Today’s lecture was all about fitting models for the electrode data. I started by showing them how one could hand-sketch Bode plots, at least for RC and RL circuits. We did a hand plot and a gnuplot plot for the model with arbitrary values, showing the initial horizontal , the final horizontal , and the diagonal at .
In class I went through trying to do fits to data collected for stainless-steel electrodes, and showing how to debug various problems (it was all live-action plotting—I did not script my actions). The biggest problems were getting very bad fits (in one case from taking the log of the function but not the log of the data, in another case from having bad initial values) and singular matrices (mainly from having variables in the function that didn’t affect the fit, though in some cases from trying to fit complex models to real data without taking absolute value of the complex model).
It turns out that the standard R+(R||Z_C) model is very hard to fit to the data we collected for the stainless steel electrodes. The oxide coatings don’t leak much current, so we had no low-frequency plateau for estimating the parallel resistance from. I suggested making the parallel resistance infinite and using a simple R+Z_C serial connection. That can model the data well at high frequencies, where the change in |Z| is fairly small, but at low frequencies the model is poor.
I came up with a different model on the spur of the moment (not one I had ever tried before on electrode data): with a capacitor-like element having a smaller slope that the normal 1/f slope of a capacitor (about 0.6). This turned out to fit the data quite well. I don’t have a convincing physical explanation for the exponent α, but I suspect it has to do with diffusion times for ions near the surface of the electrode and depletion regions in the electrolyte.
In the new model, the R term probably corresponds to the bulk properties of the electrolyte solution and the term to the surface chemistry at the electrode, so 1/R should be proportional to the concentration of the NaCl, I think. I wonder whether students will get that result in their fits. I’m thinking that I should rewrite some of the book to incorporate this model.
I ended by trying to model some of the data collected by students that did not work well—they had a huge inductance uptick at high frequency (fitting nicely to something like a 3mH inductance). I’ve no idea how they got that data, as I saw their setup and they couldn’t have had more than a few µH of stray inductance. Other students had small upticks at the high frequencies that were almost certainly stray inductance, since moving the voltmeter leads to connect directly to the electrodes eliminated the uptick, which did not happen with the students whose data I tried modeling. I showed students how to model the uptick with an additional inductor, but I really don’t know what went wrong with the student data—I didn’t see any problems with their setup or recording, so I can only assume we all missed something.
Some of the students at least are getting the idea that modeling is not forcing your data to fit the theory in the book, but looking for regularities in the data.