Electrical interactions in ionic solutions

So far we’ve considered two kinds of environments in which electrical interactions occur. We began with considering the electric forces and field produced by charged objects in vacuum. We found in this case that the electric field of a spherical charged object, such as an ion or a globular protein, depends on distance away according to $1/r^2$, and the corresponding electric potential depends on distance according to $1/r$:

$$E(r) = \frac{k_CQ}{r^2} \quad\quad V(r) = \frac{k_CQ}{r}$$

We also considered what happens when a very good conductor, such as a metal, placed in an external electric field (“external” just means its sources are outside the conductor). (See Electric fields in matter.) We reasoned there that charge will rearrange on the conductor so that the total electric field in the bulk of the object is zero: the field due to the rearranged charge perfectly cancels the external field. (“In the bulk” means everywhere in the material except right at the surface.) This happens because if there were a nonzero electric field in the bulk of the material, then that field would exert a force on the mobile charge carriers. These carriers therefore would move until the external field and the field due to the rearranged mobile charge carriers add to zero everywhere inside the material. Once there is no field inside the material, and hence no force, there is no further rearrangement.

Our treatment of charges and a conductor is a toy model. In metals, for reasons that require some pretty sophisticated physics to explain, we can ignore the thermal motion of electrons in conductors, and explain the electrical behavior of electrons in metals entirely in terms of electrical forces.* 

In an ionic fluid, like salt water, the relevant charges are not free electrons. What matters are the many dissolved ions (K⁺, Na⁺, Cl^-, OH^-, H⁺, ...) found in biological fluids. In this case, an additional effect matters. We have to take into account both the electrical forces on the dissolved ions, and the thermal energy that the water molecules and ions have that keeps them in steady motion. Even if two charged particles are attracted together electrically, thermal motion can separate them if the thermal energy is greater than the electric potential energy associated with keeping them together. The effect that tends to spread the ions out, is entropy.

Consequently, when a charged object, such as a protein (most proteins are negatively charged under physiological conditions) or a DNA molecule, is immersed in an ionic solution, a process happens called screening or shielding.

Let’s consider a negatively charged DNA molecule in an ionic salt solution. Without the DNA molecule, the salt solution will have equal concentrations of positive and negative ions everywhere, so that there are no electric fields (on the average). However, with the DNA molecule, positively charged ions are attracted to it and negatively charged ions repelled.

Without considering temperature and entropy, we’d expect the same scenario as with the charged object embedded in metal: positively charged ions would attach to the DNA molecule all along its length until it was electrically neutral. However, bringing these ions closer to the DNA molecule modifies the even distribution of positive and negative ions throughout the solution. But that even distribution has the highest entropy and thus is the most entropically favored.

Consequently, what actually happens is a balance of these two effects as shown in the figure.** A “cloud” of positively charged ions clusters near the DNA, but there aren’t enough to completely neutralize the DNA. Thus outside the cloud, there is a weak electric field produced by the combination of the negatively charged DNA and this cloud of positively charged ions. This weak field in turn attracts in some more positively charged ions, and the field of these ions further reduces the electric field. The result is a high concentration of positively charged ions immediately adjacent to the DNA, and a decreasing concentration as distance from the DNA increases.

Not shown in the picture is that correspondingly, somewhere else there are extra negative ions. However, there is such an enormous volume of salt solution that can accommodate these extra negative ions that they can just spread out basically evenly and have very little effect.

This means that the electric field produced by a charged biological macromolecule in salt solution is different from the electric field it would produce in vacuum. Instead of decreasing with distance according to $1/r^2$, we can see from this picture that the field should decrease more rapidly with distance, and at distances greater than the size of this collection of positive ions there should be no electric field at all.

The distance from the charge that it takes for the ionic fluid to rearrange to effectively cancel the E field is called the Debye length. The math of this is worked out in the follow-on.

* When there is a study current flow through materials, the electrical resistance and conductance of metals does depend on temperature — but that is because of the thermal motion of the atomic nuclei, not the electrons.

** R. Phillips, J. Kondev, & J. Theriot, Physical Biology of the Cell (Garland Science, 2008) p.343.

Catherine Crouch, Vashti Sawtelle 2/16/12 & Joe Redish 5/4/19