatic interactions
will eventually lead to a net attractive interaction. This is true for a model
system of two spheres, each with a single net charge, but the same mechanism is
also operating between two protein molecules with discrete charge distributions
of irregular form, 18 or between two DNA molecules. 19