7. Ionic screening¶
The goal of this exercise is to estimate the length over which electric fields can extend in electrolyte solutions. It is known since the experiments of Charles-Auguste de Coulomb that the electric force between two charged spheres depends on the inverse distance squared, or,
and so does the electric field around a charged sphere or point charge. In vacuum or air, there is no typical “range” of the electric field. The length over which you could measure the field would depend on the quality of the device you are using, or the strength of the charge. This is different for electric fields in electrolyte solutions, where electric fields drop to zero after a well-defined length, because of the influence of the ions. This length is usually much smaller than a micrometer in water, and depends on the concentration of electrolytes.
In this exercise we will calculate the density of ions near a charged plate, e.g. an electrode. The plate has a surface charge density \(-\sigma\), is immersed in a solution with dielectric permittivity \(\epsilon_\mathrm{w}\), and salt concentration of 0.1 M. The ions are monovalent, i.e. their charge is \(+e\) or \(-e\), with \(e\) the elementary charge. A schematic picture is shown in Fig. 7.1.
Fig. 7.1 Schematic depiction of a distribution of ions near a flat surface with negative charge.¶
Express the law of Gauss
\[ \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_\mathrm{w}} \]in terms of the electric potential \(\psi\), with \(\vec{E} = -\vec{\nabla} \psi\). This law is referred to as Poisson’s law. The symbol \(\rho\) stands for the charge density.
What is the potential energy of a positive ion, and what of a negative ion, at a distance \(z\) from the plate? (Express the answer in terms of the electric potential \(\psi(z)\).)
What would be the density of ions as a function of \(z\), in terms of the potential \(\psi(z)\), if one would use the Boltzmann distribution? (This is actually a terribly difficult question, which has not completely been solved after a century of literature. A couple of hints will be given: Remember that the Boltzmann distribution is
\[ P_n = \frac{1}{Z}e^{-\beta E_n} \]where \(n\) is the state of the system, and \(E_n\) the energy of that state. If the particles would not interact with each other, then the density of positive ions would be
(7.1)¶\[ \rho_+(z) = \rho_{+0} e^{-\beta V_+(z)} \]with \(\rho_+(z)\) in units of number per volume, \(V_+(z)\) the potential energy of a positive ion at position \(z\), and \(\rho_{+0}\) the density far away from the plate, where \(V_+(z)\) is set to zero. Of course, the ions do interact with each other, and that makes the question very difficult. A more appropriate distribution would be
\[ \rho_+(z) = \rho_{+0} e^{-\beta (V_\mathrm{plate}(z) + V_\mathrm{int}(z))} \]where \(V_\mathrm{plate}(z)\) is the potential energy of an ion, caused by the plate, and \(V_\mathrm{int}(z)\) is the average potential energy from the interaction between an ion at position \(z\) and all the other ions. This second term is very difficult to calculate, without making assumptions. For now, we will assume that we can use (7.1), with \(V_+(z)\) as found in the previous question.)
What would be the charge density at position \(z\)?
Argue why the constants \(\rho_{+0} = \rho_{-0}\). Call \(\rho_{+0}\equiv\rho_0\).
Express Poisson’s law in terms of \(\rho_+(z)\) and \(\rho_-(z)\), and derive the second order differential equation for \(\psi(z)\)
(7.2)¶\[ \psi''(z) = \frac{2e\rho_0}{\epsilon_\mathrm{w}}\sinh{\beta e \psi(z)} \]using the relation \(2\sinh{x}=e^x - e^{-x}\).
Rewrite the equation in terms of the variable \(\phi(z)\equiv\beta e \psi(z)\) (which can be interpreted as the potential energy of a positive ion at position \(z\) in units of \(k_\mathrm{B}T\)) in the form:
(7.3)¶\[ \phi''(z) = \kappa^2 \sinh{\phi(z)} \]This equation is referred to as the “Poisson-Boltzmann equation”, because it combines the Poisson equation (for the electric potential) with the Boltzmann distribution (for the ions). The Boltzmann distribution of the ions is approximated here.
What is \(\kappa\) in terms of the system parameters?
If \(\phi < 1\) one can approximate \(\sinh{\phi}\approx \phi\). Find the general solution of the differential equation for small \(\phi<1\).
Use the boundary conditions to find the solution specific to this system. Hint: make sure that the electric potential goes to zero for large \(z\) and that the electric field at \(z=0\) obeys \(E_0 = \frac{\sigma}{\epsilon_\mathrm{w}}\).
Sketch this function, and find the distance over which the electric potential drops by a factor \(e^{-1}\). This distance is known as the “Debye screening length” named after Pieter Debye.
Sketch the density of positive and negative ions near the plate. Hint: you can use the assumption that \(\phi\ll1\).
Calling this distance \(\lambda_\mathrm{D} \equiv \kappa^{-1}\), what is the screening length in water, with a concentration of 100 mM sodium chloride? And what for 1 mM? Hint: to convert molar density to number density, one simply has to multiply by Avogadro’s number \(6.022\cdot10^{23}\). It should be roughly between 0.1 and 100 nm, in case the calculated number would fall outside this interval.
In this specific case, there is also an analytical solution to equation (7.3), but the solution is only different from the one you calculated if \(\phi>1\), which can only occur close to the plate, say \(z < \lambda_\mathrm{D}\). Even though we made a convenient assumption in part (c), the expression for the screening length that you calculated is still a very useful estimate for how far electric fields extend in electrolyte solutions, and how far charged particles such as proteins can interact with each other. What happens within that distance can be very complicated, especially when the electrolytes interact more strongly, such as Ca\(^{2+}\) ions and ions in other solutions with a smaller dielectric constant. Also, it takes some time for the ions to screen a charged object, because they have to diffuse over some distance (of the order of a few nanoseconds at 0.1 M). Therefore, quickly varying fields can penetrate deeper into an electrolyte solution, and can also be disruptive for cell membranes. This phenomenon is known as electroporation. And there are many more interesting effects that rely on the response of electrolytes, that the body uses for many purposes. But that is a story for another time…