Logarithmic Transients: High-Field Ionic Conduction

If a capacitor is discharged by a current that depends exponentially on the potential difference across the capacitor, the resulting transient is logarithmic rather than exponential. The discharge equation can be written:

where i_o and V_o are parameters with the dimensions of current and potential, and V* is the potential at which the capacitor is fully discharged. The solution of the equation is

V = V₁ - V_o log(1 + t/t)

t = CV_o/i₁

The data for the example were obtained from an experiment on a vanadium electrode immersed in an organic electrolyte with a low water concentration (vanadium oxide dissolves in water). The experiment consists of five steps. In the first, a constant anodic current is applied to the electrode to grow an oxide film at a uniform rate until a specified potential is reached. In the second, the current is reduced by a factor of two and made cathodic to convert the film to a reduced phase. In the third, the original current is reapplied to oxidize the film. The current remains constant through the fourth and fifth steps, the fourth converting the film back to the original oxide, and the fifth recovering the thickness lost during step three. The optical data from this experiment along with a model of the electrochemical processes are shown in the section on applications of ellipsometry. Here our object is to show how open-citcuit transients applied at specified intervals can provide information about high-field ionic transport processes. The ionic transport model itself is shown on the plot of the optical data. A potential-time plot for the experiment is shown below.

Java Source   Liberty Basic Source

The points at which transients were applied are shown in red for step one, yellow for step two, cyan for step three, and green for steps four (the first point) and five (the remaining three).

The analysis program lets you look at analysis plots similar to those in the capacitor tester for the 31 transients. You may choose to fit the initial region ("Init", points 2-9), the full transient ("Full", all points except the first), or full fit that includes dielectric constant variation ("Disp", see note below). The initial point is excluded because the potential includes the iR drop in the electrolyte. (The fit is extrapolated back to zero time to check the iR drop.) The transient is plotted in green on a linear time scale, and in cyan (with a fitted line in blue) on a logarithmic time scale. The scatter plot is in red. You will see that a good fit is obtained over the initial region, but a systematic error is observed over the full range unless "Disp" is chosen. The scatter plot for "Init" and "Disp" analyses uses 10-mV (3 lsb) scale divisions, and the scatter plot for "Full" uses 100-mV divisions. The values determined in the analysis are tabled beside the plot.

You may also choose to look at a plot of V_o or 1/C for the entire series with least-squares lines fitted to the results for steps one, two, and three. If it is the field in the film that controls the conduction process, V_o as well as 1/C should be proportional to film thickness.

Java Source   Liberty Basic Source

Note added 30 October 2011:

The original Applet did not have the "Disp" option, and offered no explanation for the systematic error in the "Full" analysis. When the anodic oxide is grown at constant current density, the field in the oxide is constant, and potential increases linearly with thickness. When the circuit is opened and the field decreases, the thickness of the film decreases slightly and its density increases. This in turn increases both the refractive index and the dielectric "constant". The changes are small (on the order of the thickness change) at optical frequency, but in high dielectric constant oxides the changes can be substantial at low frequency.

In terms of the displacement D (the electric field E multiplied by the permittivity), an open-circuit transient is one in which the displacement current and the ionic current sum to zero. If the ionic current is controlled by the effective field in the oxide, and if that field is proportional to the dielectric constant, D appears in both terms in the discharge equation, and the logarithmic time analysis can still be used. First the potential is transformed to be proportional to D. Here a linear variation of dielectric constant is assumed. The same vertical plotting scale is used, and a narrower fitting range results because the dielectric constant increases as E decreases. The logarithmic time scale displaces points horizontally as usual.

This note is overly terse, but I doubt many people will read it. To those that do, I apologize.

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