The Self-Made Tapestry Phần 3 pot

The idea that chemical reactions can develop travelling waves goes back a long waybefore, even, the
theory of oscillating reactions (which, as we saw, started with Lotka in 1910). At a meeting of German
chemists in Dresden in 1906, Robert Luther, director of the Physical Chemistry Laboratory in Leipzig,
presented a paper on the discovery and analysis of propagating chemical wavefronts in autocatalytic
reactions. Sceptics were apparently quelled by Luther's demonstration of the phenomenon before their
very eyeshe showed chemical waves in a reaction between oxalic acid and permanganate ions, projected
onto a screen in front of the audience.
Luther suggested that the waves arose from a competition between an autocatalytic reaction and the
process of diffusion that transports the chemical reagents through the reaction medium. Diffusion is a
random processmolecules of the reacting molecules are buffeted from all directions by collisions with
molecules of the surrounding solvent (generally water), and as a result they execute a convoluted,
meandering path often likened to a drunkard's walk. Despite this randomness, the molecules do actually
get somewhere rather than just meandering a little around their initial positionsbut the direction they
take is random, and the distance travelled from some initial location increases only rather slowly as time
progresses. (Whereas the distance covered by walking along a straight path at constant speed increases
in direct proportion to the time elapsed, the distance travelled by a random walker is proportional to the
square root of the elapsed time.) Random walks owing to diffusion were much studied at the beginning
of the century, notably by Albert Einstein.
When a chemical reaction is conducted under conditions where the concentrations are not maintained
uniformly throughout the medium by vigorous mixing, diffusion becomes important, since it limits the
rate at which a reagent that has become used up in one region can be replenished from elsewhere to
sustain further reaction. This is particularly important for autocatalytic reactions, since they can use up a
reagent locally at an extremely rapid rate. If diffusion cannot keep pace with this, the reaction runs into
problems. This is precisely the situation that I described earlieralthough not quite in these termsin the
vicinity of a wavefront in the BZ reaction. The inadequacies of diffusional transport create the
refractory period in the medium just behind an advancing wavefront, where the reaction has exhausted
itself but has not yet been replenished with fresh reagents. The poorly mixed BZ reaction is thus an
example of a so-called reaction-diffusion system, which is now clearly recognized as one of the most
fertile generic pattern-forming systems that we know of.
After Luther's pioneering studies, the theory of reaction-diffusion systems was placed on a firm
mathematical footing by the eminent population biologist Ronald Fisher and by the Russian

mathematician Andrei Kolmogoroff and co-workers, both of whom published seminal works in 1937.
Fisher was interested


Page 59
in reaction-diffusion processes for modelling the spread of an advantageous gene in a population, not
with their manifestation in chemistrya curious repetition of Volterra's assimilation of Lotka's ideas on
oscillating chemical reactions into mathematical biology earlier in the century. It is almost as if
chemists were for decades unwilling to face up to the existence of these complex and surprising
phenomena in their own field!
All the same, studies of waves in chemical media were conducted in parallel with, but independently
from, work on oscillatory reactions since the beginning of the century. In 1900 the German physical
chemist Wilhelm Ostwald described travelling pulses in an electro-chemical system. When he used a
zinc needle to prick the dark coating of oxidized iron on the surface of an iron wire immersed in acid,
Ostwald saw a colour change that propagated away from the point of contact at high speeds. From the
1920s onwards, many researchers studied this simple system as an analogue of nerve impulses (which
are also propagating electro-chemical waves), and in the early 1960s Jin-Ichi Nagumo and co-workers
in Tokyo observed spiral waves on the surface of a two-dimensional grid of iron wire subjected to this
treatment. But this work, published in Japanese, met the fate so common for studies that are not
reported in the English languageit was ignored in the West, until Zhabotinsky's efforts had established
the significance of this sort of wave activity.
The ripples spread
The BZ reaction is by no means unique: several other chemical mixtures share the same general
features of autocatalysis, feedback and competing reactions that lead to excitable and oscillatory
behaviour. It has been seen too in many biochemical processes, including, rather pleasingly, the
glycolytic cycle of metabolism that Belousov had first set out to emulate. Similar effects crop up in
some corrosion and combustion reactions. When these processes take place in poorly mixed conditions,
spatio-temporal patterns can arise whose forms are attractively diverse.
Fig. 3.7
Oscillations in the reaction of carbon

monoxide and oxygen on a platinum surface.
The reaction produces carbon dioxide.

Fig. 3.8
Target (a) and spiral (b) waves in the reaction of carbon monoxide and oxygen
on platinum. The images are all several tenths of a millimetre across.
(Photos: Gerhard Ertl, Fritz Haber Institute, Berlin.)
One of the functions of a catalytic converter in automobiles is to reduce emissions of carbon monoxide
(CO), a poisonous gas, in the exhaust fumes. This is done by combining CO with oxygen gas in the
converter to create carbon dioxide (CO
2
), a reaction that is


Page 60
speeded up by the use of a metal catalyst consisting of a mixture of rhodium and platinum. The reaction
takes place on the metal surface, where the chemical bonds in the reactant molecules are broken or
loosened up. So the reaction between CO and oxygen on a platinum surface is of considerable
technological interest. There is no obvious mechanism for autocatalysis here, howeverthe product is
simply CO
2
, which is not then involved in subsequent reactions.
So it was a surprise to Gerhard Ertl and colleagues at the Fritz Haber Institute in Berlin when they
found oscillatory behaviour in the rate of this reaction in 1985 (Fig. 3.7). And when in the early 1990s
the Berlin group developed a new kind of microscope to look at the way that the CO and oxygen were
distributed on the surface, they saw spiral and target patterns just like those of the BZ reaction, albeit
just a fraction of a millimetre across (Fig. 3.8). The bright regions in this figure correspond to parts of
the metal surface covered with CO molecules, and the dark regions are richer in oxygen atoms. Ertl's
team deduced that the molecules of CO that became stuck to the metal surface were altering its
structure, and thereby its catalytic behaviour, in a way that introduces feedback into this apparently

simple reaction.
Fig. 3.9
(a) The atomic structure of the 1 × 1 surface phase of platinum.
(b) In a vacuum, this surface will rearrange itself to the 1 × 2
reconstruction.
Platinum metal is a crystal: its atoms are packed together in a regular array like oranges on a fruit stall.
On a clean platinum surface exposed by cutting through the metal, the arrangement of atoms depends
on the angle at which the cut is made; for one particular cleavage plane, the surface looks like that in
Fig. 3.9a. This is called the {110} surface, and the arrangement of surface atoms is termed the (1 × 1)
phase. In a vacuum, the top-most atoms of a freshly exposed platinum (1 × 1) surface will
spontaneously shift their positions to create a different surface structure with a lower surface energy.
This is called the (1 × 2) phase, and has a 'missing' row of surface atoms (Fig. 3.9b). The rearrangement
process is called a surface reconstruction.

If CO molecules become attached to the reconstructed (1 × 2) surface of platinum, the balance of
energies gets shifted around, and the original (1 × 1) phase becomes more favourable. This means that,
as the reaction between CO and oxygen atoms on the platinum {110} surface proceeds, the surface does
not remain passive but shifts its structure between the (1 × 2) and (1 × 1) phases, depending on the
amount of CO on the surface.
Now the point is that these two surface phases have different catalytic abilities: the (1 × 1) phase is
considerably better at speeding up the reaction with oxygen than is the (1 × 2) phase. We can now see
the possibility of some subtle and complex interactions, which can give rise to feedback. The more the
bare (1 × 2) surface becomes covered in CO, the greater the extent of reconstruction to the (1 × 1) phase
and the more the catalytic potential of the metal is enhanced. But as the reaction proceeds, the CO gets
converted to CO
2
, which departs from the surface and leaves behind a bare (1 × 1) surface. On its own,
this prefers to revert to the reconstructed (1 × 2) phase.
Gerhard Ertl, David King at Cambridge University, and their co-workers have devised a six-step
reaction scheme that is akin to the Oregonator of the BZ reaction, which incorporates these various

processes for reactions on platinum surfaces. It includes an autocatalytic process in which the reaction
between CO and oxygen on the (1 × 1) surface creates new 'bare' catalytic sites. They have found that
this scheme produces oscillatory behaviour of the various reaction parameters, such as the rate of CO
2

formation or the surface coverage of CO (Fig. 3.10). Like the Oregonator, the process jumps between
two branchesessentially a low-reactivity branch involving the (1 × 2) surface and a high-reactivity
branch involving the (1 × 1) surfacewith the autocatalytic steps providing a mechanism for rapid
switching between the branches. It is easy to see that sites of non-uniformity in these surface reactions
can act as the centres for the formation of travelling waves like those shown in Fig. 3.8.
Several other metal-catalysed surface reactions are now known to show oscillatory behaviour. One
difference between these essentially two-dimensional processes and those in flat dishes of the BZ
mixture is that for the latter the medium is isotropic: it looks the same in all directions. For surface
reactions taking place on metal crystals, on the other hand, all directions are not the same, because the
metal atoms are lined up in a regular checkerboard-like array. This means that the


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Fig. 3.10
The oscillations in the surface reaction
of CO and oxygen can be reproduced by
a theoretical model that includes the autocatalytic
processes. Oscillations are seen in both
the rate of reaction (a) and the amount of
carbon monoxide on the surface (b).
ability of the reacting molecules to move about can be similarly anisotropic (direction-dependent). It is
for this reason that the target and spiral patterns in Fig. 3.8 are elliptical rather than circularthe speed of
the chemical wave fronts differs in different directions. In extreme cases, this anisotropy means that the
symmetry of the underlying metal crystal surface can leave itself imprinted on the spatial patterns that
arise. For example, Ertl's colleague Ronald Imbihl has seen square travelling waves in the reaction of

nitric oxide and hydrogen on a rhodium surface, an echo of the square symmetry of the metal crystal
surface (Fig. 3.11).

Fig. 3.11
The spiral waves of the oscillatory reaction of nitric oxide and
hydrogen on a rhodium surface have a square appearance
which derives from the square symmetry of the underlying atomic
lattice. (Photo: Ronald Imbihl, Fritz Haber Institute, Berlin.)
Rock art
If you are a rock collector, the target patterns in Plate 5 may look familiar. They are reminiscent of the
stunning concentric bands displayed by agates (Plate 6). Agates are formed when water from rain or
snow permeates through fissures in cooling basaltic lava, dissolving metal ions as it goes. Once the
body of lava has cooled sufficiently, the ions precipitate out of the mineral-rich solution as agates. This
is a process of crystallization occurring far from equilibrium, and so we should perhaps not be too
surprised that it can lead to pattern formation.
Periodic patterns due to non-equilibrium crystallization and precipitation have a history that predates
the discovery of oscillating chemical reactions. In 1896, the German chemist Raphael Eduard Liesegang
performed experiments in which he reacted silver nitrate with potassium chromate in a gelatin gel. This
reaction generates insoluble silver chromate, which precipitates as a dark deposit. In solution, this
precipitate would all be flushed out at once, as the two salts would mix very quickly. But in a gel, the
mixing is much slower, limited by the slow diffusion of the ions. Liesegang saturated the gel with
potassium chromate, and then allowed a drop of silver nitrate solution to diffuse through it. He found
that the dark precipitate appears in a series of rings behind a reaction front that advances through the
reaction vessel. Many other chemical reactions that generate an insoluble compound show the same
behaviour when limited by diffusion through a gel (Fig. 3.12), and you can try it for yourself using the
recipe in Appendix 4.

Liesegang's experiments are not nearly so obtuse as they might sound. The precipitation of silver metal
and salts in gelatin gels became a subject of intense interest in the late nineteenth century owing to its
relevance to photography: black-and-white photographic emulsion is essentially a gel containing a

silver salt, which is converted to a dark, fine precipitate of silver metal on exposure to light. Indeed,
Liesegang's father and grandfather were both early pioneers of photography. Raphael Liesegang himself
was by all accounts a remarkable, not to say eccentric, character, with interests every bit as catholic as
D'Arcy Thompson's. He


Page 62
wrote about the possibility of television in 1891 and, as well as his work on photography, he pursued
research on bacteriology, chromosomes, plant physiology, neurology, anaesthesia and the disease of
silicosis.
Fig. 3.12
Liesegang bands, a signature of oscillatory
precipitation at an advancing diffusion front.
Here the bands are produced by cobalt hydroxide
as hydroxide ions diffuse down a column of
cobalt-laden gelatin. (Photo: R. Sultan, American
University of Beirut.)
Liesegang's rings (only later was the reaction performed in cylindrical test tubes, so that the
precipitation fronts appeared instead as a series of band-like disks) captured the imagination of many of
the leading scientists of the time, including Lord Rayleigh, J.J.Thompson and Wilhelm Ostwald. Some
early enthusiasts around the turn of the century suggested that in the bands and rings one might be
seeing a simplified version of the stripes of tigers and zebras or the patterns on butterfly wings. In this,
remarked one critic in 1931, 'enthusiasm has been carried beyond the bounds of prudence'. But as we
will see in the next chapter, on one level at least such scepticism is misplaced (although given what was
known at the time about chemical pattern formationnext to nothingwe can't really regard these
speculations as anything more than a lucky guess).

As the gel medium of the Liesegang process evidently makes diffusion a critical aspect, it's not hard to
guess
from the preceding discussion that a reaction-diffusion process lies behind the pattern formation. But

while this is no doubt the case, the phenomenon is not fully understood even today. One idea, which
was first proposed by Ostwald a year after Liesegang published his findings, is based on the proposition
that the reaction product does not precipitate until the solution becomes supersaturated above some
critical threshold concentration. Precipitation can potentially occur as soon as the concentration of the
reaction product becomes too high for the solution to bearas soon as it becomes supersaturated. But in
practice, particles of the insoluble product will grow large enough to precipitate only after they have
first attained a certain critical size. This is one of the basic tenets of the theory of crystal growth, which
Ostwald helped to establish. If the product molecules cannot cluster into these 'critical nuclei', the
solution can become highly supersaturated.
Ostwald suggested that in Liesegang's experiments, formation of the critical nuclei was slowed down by
the fact that the reaction product diffuses only slowly through the gel. The reaction is all the while
increasing the degree of supersaturation, however, and once this exceeds a threshold, the concentration
of the product is at last great enough everywhere for nucleation to occur. Then the nuclei grow rapidly,
accreting the reaction product from the solution around it and precipitating as a dark band. Precipitation
leaves the reaction front depleted in the product, and so precipitation stopsand it takes some time for it
to build up again to the critical threshold, by which time the front has moved forward. This cycle of
nucleation-precipitation-depletion dumps a train of bands in the wake of the front.
Ostwald's theory was refined in 1923 by K. Jablczynski, who showed that it could be used to predict the
spacing between successive bands. Jablczynski's spacing law states that the ratio of the positions of two
consecutive bands (defined relative to, say, the first band) approaches a constant value as the number of
bands gets larger. The theory was further refined by S. Prager in 1956, who turned it into a well-defined
mathematical model; but unfortunately Prager's model predicted that the bands will be infinitely
narrow, which is certainly not what is observed. Peter Ortoleva at the University of Indiana and co-
workers made further improvements to the theory in the 1980s to overcome this shortcoming. More
recently, Bastien Chopard from the University of Geneva and colleagues have devised a cellular-
automaton model which takes into account some of


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Fig. 3.13

Liesegang bands generated in a cellular automaton model of a precipitation-diffusion process. (Image: Bastien
Chopard, University of Geneva.)
the microscopic processes that control the diffusion, nucleation and precipitation of the reacting species
in Liesegang systems. Their model is able to produce precipitation bands (Fig. 3.13) which obey
Jablczynski's spacing law.
But the trouble is that the band spacings in real experiments don't by any means always observe this law:
several different relationships have been reported, and there seems to be no general law that applies to
all Liesegang-type experiments. Other explanations for the banding have been put forward, many of
which generally involve processes that take place after nucleation has occurred. But with such a
diversity of observations, it isn't hard to find results that will fit most models, while preventing
unambiguous discrimination between them.

Fig. 3.14
Liesegang banding at very small length scales in iris quartz
gives it an iridescent appearance. (a) The bands here are
about seven micrometres apart, and are caused by periodic
differences in the concentration of defects in the crystal structure.
(b) At a larger scale, thin bands of quartz alternate with
thicker bands of chalcedony. The bands run from top to bottom;
the horizontal striations have a different origin, caused by the fibrous
structure of the mineral. The image here is about 2.5 mm across.
Banding is also evident on scales of about a centimetre or so
(Plate 6). (Photos: Peter Heaney, Princeton University.)

Liesegang realized that the banded patterns he saw were similar to those found in certain rocks. There is
now good reason to suppose that many banded rock formations do indeed arise from cyclic precipitation
as mineral-rich water infiltrates a porous rock and reacts to form an insoluble product. Amongst the
mineral patterns that have been attributed to Liesegang-type processes are the bands seen in some iron
oxide minerals, the wood-grain texture of cherts, the striations of a mineral called zebrastone, and
perhaps most familiarly of all, the bands of agates. And Ostwald's idea is just one of a whole class of

models involving particle transport, nucleation and precipitation that have been put forward to explain
such formations. To take just one example:geologists Peter Heaney from Princeton University and
Andrew Davis from the University of Chicago showed in 1995 that Liesegang precipitation-diffusion
cycles can account for the iridescence of iris agates. Whereas the colour banding shown in Plate 6 is
perhaps the most spectacular feature of these and other agates, the iridescence comes from a periodic
banded structure on a scale too small to see by eye. The bands vary in width from about a tenth of a
micrometre to several micrometres


Page 64
(Fig. 3.14a), and this banded 'grating' scatters visible light (because the light has wavelengths of
comparable dimensions), creating the iridescent effect. Heaney and Davis showed that these bands
correspond to differences in the crystal structure of the mineral: regions of highly crystalline quartz
alternate with regions in which a high concentration of defects disrupt the regularity of the crystalline
lattice. They postulated that the defective regions formed by the initial linking together of soluble
silicate ions into long chains, which precipitate to give a poorly crystalline form of the mineral
chalcedony. The highly crystalline regions, meanwhile, are formed by precipitation of individual
silicate ions as quartz. This latter is possible when the concentration of silicate ions at the crystallization
front is low. But because quartz precipitates slowly, silicate ions diffuse towards the front more rapidly
than they are removed by precipitation, and eventually the concentration builds up to a degree that
allows their linking into chains. Then the more rapid precipitation of chalcedony takes precedence, until
this depletes the silicate solution once again.
Heaney and Davis pointed out that, while this mechanism could account for the iris banding, the agates
are in fact patterned on several length scales. There are also oscillations between defect-rich chalcedony
and defect-poor quartz with wavelengths of several hundred micrometres (Fig. 3.14b) and of a
centimetre or so (Plate 6), suggesting that there are several hierarchical mechanisms for oscillatory
patterning at play here. This kind of hierarchical repetition of pattern over several length scales is a
feature of some of the patterns that we will encounter in later chapters.
Burn up
I have already mentioned that combustion processes are autocatalytic; but normally this doesn't produce

anything more interesting than a big bang, because there is nothing to keep the process in check. When,
however, an explosive combustion process such as the burning of hydrogen in air is conducted under
experimentally well controlled conditions, oscillations in the reaction rate can arise. The overall
reaction looks simple enough: two molecules of hydrogen combine with one of oxygen to form two
molecules of water:
2H
2
+ O
2
→ 2H
2
O. (3.6)

But the detailed evolution of this reaction is rather complicated, involving short-lived, reactive
intermediate species such as lone hydrogen and oxygen atoms and the hydroxyl free radical, OH. In an
autocatalytic process, three molecules of hydrogen and one of oxygen can react with a lone hydrogen
atom to produce two molecules of water and three hydrogen atomsthus the products of this process
represent a multiplication of the reactants. This autocatalytic process arises because hydrogen atoms are
less reactive, and so hang around for longer, than oxygen or hydroxyl radicals. When the reaction of
hydrogen and oxygen is carried out in a stream of flowing gases in a CSTR, the result of these
autocatalytic processes is an oscillatory variation in the burning rate, which shows up as a rise and fall
of the temperature generated in the combustion flame (Fig. 3.15). In effect, the mixture of gases
repeatedly ignites and then subsides into an unreactive state. The reaction between carbon monoxide
and oxygenthe same reaction as that studied by Ertl's group, but this time carried out by burning the free
gases rather than bringing them together on a catalytic metal surfacealso shows oscillatory behaviour in
a CSTR.
Fig. 3.15
Oscillations in the combustion of hydrogen in
a flow reactor, revealed by variations in the
temperature. (After: Scott 1992.)

Can these combustion processes also generate spatial patterns if they are not well mixed? Well, it has
been known for a long time that when a hydrocarbon fuel such as butane is burnt in a flame under
carefully controlled conditions, the flame can become very nonuniform, separating into a number of
distinct cells. The cells are bright regions, separated by darker boundaries where the temperature is
lower and there is less emission of light by the excited gas molecules. These cellular flames were first
reported by A. Smithells and H. Ingel in 1892, who described a flame that separated into petal-like
segments that rotated around the flame's axis. George Markstein made a careful study of such flames in
the 1950s, and in 1977 G.I. Sivashinsky showed theoretically that the cell patterns could be the result of


Page 65
a reaction-diffusion process. A burning flame requires both fuel (generally a hydrocarbon like butane)
and oxygenthe former cannot burn without the latter. But molecules of these two compounds travel
(diffuse) through the gas mixture at different ratesthe oxygen molecules are lighter and so diffuse more
rapidly. The combustion reaction can be sustained only as long as the rates of diffusion of both species
are sufficient to

Fig. 3.16
Cellular patterns in a shallow cylindrical flame seen from above. The temperature of the
flame is lower in the dark regions. (These dark regions are not truly dark to the eye; they are
simply a result of the limited dynamic range of the video tape on which the images were
recorded.) The cellular flames adopt ordered states. Here I show a sequence of ordered states
of increasing complexity as the rate of gas flow in the flame is increased. (Photos: Michael
Gorman, University of Houston, Texas.)


Page 66
feed it. Spatial irregularities can arise when the oxygen in one region burns up the fuel more rapidly
than it is replenished by diffusion, and combustion cannot then continue until more fuel diffuses there.
Another theory was developed around much the same time, however, which held that flow effects in the

gas streams, not molecular diffusion, were responsible for cellular flames.
Sivashinsky showed that the diffusive mechanism could in principle produce ordered hexagonal
arrangements of cells. But the cells that Markstein and others had reported were irregularly shaped and
were in constant, disordered motion, breaking up and coalescing. It was not until 1994 that Michael
Gorman and Mohamed El-Hamdi of the University of Houston in Texas and Kay Robbins of the
University of Texas at San Antonio found the first clear examples of regular patterns in cellular flames.
They studied flames of a butane-oxygen mixture passing through a stainless steel plug in a cylindrical
chamber. The flame appeared as a luminous disk just half a millimetre thick. As the researchers
increased the rate of gas flow through the disk, they saw an initially uniform disk-shaped flame break
up into a ring of cells (Fig. 3.16). When the cells first appeared, there were four of them; but as the flow
rate was increased, a fifth and then a sixth cell appeared. Increasing the rate still further created a new
inner ring of cells, which multiplied from one to six and then spawned a third ring (Fig. 3.16j). An
hexagonal array of cells appeared as the last ordered structure; for higher flow rates the cells began to
exhibit rapid, chaotic motion in which no recognizable pattern was apparent.
The overall number of cells in each concentric ring stays steady for a given flow rate, but their positions
keep altering. In some experiments, each ring of cells seemed to rotate independently in a kind of
hopping motion: the cells would stay put for a while, then the whole ring would abruptly rotate like a
gear wheel shifting position. Under other conditions the motion was more chaotic: the ordered states
might disappear intermittently into randomly shaped cells, and then reappear (Fig. 3.17). Sometimes the
innermost cells took on a spiral shape, which circulated like a rotating yin-yang symbol.

Fig. 3.17
Under some conditions, ordering in
the cellular flames is intermittent,
being interrupted sporadically by the
appearance of more-random cell
arrangements. Time advances here from
(a) to (e) (Photos: Michael Gorman.)

The Texas researchers concluded that Sivashinksy's diffusion mechanism, not gas-flow effects, lies

behind the ordered patterns that they sawmainly because the latter mechanism was expected to produce
bigger cells than those observed. These ordered cellular flames represent an unusual kind of dynamic
reaction-diffusion pattern, however, because they are all unstable, undergoing intermittent
rearrangements via more chaotic states. On the other hand, Howard Pearlman of NASA's Lewis
Research Center and Paul Ronney of the University of Southern California have observed flame
patterns that look very much like the spiral waves of the BZ reaction, reinforcing the idea that they are
the products of a reaction-diffusion process.


Page 67
Fig. 3.18
As the flow rate of a BZ mixture in a continuous stirred-tank reactor is increased, the oscillations
double up, a phenomenon called periodic doubling (a). The limit cycle of the period-doubled oscillations
develops two loops (b).
Going wild
Another thing that this combustion process and the BZ reaction share in common is that gradual
changes in the flow rate of the reacting molecules do not induce correspondingly gradual changes in
behaviour; rather, there are abrupt jumps to a new mode of behaviour when a certain threshold is
reached. In the well-mixed BZ reaction conducted in a CSTR, I indicated that the oscillatory behaviour
defines a certain limit cycle that remained robust in the face of changes in, say, flow rate or initial
concentrations of the reagents. But this is true only up to a point. If the flow rate is increased far
enough, the colour-changing mixture suddenly starts to exhibit a new temporal pattern. It alternates
between blue and red, sure enough, but if we were to time the colour changesor better still, to measure
the rise and fall in concentration of one of the intermediates such as the bromide ionwe would find that
something new has happened. The switching now appears to have a double pulse (Fig. 3.18a). Plotted
as a limit cycle, this behaviour manifests itself as a double loop (Fig. 3.18b). The system has to traverse
both lobes before it repeats itself.
This is called a period-doubling bifurcation, and it was observed in the BZ reaction in the 1980s by J.C.
Roux and co-workers. 'Period-doubling' is obvious enoughthe system now has two stable pulses or
periods. 'Bifurcation' simply means that the stable, oscillating state of the system has forked into two,

with each state corresponding to a loop of the limit cycle.

I should point out that the initial oscillatory state of the BZ mixture in a CSTR is itself the product of a
bifurcation, because the flow rate has to reach a certain threshold before the indefinitely oscillating
colour change is stable at all; below this flow rate, the reaction will simply go through a series of
transient colour changes before settling down into an unchanging, uniform state. (Admittedly, this may
take some time, which is why 'clock' reactions like this still look temporarily like regular oscillators
even in a closed vessel.) This kind of abrupt transition from a stable, steady state to an oscillatory one
was first identified mathematically by the German Eberhard Hopf, long before anyone knew about
chemical oscillators. It is therefore called a Hopf bifurcation. Hopf bifurcations are a common source of
periodic motion from initially steady motionthe mathematicians Ian Stewart and Martin Golubitsky
describe them appealingly as the onset of a wobble.
It does not yet seem to be clear whether the transitions seen in cellular flames are indeed Hopf
bifurcations or some other kind of bifurcation. All the same, both these and the period-doubling jumps
seen in the BZ mixture in a CSTR share the characteristic that they just keep on coming as the flow rate
is increased, giving patterns of ever more complexity. In the latter case, a further increase in flow rate
induces another bifurcation into a limit cycle with four lobes, and then eight, and so forth. In cellular
flames, each jump adds more cells or even a new ring of cells. With each jump, the amount that the
flow rate has to be further increased to induce another bifurcation decreases: the jumps get closer and
closer together and the patterns become more and more


Page 68
complex (which is to say, of lower and lower symmetry). There eventually comes a point at which all
pretence of pattern is thrown to the winds and the system descends into chaos. For the BZ reactor, this
means that the oscillations in concentrations no longer show any sign of periodicity at allthey appear to
be random (Fig. 3.19). The cellular flames, on the other hand, dissolve into a random pattern of
irregular cells that is forever shifting.
Fig. 3.19
At high flow rates, the oscillations of the BZ

reaction become apparently randomthe system
becomes chaotic.
The route to chaotic behaviour through a series of period-doubling bifurcations is a common one, seen
in many diverse systems that exhibit chaos, including lasers and populations of predators and their prey
(Chapter 9). Very clear period-doubling bifurcations leading to chaotic oscillations have been seen in
the combustion of carbon monoxide and oxygen gases in a CSTR (Fig. 3.20). There are other ways for a
chemical system to become chaotic too, and Jack Hudson and colleagues at the University of Virginia
identified one such in the late 1970s. In studies of the BZ reaction in a CSTR at high flow rates, they
saw 'mixed-mode' oscillations in which a single cycle involves both small- and large-amplitude
oscillations (Fig. 3.21a). Typically, each large-amplitude oscillation (in the concentration of, say,
bromide) is accompanied by a little train of small oscillationsperhaps just one, perhaps more. Under
some conditions these mixed sequences keep repeating regularly, but in other cases different mixed
modes may alternate with no apparent periodicity (Fig. 3.21b). Although these non-periodic states
satisfy all of the mathematical criteria for chaos (which distinguish them from purely random
processes), there was much debate initially about whether they were genuine examples of 'chemical
chaos' rather than effects induced by poor mixing in the experiments. But it is now clear that theoretical
models of oscillatory reactions (which don't have to suffer any experimental deficiencies) can gener-

Fig. 3.20
Period doubling and the transition to chaos in the reaction of carbon monoxide
and oxygen in a flow reactor. (After: Scott 1992.)
Fig. 3.21
Mixed-mode oscillations in the BZ reaction consist of a mixture of large-and
small-amplitude oscillations (a). Mixed modes may alternate apparently at random
(chaotically) when the flow rate is high (b).


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ate chaotic mixed modes, so this seems to be a bona fide example of a new route to chaos.
Rhythms of life

Chemical waves are not merely a curiosity conjured up in laboratories under highly specialized conditions.
For many living organisms, ourselves included, they are a matter of life and death.
Heart attacks are the leading cause of death in industrialized nations. The majority of these result from a
pathological condition of the heart called ventricular fibrillationa medical term which, roughly translated,
means that the heart forgets how to beat. Instead of acting in a coordinated manner to generate a regular
pumping motion, the tissues of a heart that has entered into ventricular fibrillation lose their ability to
execute large-scale coordinated contractions, and the heart appears to flutter feebly to no great effect, like
a frightened bird. During the onset of ventricular fibrillation, the heart enters a kind of behaviour called
cardiac arrhythmia, which, despite the name, actually denotes a new rhythmic activity in which the regular
beats of about one per second give way to rapid pulsations about five times faster. These eventually
dissolve into uncoordinated fibrillation, leading to sudden cardiac death. That eventual heart stoppage in
such cases is preceded by this frenzied activity was recognized as early as 1888, when J.A. MacWilliam
described the fateful events in colourful terms: 'The cardiac pump is thrown out of gear, and the last of its
vital energy is dissipated in a violent and prolonged turmoil of fruitless activity in the ventricular wall'.
These changes in heart activity can be seen in electrocardiograms, which record the change in electrical
voltage in a region of the heart tissue (Fig. 3.22).
How do cells in a healthy heart act in synchrony in the first place? Each heartbeat corresponds to a
travelling wave of electrical activity, which begins at a pacemaker region of the heart called the sinoatrial
node and travels throughout the heart tissue. At the front of this travelling wave, the electrical voltage
across the cell walls alters as electrically charged ions move from one side to the other. The wave of
electrical activity (which is akin to a nerve impulse) induces muscle contraction, causing the heart to pump
blood. Once the wavefront has passed, the cells become refractory (immune to a further pulse of electrical
activity), while they 'reset' their across-membrane voltages by redistributing the ions. Thus, heart tissue is
an excitable medium, and the heartbeat is induced by a spatio-temporal patterna travelling wavevery much
akin to that of the BZ reaction. This is one of the major reasons why the BZ reaction has attracted such
interest: scientists are interested in the patterns not only for their own sake (pretty as they are) but because
they might provide us with a model to help understand some aspects of heart behaviour.
It now seems clear that the fatal condition of ventricular fibrillation is associated with the initiation of
spiral waves in the heart. You can see from Plate 5 that spiral travelling waves in an excitable medium
tend to have a shorter periodicity than target waves (adjacent wavefronts are closer together). The

consequence of this is that, once they are created, spiral waves come to dominate over target waves,
because they 'jump in' to excite the medium more quickly. A clue to the role of spiral waves in ventricular
fibrillation (VF) is given by the fact that, when cardiac arrhythmia is initiated, the frequency of the heart's
oscillations increases.
It is now possible to see these lethal spiral waves directly in beating hearts. Early experiments involved