Working Content

Prerequisites:

• Electric fields in matter
• Screening of electrical interactions in salt solution  

In the previous reading on Screening of electrical interactions in salt solution, we found that the charge on a a DNA molecule in a salt solution experiences a screening effect as a result of the salt ions clustering near the DNA. As a result, the electric field produced by a charged biological macromolecule should be different than it would produce in a vacuum.

To make this quantitative we would like to know two things:

• How large is the volume occupied by the positively charged ions that neutralize the negatively charged DNA — the “screening cloud”? Put differently, how far away from the DNA do we have to go before the electric field will drop essentially to zero?
• What is the mathematical function describing how the electric field and the electric potential depend on distance?

Let’s begin by reasoning about the physical parameters that determine the size of the screening cloud. Specifically, let’s model the screening cloud as extending a distance λ_D, known as the “Debye length,” away from the DNA at every point. (The Greek letter λ is named “lambda” and corresponds to our letter “L” so it is commonly used in science to denote lengths.) Inside the cloud there are many more positive ions than negative; outside the cloud there are equal concentrations of positive and negative ions. What determines this size λ_D?

• If temperature increases, the effects of entropy and temperature also increase. We'd expect the screening cloud to spread out due to the increased molecular motion. Therefore, λ_D should increase as T increases.
• If the undisturbed concentration c₀ of salt ions increases, then the number of ions that have to be moved away from an even distribution in order to form the screening cloud is a smaller fraction of the total. Therefore, the fractional change in concentration required to screen the DNA is smaller. This means that the entropic cost of forming the screening cloud decreases as the salt concentration increases. Therefore, λ_D should decrease as c₀ increases.
• If the charge of the individual positive ions increases, fewer of them are needed to screen the DNA and again the entropic cost of forming the screening cloud is less. Therefore λ_D should decrease as Ze, the charge of one of the positive ions, increases.

The mathematical expression for the size of the screening cloud λ_D is determined by solving differential equations to find the arrangement that balances electric potential energy and entropy. We won’t do that in this class, but we’ll make use of the result.

Let’s look a little more closely at this expression. The central charge we are trying to screen drags in ions and they have extra PE because they are now closer.  The electrostatic potential energy for two charged particles of charge ze separated by a distance r is given by (using the dielectric constant to reduce the strength of the field due to simple polarization)

The thermal energy of their motion tends to push them apart. Let's try estimating the amount of PE against the amount of thermal energy. Consider the ions in a volume of (λ_D)³. Since the concentration is c₀, we have about c₀(λ_D)³ ions in that volume. If total PE is about k_CQ²/λ_D and we balance that against k_BT for each ion, we'll get something like

Solving for λ_D gives

So the numerator inside the square root is the thermal energy k_BT, and the denominator of our expression for λ_D is proportional to the electric potential energy with the length left out.

A more mathematically rigorous treatment gives

In addition, solving the differential equations gives the following form for the electric potential of a charged particle or sphere in salt solution:

So we see that the potential decreases exponentially compared to the potential of a charged particle in vacuum, and the distance over which the potential decays to 1/e of its value without screening is λ_D. The corresponding electric field is slightly more complicated (it’s the derivative of V with respect to r) but is essentially the same in being exponentially screened compared to the electric field of a charged particle in vacuum:

Catherine Crouch, Swarthmore College  2/12/12