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.
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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):
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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:
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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:
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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.