Chemical Oscillations & Waves: The Belousov-Zhabotinsky Reaction


Course Outline:

This semester the course considered the BZ reaction in a closed reactor (glass beaker) with continuous stirring (spatially homogeneous reaction). We performed the experiment and modeled it with a system of ordinary differential equations. First, the students learned how to use the Law of Mass Action to derive ordinary differential equation models of chemical reactions. Then, nonlinear-dynamics theoretical tools for the analysis of the resulting finite-dimensional dynamical system of ordinary differential equations were employed to analytically study the resulting models. Also, the students employed potentiometric methods to directly measure the concentrations of the important components in this reaction using special electrodes and the LabPro equipment; the experimental  results were compared to the  theoretical model result.

Also, the BZ reaction in Petri dishes (spatially inhomogeneous reaction) was considered. The students performed the experiment and modeled it with a system of parabolic partial differential equations. Some theoretical tools for the analysis of the resulting infinite-dimensional dynamical system comprised of reaction-diffusion partial differential equations were presented, and some numerical tools to solve such models of the experiment were employed. Also, a video camera was used to record the evolution of target patterns and spiral waves in an attempt to compare the observed chemical wave speed with that obtained through theory and computational experiments.

Course Objectives: 
  • To learn how differential equations arise in the modeling of chemical reactions.
  • To learn how to measure quantities of interest while a chemical reaction is occuring.
  • To learn how useful the Belousov-Zhabotinskii reaction is in diverse scientific areas.
  • To see how one can employ mathematical models to simulate experiments.
  • To see how one can employ mathematical analysis to guide experiments.
  • To gain more experience in writing a scientific report and constructing a public presentation of scientific results.


Experimental Work:

A) Pattern Formation in the Unstirred Belousov-Zhabotinsky reaction: We tested the iSight camera (thank you Prof. Muratov) setup to capture time-lapse video of pattern formation (chemical waves) when the BZ reaction takes place in a Petri dish. The software used was an evaluation copy of BTV Pro. Here is a picture of the kit and of the setup:




It turned out that the iSight camera was picking up a 60 Hz variation in the illumination amplitude. So after some initial runs of the experiment, and the taking of some images (see below), this camera was abandoned in favor of a digital camera (e.g., Canon A85) which could be controlled from software installed on a PC (in the case of the A85 the software in question is RemoteCapture accessible from ImageBrowser). See below for more info on the settings used and other issues that arose in this case.

Here's a gif animation of 55 captures spaced 30 seconds apart (5 frames per capture are averaged). The image capture was done with the iSight camera and the Petri dish was placed on a white sheet of paper. Lightning was provided from the room's overhead fluorescent lights (this was our first ever try of this experiment).



Following we see two images captured with the iSight camera. The Petri dish was on the lightbox and what's shown below are the best images of a sequence of a few hundred. The variation of the background illumination due to the 60 Hz effect described above was very deleterious. The iSight setup was abandonded after this run. One can discern spiral waves in these images.




A word of caution: the Petri dish that comes with the kit is not suitable as it has a very uneven bottom. You can see from the images above that the center area appears more illuminated than the area close to the rim; this is because the dish becomes shallower towards the center. We attributed the development of patterns predominantly near the rim to this manufacturing imperfection. We tried to create target patterns in the center of the dish but we were not successful. Thus, the supplied Petri dish was put aside in favor of a plastic square Petri dish with a perfectly flat bottom. We employed the square Petri dish in the following way: A drop of 0.1 % solution of Triton-X was added to the solution prior to its introduction into the dish and regular dish soap was spread with a paper towel on the inside of the dish cover to eliminate fogging. With this dish we have not observed any bubbles, and the uniformity of depth keeps the solution in an excited state (red) for many tens of seconds. The dishes have a grid etched on the outside of the bottom hence facilitating the use of a silver wire to initiate target patterns at the exact center of the dish. The dish being square also helps when the experiment is simulated on a square domain with no-flux boundary conditions (makes it easier to convince a doubting audience of the correspondence between modeling and experiment without having to explain much). Here are two images captured with the Canon A85; They show a pacemaker center that produced three bands of oxidation and then died out (the group of bands then expanded out at a constant speed with the region behind the bands returning to the reduced state):




The images above were taken 49 seconds apart. NOTE: When the dish cover is in place the digital camera (when set to auto-focus) likes to focus on its reflection off the dish cover top face hence the stuff happening 4 mm below the cover tend to be out of focus; one solution is to manually focus the camera with the cover off and then to place the cover, the other solution is to simply not cover the dish. The following graphs show the result of post-processing a number of images from this sequence, and from another sequence where one band of oxidation moved out from the center which did not produce any more waves. ImageJ was used (along with the Radial Profile plugin) to extract information from the relevant images:




Obviously, the fronts move with constant speed. In both graphs above, a low value for the integrated intensity corresponds to the red state of the solution (reduced) while the peaks correspond to the blue state of the solution (oxidized, see 2D images further up). In all our experiments, the mixed solution almost always undergoes 1.5 bulk oscillations prior to being poured into the dish (red-->blue-->red). Once in the dish, the swirling/jostling involved in centering the dish under the camera produces another 1.5 bulk oscillations (red-->blue-->red).

Finally, near the end of a particularly good run of target patterns appearing all over the place, it was observed that a particular  pacemaker site dominated the whole domain (all other pacemakers disappeared). The following animated gif file shows a bifurcation of the frequency of the pacemaker. We have no idea how to investigate this mathematically...if you, the reader, can help us then send us an e-mail. The following short loop is cropped out of a larger sequence of larger-in-size images:



B) Temporal Oscillations in the Well-stirred Belousov-Zhabotinsky reaction: Here are some images of the setup and of the oscillations observed thus far. In the bottom row images the time from blue to blue is about 25 seconds. NOTE: The Weiss Bromide-sensitive and Platinum electrodes are connected to the Vernier LabPro interface (not shown in these shots) through Vernier Electrode Amplifiers.








No chemical potentials were measured for the solution shown above. Below we show some pictures of our first attempt to measure the catalyst concentration (Pt electrode) in the well-stirred BZ reaction:






In the above 2X2 panel, the top row shows oscillations in the Cerium catalyzed BZ reaction (color oscillates between clear and yellow); the bottom row shows the color oscillation after adding 1 mL of Ferroin (color now oscillates between  red-brown  and blue-green). The graph below shows the electrode potential measurements before and after the addition of the Ferroin in a run with the same recipe as in the pictures above but with the Bromide electrode present:


We also tried a recipe due to Tyson (no KBr used). For this one we only measured the cerrium ions:




The Figure above shows long-time measurements of the concentrations in the Tyson recipe. As the reaction drifts in phase space the measurements indicate the presence of intermittency.

Numerical Simulations:

A) Pattern Formation in the Unstirred Belousov-Zhabotinsky reaction:

Using the Mathematica package Reaction-Diffusion lab the students attempted to simulate the BZ pattern formation experiment. The Figure below shows simulation results obtained by solving the Tyson-Fife 2D reduction of the Oregonator with no-flux boundary conditions on a square domain. The picture on the left shows a fast-rising oxidation front, propagating in a reduced medium (black), which relaxes back to a state that is not as reduced as ahead of the front.


The experimental image below resembles the above-left simulated image.



The following image shows an experimental determination of the wave speed of the first four circular waves emitted from a pacemaker center. The initial front is faster than subsequent fronts in accordance with the theory presented by Tyson and Fife. The slope of the lines represents the wavespeed in units of mm/sec.



B) Temporal Oscillations in the Well-stirred Belousov-Zhabotinsky reaction:

  The Figures below show the computed (left) and the measured (right) Bromide concentration for Tyson's recipe.




The Figures below show the measured induction periods for Bromide (left) and Cerium (right) in Tyson's recipe.




Below, the Figure on the left shows the  measured limit cycle between Bromide and Cerium while the Figure on the right shows a reconstruction using only the measured Cerium potential.