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CHEMISTRY provides some of the most well-defined examples of chaos.
The dynamics of chemical reactions can show the same kind of periodic and
chaotic behaviour that appears in population dynamics in biology. But the
transition from order to chaos is easier to study because we can readily
control the conditions of the experiments. The results are often visually
dramatic, with sudden colour changes – of course, the timescales are much
shorter.

The equations that describe how fast a chemical reaction proceeds are
nonlinear. For a chemical reaction to happen at all, molecules have to approach
each other, so the more molecules there are the faster the reaction goes.
The rate of reaction, therefore, depends on the concentrations of the reacting
molecules, normally raised to some power (the square of the concentration,
for example, if two molecules are involved in the reaction). Sometimes,
the reaction rate also depends on the concentrations of the molecules formed
from the reaction. These molecules may speed up the reaction, so acting
as autocatalysts, resulting in positive feedback.

Combustion, or oxidation, provides a good example of positive feedback.
In this case, it is the heat produced by the reaction that ever-increasingly
speeds it up. Everyone knows that above a certain temperature objects can
suddenly burst into flame, burn furiously and then, when the heat produced
drops off, the flames can suddenly cease. Sometimes, the reaction can choose
between two stable routes. For instance, carbon monoxide will burn in oxygen,
in the presence of a platinum catalyst, to give carbon dioxide. (This is
the way catalytic convertors in car exhausts work.) The reaction varies
with the catalyst’s temperature to give two branches: one corresponding
to high reaction rates, the other to low rates. Over some range of temperature,
these branches overlap, leading to two possible rates for identical experimental
conditions. The reaction can also oscillate between the two routes, responding
to minute changes in the experimental parameters. In similar reactions,
chaos can ensue – as we shall see later.

Another kind of chemistry where the rate can increase or decrease dramatically
and sometimes discontinuously in response to small smooth changes in experimental
conditions are the intriguing ‘clock’ reactions. An example is the Landolt
reaction in which dissolved iodate (IO₃–) and iodide (I–) ions
react together to form iodine (I2). This product does not survive long,
however, because it oxidises the other main ingredient of the reaction,
bisulphite ion (HSO₃2–) to form sulphate (SO42–) and is itself
reduced back to (more) iodide. This provides positive feedback; the rate
of the first reaction increases as the concentration of iodide increases.
As the reaction proceeds further, the bisulphite keeps the concentration
of iodine low, until the bisulphite is almost all consumed. The moment that
the bisulphite is exhausted there is a rapid build up of iodine. If we add
starch, as an indicator for iodine (starch and iodine form a deep blue compound),
we can see what happens. Initially, the solution is colourless, and stays
so for a while, then suddenly goes blue as the iodine forms. If bisulphite
is in excess, the high concentration of iodine is only transient, and the
solution ‘blinks’ blue and then turns colourless again.

Positive feedback can also cause chemical reactions to oscillate spectacularly.
The famous Belousov-Zhabotinskii, or B-Z, reaction not only shows oscillations
but also displays chaos under certain conditions. Although chemists have
studied its intricate dynamics for many years, they still do not fully understand
the underlying mechanism.

Thirty years ago, Boris Belousov, who was at the USSR Ministry of Health
in Moscow, noticed quite by chance that mixing a cocktail of chemicals –
sulphuric acid, potassium bromate, cerium sulphate and malonic acid – produced
oscillations in the concentrations of bromide (initially present as an impurity
but also produced in the reaction) and cerium ions. But poor Belousov had
trouble convincing other chemists that the periodic behaviour was a genuine
phenomenon; they blamed inefficient mixing of the reactants. The problem
was that chemists assumed that such behaviour – where a reaction couldn’t
make up its mind which way to go, refusing to settle down to an energetically
stable state – contravened the second law of thermodynamics.

When Belousov submitted his work to an academic journal for publication,
he was told that his ‘supposedly discovered discovery was impossible’ and
that he would have to do a lot more work before the journal could accept
the paper. Six years later, after much extra work and revision, the paper
was rejected as ‘too long’. Thoroughly disheartened, Belousov vowed never
to publish, although he was persuaded to produce a short report which appeared
in an obscure conference proceedings on radiation medicine.

Some years later, Anatol Zhabotinskii at Moscow State University learnt
of the curious reaction and, through careful research, was able to show
that the reaction was genuine. The work was then published. Now there are
many volumes on such clock reactions. Belousov, alas, did not live to see
his reaction accepted, nor to collect the 1980 Lenin prize awarded to him,
with Zhabotinskii and three others.

Basically, the B-Z reaction is a more complicated version of the Landolt
reaction. It involves oxidising an organic compound instead of bisulphite,
this time with potassium bromate and bromide in the presence of a catalyst
(see Box 1 for a recipe and details of the underlying chemistry). The catalyst
is usually a mixture of cerium ions in two oxidation states, cerium(III)
and cerium(IV), and an iron compound called ferroin. This acts as a visual
indicator for the reaction, being magenta when the iron is in its reduced
state (Fe2+) and blue when it is oxidised (Fe3+). Figure 2 shows a typical
B-Z mixture at different times during a period of about 80 seconds, oscillating
in composition and thus colour.

Chaos in the swing of chemistry

The simplest way of carrying out the B-Z reaction is to operate under
‘batch’ conditions. Here we just mix the reacting materials in a beaker
at the start of the reaction and watch what happens. This produces long
trains of perhaps 100 oscillations. This approach has important disadvantages.
Eventually, the reactants get used up, so the reaction is proceeding against
a continuously, if only slowly, changing background. Each oscillation will,
therefore, be slightly different from its predecessor. Eventually, the oscillations
will stop completely as the reaction approaches its end.

If we want to study the dynamics in any detail, we obviously want to
keep the reaction going as long as possible by maintaining it far from equilibrium.
To do this, we drive the reaction by continually pumping in fresh reactants,
into what is called a continuous flow reactor (with a corresponding outflow
of products to maintain a constant volume). Flow reactors are ideal for
studying autocatalytic reactions. In a series of experiments, we might vary
the concentration of each of the reactants flowing in, or the rate at which
the solutions are being pumped through the reactor (the flow rate).

Under these conditions, the B-Z reaction can show immense complexity,
and we can study the potential routes, through periodic oscillations to
chaotic dynamics. At low rates of flow, the oscillations resemble those
shown in Figure 2 from a batch reactor – simple, large-amplitude, low-frequency
waves. At high flow rates, the waves have a much smaller amplitude and higher
frequency. If we increase the pumping rate sufficiently, the oscillations
may become damped out, so obtaining a steady reaction. It is between these
extremes of flow that we find the most interesting behaviour. We can, for
example, obtain the kind of period-doubling found in fluid flows and biological
populations, as other articles in this series have described (see Figure
5). But more significant is the route to chaos through ‘mixed-mode’ oscillations.
The steps up in complexity can be neatly represented by a so-called ‘Devil’s
staircase’ – a fractal object with chaos in between its treads . Mixed-mode
oscillations consist of large and small peaks, and the Devil’s staircase
plots how the fraction of small peaks varies with the experimental conditions.

The evidence that chaotic patterns appear in such experiments now seems
incontrovertible. Researchers have found ‘strange attractors’ and other
phenomena typical of chaotic dynamics described in other articles in this
series. There is still an active debate as to where this chaos really comes
from. The question some chemists are asking is whether the chaos arises
solely from the chemistry. No experiment is perfectly controlled; the inefficient
mixing might be causing the chaos. The pumps used in flow reactors impose
a small oscillatory pulse of their own, which we can minimise but not eliminate.

The chemical mechanism for the B-Z reaction is now well established.
Richard Field, Endre Koros and Richard Noyes at the University of Oregon
developed a scheme in 1974, now called the Oregonator scheme, which has
explained the details of the reaction very well. Modelling the reaction
on a computer, however, has raised doubts – at least in the minds of some
chemists – about whether the B-Z reaction can show genuine chaotic dynamics.
Computations for the flow systems have reproduced many of the most complex
patterns seen in experiments, but they do not predict chaos convincingly.
Some chemists have suggested that the reaction is always trying to be periodic,
but that external factors turn this into ‘noise’.

Researchers are hotly debating chaos in the B-Z system. Different research
groups are re-running each other’s computations, often not just drawing
different conclusions but obtaining completely different results. At this
stage, it is probably wisest to go back to the chemistry of combustion and
see if we can find chaos there.

As I hinted earlier, combustion can produce oscillations in reaction
rate. The simplest combustion reaction is the oxidation of hydrogen to produce
water. At low pressures and temperatures of between 700 and 800 K, mixtures
of hydrgen and oxygen ignite spontaneously. In a batch system, such as a
closed bulb, this ignition is a one-off event because all the fuel is consumed.
In a flow system, there is a fresh supply of reactants. After an ignition,
the water formed is pumped out and the vessels refilled with hydrogen and
oxygen, and so the process can start again. In this way, the ignition starts
to oscillate. The water formed inhibits the reaction, so providing negative
feedback and preventing a steady flame from being established. At the lowest
temperature in this range, the oscillations have a large amplitude and a
long period. At higher temperatures, we encounter oscillations with a small
amplitude and a high frequency. As with the B-Z reaction, KK the transition
between the two kinds of oscillation at intermediate temperatures can give
rise to a complexsequence of mixed-mode patterns. We still do not know ifany
of these patterns are chaotic but many of the warningsigns are there.

Oscillations are also common in the burning of the most familiar fuels,
the hydrocarbons. An important intermediate in the oxidation is acetaldehyde
(ethanal, CH₃CHO). This shows complex ignitions that can lead
to engine ‘knock’ in cars. Computer modelling suggests that the dynamics
are extremely sensitive to the initial conditions of the experiment, with
chaos appearing when the temperature of the experiment is changed by only
a fraction of a degree. At the moment we cannot test these predictions because
they are beyond the limits of control in equipment available today.

The dynamic subtleties of such industrially significant reactions can
have economic consequences. The chemical industry uses metals as catalysts
in many important processes, for example; catalysts can speed up a chemical
reactionmany thousandfold. Using them efficiently can save a lot ofmoney.
Specific sites on the metal’s crystalline surface act astemplates for the
reacting molecules, bringing them closetogether, so lowering the energy
of the reaction and making ithappen more easily. Usually the catalyst works
better whenprepared as a fine powder of tiny crystals, say between 4 and10
nanometres across (a nanometre is a millionth of amillimetre). When spread
evenly onto an inert support,such as zeolite, they provide the maximum surface
area onwhich to catalyse the reaction.

The interesting thing about this set-up is that although the crystals
are separate from each other, they do not behave independently. How one
metal site behaves affects those close to it. The metal sites can exist
in either of two different crystalline forms: one is chemically active so
can catalyse the reaction, the other is not. When the catalyst is working,
each site repeatedly changes from one form to another.

The rate at which this change happens at any one site depends on the
state of its neighbours. It may be that the heat produced by a reaction
at an active site passes on to a neighbouring inactive one, and there causes
a ‘phase transition’ to the active form. As you might expect, the reaction
rate for the whole catalyst depends at any moment on the number of active
sites operating.

It also won’t surprise you to learn that this kind of cooperative behaviour
can lead to chaotic dynamics. To try and model what is going on, we use
a computer technique call cellular automata. This approach, where a simple
set of rules model evolving complex behaviour, is used in many areas of
research dealing with cooperative but potentially chaotic systems (see ‘The
life and times of cellular automata’, Greg Wilson, New ÐÓ°ÉÔ­´´, 8 October
1988).

To see how it works, think of a chessboard with the squares representing
individual catalytic sites. A given square is coupled to its nearest neighbours
– the four squares sharing an edge with it. We represent the state of the
site – active or inactive – by a whole number, i. If i is zero, the site
is inactive, otherwise it is active. To play the cellular automaton ‘game’,
we must specify a set of rules that determine how i for each square changes
from one moment to the next. This means we have to establish how the rate
at which sites change from one state to another depends on what proportion
of sites areactive or inactive. When the value of i reaches some predetermined
value, the site becomes inactive at the time step, withi set equal to zero.

Catalysis on a chessboard

Figure 5 shows some of the possible responses of the automaton. At certain
values of i, the fraction of active sites oscillates in a distinctly irregular
way that is reminiscent of what we find experimentally. In other cases,
the waveforms become much simpler. The fractional activity of the catalyst
is one feature of the system, but we can also wonder how that activity is
distributed around the zeolite chessboard. This turns out to be neither
random nor uniform. Instead, distinct spatial patterns can emerge. If we
ascribe a colour to the value of i on each site, then we get the spectacular
spirals shown in Figure 5. The rings propagate outwards with time, giving
‘target’ patterns, and the spirals unwind. We see similar patterns in the
unstirred B-Z mixtures.

Oscillations and chaos in inorganic or gas-phase chemical reactions
are interesting but it would be even more exciting to find chaos in biochemical
systems. As Robert May described in his article ‘The chaotic rhythms of
life’ (18 November), there are many natural rhythms in living organisms:
the heartbeat and the 24-hour ‘body clock’ of circadian rhythms, and so
on, where feedback is important, so giving the potential for chaos. So we
would expect to find similar behaviour in the underlying biochemistry.

A well-established and much studied oscillation is that inthe glycolytic
pathway for digesting glucose. Part of thisprocess involves repeatedly adding
and removing hydrogenfrom an important ‘coenzyme’ involved in the reaction
– nicotinamide adenine dinucleotide, mercifully known as NAD, or NADH when
hydrogenated. A standard way of detecting NADH is to tag it with a chemical
marker that fluoresces in ultraviolet light. In extracts from yeast cells,
the emission of fluorescent light oscillates with a period of about five
minutes. Biochemists understand the mechanism quite well and have modelled
it mathematically. The process involves an ‘allosteric’ enzyme, which consists
of several subunits, each having its own ‘active site’ where a particular
substrate molecule binds and then reacts. The binding of a substrate at
one site may influence what happens at others, either activating their ability
to catalyse a chemical reaction or inhibiting it. In glycolysis, one of
the molecules produced by the reaction activates the sites.

Biochemical oscillators also appear in the nervous system. Transmission
in nerves is a very complex process and there is a lot of scope for feedback
mechanisms. Take one important reaction involved in the so-called secondary
messenger network. In neurons, calcium ions combine with a protein called
calmodulin. At low intracellular concentrations of calcium, the ions combine
with calmodulin to form a complex that goes to produce a molecule called
cyclic-AMP. This causes the calcium ions to be released from the complex.
Eventually, at high concentrations of free calcium ions, the enzyme stops
working. The same concentration of calcium ions can either activate or inhibit
the reaction, depending on the system’s previous history.

Although biochemists are very aware of these kinds of complex feedback
mechanisms, many chemists find it difficult to accept that even quite simple
autocatalytic reactions can show chaotic behaviour. Chemists nearly always
assume that reactions proceed under conditions of thermodynamic equilibrium
directly towards a state where the products of the reaction have a lower
energy than the starting materials. They do not even expect concentrations
of intermediate products to oscillate back and forth. But as the Nobel laureate,
Ilya Prigogine, from the Institute of Chemical Physics in Brussels, pointed
out in 1968, many reactions happen in conditions far from thermodynamic
equilibrium, producing the kind of behaviour that Belousov first noticed.
Prigogine and his group were some of the first chemists to publicise the
connection between chaos and nonequilibrium thermodynamics.

Living systems are certainly far from equilibrium, so we should certainly
expect to find plenty of chaotic chemistry there. Chemists, especially those
who study the details of organic and inorganic biochemical reactions, need
to shake off some of their traditional views and accept that, as in other
areas of science, there are also limits to predictability in chemical dynamics.

* * *

1: The amazing Belousov-Zhabotinskii reaction

HERE is a tried and tested recipe for this oscillatory reaction:

500 millilitres of sulphuric acid (1 molar)

14.30 grams malonic acid

5.22 grams potassium bromate

0.548 grams ammonium ceric nitrate

1-2 millilitres of ferroin (0.025 molar)

(Make the ferroin by dissolving 1.485 grams of 1,10-phenanthroline and
0.685 grams of hydrated ferrous sulphate in 100 millilitres of water.) Stir
the mixture continuously. The resulting oscillations between blue and magenta
with a period of about 1 minute will last for several hours.

If you pour some of the mixture as a thin layer into a Petri dish, beautiful
patterns, with blue concentric circles, called targets, on a red background,
will develop.

We can split the reaction into three overall processes (see Figure 1).
In process A, bromate ions oxidise bromide ions to produce bromine.

1. BrO₃ – + Br- + 2H+ –> HBrO₂ + HOBr 2. HBrO₂
+ Br- + H+ –> 2HOBr 3. HOBr + Br- + H+ –> Br2 + H₂O

As the concentration of bromide ions decreases, so does the rate of
step 2. The bromate ions then compete for reaction with the hypobromous
acid (HBrO₂) and switches the system to process B. In this part
of the reaction, cerium oxidises from oxidation state (III) to oxidation
state (IV). This gives the colour change from red to blue.

4. BrO₃– + HBrO₂ + H+ –> 2BrO₂ + H₂O
5. BrO₂ + Ce(III) + H+ –> HBrO₂ + Ce(IV) 6. 2HBrO₂
–> BrO₃ + HOBr + H+

Because two BrO₂ radicals are produced in step 4, and each
reacts rapidly to form an HBrO₂ molecule, this part of the reaction
constitutes an autocatalytic cycle. The autocatalysis causes the rate of
this process to increase very quickly once it has switched on, so red changes
rapidly to blue. The growth in the concentration of HBrO₂ is
limited by step 6. The switch between processes A and B will occur when
the rates of steps 2 and 4 are approximately equal.

The final stage, process C, must regenerate the bromide ion and reduce
the catalyst back to its lower oxidation state. We do not understand this
part of the reaction but we can use the following representation. Malonic
acid (MA) reacts with bromine to give bromomalonic acid (BrMA). If this
is then oxidised by the cerium(IV), we will regain bromide and cerium(III).
The oxidised form of the catalyst can also react directly with malonic acid,
so we may get fewer than one bromide ion per cerium(III) ion produced.

So we describe process C as:

7. MA + Br2 –> BrMA + Br- + H+ 8. 2Ce(IV) + MA + BrMA –> fBr- + 2Ce(III)
and other products,

where f is known as the ‘stoichiometric factor’. In the simplest computer
analyses, f is assumed to be a constant, say f ( 1.

In modelling used to match complex oscillations, we attempt to allow
f (the number of bromide ions produced as two cerium ions are reduced) to
be a function of the instantaneous concentrations of other species, such
as HOBr. Process C sees the blue change to red and resets the chemical clock
for the next oscillation.

* * *

2: Climbing the Devil’s chemical staircase

IN THE B-Z reaction, the oscillations in colour changes can be extremely
complicated. The transition in waveform from simple, large to simple, small
amplitudes, is by no means straightforward. An almost unlimited number of
‘mixed-mode’ oscillatory patterns exists.

These complex waveforms have differing numbers of large and small waves
in each complete period. Figure 3 shows some examples. To help to catalogue
the various periodicities, we can label each wave from as an LS pattern,
where L is the number of large excursions and S the number of small peaks
in a full repeating unit. Thus the patterns in Figure 3 are 2x, 28 and 14.
The first two could also be denoted 1112 and 1414 respectively.

Another quantity, related to this notation, is called the firing number,
N. We can define this as the ratio of the number of small peaks to the total
number of peaks per cycle: N = s(L + s). The simple, large amplitude oscillation
at low flow-rate, the 10 form, has N = 0; at the high flow-rate end, the
small amplitude peaks, 01, haveN = 1.

If we start with the 10 pattern and slowly increase the flow rate, the
first of the mixed-mode oscillations encountered is usually that with one
large peak and one small peak: a 11 with N = 0.5. This form will exist for
a finite range of flow rate, and will be succeeded by other patterns. We
may discover a sequence in which the repeating unit gains one extra small
peak at each stage, so we see 12, 13, 14 and so on – with N jumping through
0.66, 0.75, 0.8 respectively.

If the firing number is plotted against the flow rate, as shown in Figure
4, the result is a series of steps which form a ‘Devil’s staircase’. The
idea of adding one extra small peak to produce a new pattern is given a
firmer mathematical basis through ‘Farey arithmetic’. A Farey sum of two
rational numbers P/Q and R/S is defined by (+):

P Q P + R (+) = Q S Q +
S

Thus if we consider that our two basic waveforms are the 1 0 and the
0/1 (+) 1/1 = 1/2, corresponding to a 1 1 the one large-one small pattern.
Between this and the 0 1, the Farey sum is 2/3, or one large-two small.
Continuing in this way, we generate all the steps in the staircase. The
sequence of mixed-mode patterns formed by taking these Farey sums from the
1 0 and 0 1 waveforms is called a ‘concatenation’ of these states.

There is yet further detail. Examining the border between any two of
the mixed-mode forms reveals more concatenations. Taking 1 2 and 1 3 states,
for example, researchers have found the following patterns between them
(in order of increasing flow-rate): 1 2, (1 2)2 1 3; 1 2 1 3; 1 2(1 3)2;
1 2(1 3)3; 1 2(1 3)4; 1 2(1 3)5; 1 3. The notation (1 2)2 1 3 means one
large-two small, one large-two small, one large-three small per complete
cycle. The firing numbers are, respectively, 2/3, 7/1, 5/7, 8/11, 11/15,
14/19, 17/23, and 3/4. This sequence follows the rules of Farey arithmetic
and gives a smaller Devil’s staircase between the steps at N = 2/3 and 3/4
in Figure 4.

Similar sequences are found between each of the main treads of the staircase,
with the oscillations becoming increasingly more complex at every level
of magnification. More than this, some experiments showthat the oscillations
can become so complexthat they lose their periodicity. Such aperiodic, or
chaotic responses, have no repeating unit.

Stephen Scott is a lecturer in physical chemistry at Leeds University.
He does experimental, theoretical and computational studies of chemical
chaos.

Further reading: Oscillations and Travelling Waves in Chemical Systems,
R. J. Field and M. Burger, editors, Wiley, New York, 1985; Molecules, Dynamics
and Life, Agnessa Babloyantz, Wiley, 1986; Chaos, A. V. Holden, editor,
Manchester University Press, 1986. Oscillations and Instabilities, P. Gray
and S. K. Scott, Oxford University Press, to be published early in 1990.