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2.5.1 General Remarks

In ionic conductors, the current is transported by ions moving around (and possibly electrons and holes, too). Electrical current transport via ions, or ions and electrons/holes, is found in:

  • Conducting liquids called electrolytes.
  • Ion conducting solids, also called solid electrolytes.

Ionic conductivity is important for many products:

  • Type I and type II batteries (i.e. regular and rechargeable).
  • Fuel cells.
  • Electrochromic windows and displays.
  • Solid state sensors, especially for reactive gases.

In contrast to purely electronic current transport, there is always a chemical reaction tied to the current flow that takes place wherever the ionic current is converted to an electronic current - i.e. at the contacts or electrodes. There may be, however, a measurable potential difference without current flow in ionic systems, and therefore applications not involving chemical reactions.

  • This is a big difference to current flow with electrons (or holes), where no chemical reaction is needed for current flow across contacts since "chemical reactions " simply means that the system changes with time.

If we look at the conductivity of solid ionic conductors, we look at the movement of ions in the crstal lattice - e.g. the movement (= diffusion) of O or H+ ions either as interstitials or as lattice ions.

  • In other words, we look at the diffusion of (ionized) atoms in some crstal lattice, described by a diffusion coefficient D.
  • Since a diffusion coefficient D and a mobility µ describe essentially the same thing, it is small wonder that they are closely correlated - by the Einstein-Smoluchowski relation (the link leads you to the semiconductor Hyperscript with a derivation of the equation).

\[\mu=\frac{\text{e}\cdot D}{\text{k}T}\]

  • The conductivity of a solid-state ionic conductor thus becomes

\[\sigma= e \cdot c \cdot \mu = \frac{e^2 \cdot c \cdot D}{kT}=\frac{e^2 \cdot c \cdot D_0}{kT}\cdot \text{exp-}\frac{H^m}{kT}\]

  • with the normal Arrhenius behaviour of the diffusion coefficient and Hm = migration enthalpy of an ion, carrying one elementary charge. In other words: we must expect complex and strongly temperature dependent behaviour; in particular if c is also a strong function of T.

Ionics is the topic of dedicated lecture courses, here we will only deal with two of the fundamental properties and equations - the Debye length and the Nernst equation - in a very simplified way.

  • The most general and most simple situation that we have to consider is a contact between two materials, at least one of which is a solid ionic conductor or solid electrolyte. Junctions with liquid electrolytes, while somewhat more complicated, essentially follow the same line of reasoning.

Since this involves that some kind of ion can move, or, in other words, diffuse in the solid electrolyte, the local concentration cof the mobile ion can respond to two types of driving  forces:

1. Concentration gradients, leading to particle currents jdiff (and, for particles with charge q, automatically to an electrical current jelect  =  q · jdiff)  given by Ficks laws

\[j_{\text{diff}}=-D\cdot \text{grad(c)}\]

  • With D = diffusion coefficient of the diffusing particle.

2. Electrical fields E, inducing electrical current according to Ohms law (or whatever current - voltage - characteristics applies to the particular case), e.g.

\[j_{\text{field}}=\sigma\cdot {\color{purple} E}=q\cdot c\cdot \mu \cdot \color{purple}{E}\]

  • With µ = mobility of the particle.

Both driving forces may be present simultaneously; the total current flow or voltage drop then results from the combined action of the two driving forces.

  • Note that in one equation the current is proportional to the gradient of the concentration whereas in the other equation the proportionality is to the concentration directly. This has immediate and far reaching consequences for all cases where in equilibrium the two components must cancel each other as we will see in the next sub-chapter.

In general, the two partial currents will not be zero and some net current flow is observed. Under equilibrium conditions, however, there is no net current, this requires that the partial currents either are all zero, or that they must have the same magnitude (and opposite signs), so that they cancel each other.

  • The equilibrium condition thus is

\[q\cdot j_{\text{diff}}=j_{\text{field}}\]

The importance of this equation cannot be over emphasized. It imposes some general conditions on the steady state concentration profile of the diffusing ion and thus the charge density. Knowing the charge density distribution, the potential distribution can be obtained with the Poisson equation, and this leads to the Debye length and Nernsts law which we will discuss in the next paragraphs.