The Belousov–Zhabotinsky reaction · live simulation
Mix bromate, malonic acid, sulfuric acid and an iron indicator, and the solution flips from red to blue and back on a steady beat. Pour it into a thin layer and leave it alone, and the beat starts to travel. Waves cross the dish, wipe each other out, and wind themselves into spirals.
The dishes, the flask and the charts on this page are computed live in your browser from the equations chemists use to describe this reaction.
I · Moscow, 1951
Boris Belousov, a biochemist, wanted a test-tube model of the citric-acid cycle, the loop of reactions cells use to burn sugar. He dissolved citric acid, potassium bromate and a cerium salt in dilute sulfuric acid. The solution turned yellow, faded to clear, and turned yellow again. It kept doing this, on schedule, cycle after cycle.
Chemistry at the time said that a reaction runs downhill toward equilibrium and stops. A mixture that swings back and forth sounded like a perpetual motion machine. Belousov spent six years trying to publish and was turned down by editors who judged the result impossible. In 1959 a short abstract finally appeared, in a little-read collection of abstracts on radiation medicine.
In 1961 the graduate student Anatol Zhabotinsky took up the reaction. He replaced citric acid with malonic acid, and the reaction was soon run with ferroin, an iron complex that switches between red and blue, which made the beat easy to see. Western chemists heard about the work at a conference in Prague in 1968. Belousov died in 1970. In 1980 he was awarded the Lenin Prize, posthumously, along with Zhabotinsky and colleagues.
Six years of rejection. One short abstract in print. Seventeen years before most of the world heard about it.
The cast
| Role | Species | Typical |
|---|---|---|
| Oxidiser | bromate, BrO3− | 0.06 M |
| Fuel | malonic acid, CH2(COOH)2 | 0.02 M |
| Acid | sulfuric acid, H+ | 0.8 M |
| Catalyst & indicator | ferroin, Fe(phen)32+/3+ | ~1 mM |
Bromate, malonic acid and acid are the textbook Field–Noyes values the simulation uses. The ferroin amount is a typical recipe value.
II · Stirred
Stir the mixture and the whole flask acts as one clock. Each tick is a contest between two bromine species. Bromide ion, Br−, is the brake. Bromous acid, HBrO2, is the accelerator, and it makes more of itself. The flask, the traces and the reaction network below are all driven by the same live solution of the Oregonator model, using the textbook rate constants.
While bromide is plentiful it destroys HBrO2 as fast as it forms. The iron stays red. The bromide is slowly used up.
Once bromide falls below a critical level, about a micromole per litre here, HBrO2 doubles itself and oxidises the iron. The flask turns blue in seconds.
Oxidised iron reacts with malonic acid and its brominated form. The iron returns to red and fresh bromide is released, so the brake comes back on.
Each tick is one lap of this loop. The point creeps down the left branch of the folded amber curve while bromide holds HBrO2 down and the iron slowly returns to red. At the lower knee the brake fails and the point leaps across to the right branch. It climbs while the iron turns blue, then drops back once the reset has made enough bromide.
The stoichiometric factor f sets how much bromide each reset releases. Too little and the brake never catches; the flask stays blue. Too much and the brake never lets go; the flask stays red. In between, the steady state is unstable and the system has no choice but to cycle.
III · Unstirred
Pour the mixture into a shallow dish and stop stirring. Every spot is still a clock, but now neighbours can talk. A blue spot leaks HBrO2 a fraction of a millimetre into the red around it, pushes that patch past its threshold, and the patch fires too. The relay travels outward at a constant speed and never fades.
In a real dish you start one with a silver wire. Silver ions pull bromide out of solution as silver bromide, which releases the brake on the spot.
Water waves pass through each other. These cannot. Behind every front the medium is refractory: the iron is still oxidised and bromide is high, so nothing there can fire until it recovers. When two fronts meet, each runs into the other’s exhausted wake, and both stop dead.
A speck of dust or a scratch can make one spot fire on its own schedule, sending out rings like a target. Here two pacemakers compete, one firing every …, the other every …. Each collision line drifts toward the slower source, because the faster one gets more waves into the gap. Eventually its rings reach the slow pacemaker before it can fire, and the dish beats to one rhythm.
IV · Broken waves
Cut a front and its two loose ends cannot heal. Each end curls around itself and keeps turning, and a pair of spirals is born, one turning each way. Arthur Winfree photographed them in this reaction in 1972. A spiral needs no trigger. It turns faster than an ordinary pacemaker, so its waves take over the dish.
Try it yourself: drag across any wave on the dish.
Now each point is coloured by where it is in its own cycle. At the tip of every spiral, every colour meets, the way all time zones meet at the North Pole. These points are phase singularities. Each carries a sign, +1 or −1, set by its sense of rotation. They are born in pairs and die in pairs, so the total changes only when a tip runs into the wall.
So far each tip has traced a tight circle. Change the recipe, with more malonic acid relative to bromate so the fast and slow chemistry are less separated (larger ε) and a little background light, and the tip begins to wander along flower-shaped paths. This is meander, mapped out in the Oregonator by Jahnke and Winfree in 1991. The trails show where the tip has been.
V · Same equations, different tissue
Heart muscle is an excitable medium. Each cell fires once, stays refractory for a while, then recovers, and the electrical wave passes from cell to cell the way HBrO2 spreads through the dish. The sinus node, the heart’s pacemaker, starts each beat. The trace under the dish is a probe electrode.
The units now follow heart tissue: the same pattern, rescaled to a conduction speed of about 0.5 m/s and the 0.18 s spiral rotation measured in slices of dog and sheep ventricle. The dish uses the light-sensitive ruthenium catalyst for this act.
Block part of one beat and its broken ends curl into a pair of spirals, exactly as in the dish. Cardiologists call this figure-of-eight reentry. The spirals turn faster than the sinus node fires, so the pacemaker’s waves never get through. Rotating waves like these drive many ventricular tachycardias.
Break the waves in many places and many spirals compete. Fronts collide and break again, tips wander and annihilate, and the electrode records rapid, irregular activity. Many interacting spirals are one of the leading explanations for ventricular fibrillation, in which the heart quivers instead of pumping.
A defibrillator shock excites nearly the whole heart at once. With no resting tissue left to move into, every spiral dies, and the sinus node takes over again. Here a flash of light does the same job from the other direction: light makes the ruthenium catalyst produce bromide, which blocks excitation everywhere for a moment. When it fades, only the pacemaker is left.
VI · Free play
Every tool is unlocked. Change the recipe while the dish runs and watch the waves respond.
Below about φ = 0.006 the resting dish is itself oscillatory and the whole plate flashes on its own. Above about φ = 0.05 waves can no longer propagate.
VII · The fine print
Belousov’s editors were right to distrust anything that runs in circles, but this reaction never runs uphill. In a sealed dish the bromate and malonic acid are used up steadily, carbon dioxide bubbles off, and the free energy of the mixture falls the whole time. What oscillates are intermediates present in tiny amounts, riding on that one-way slide.
Close to equilibrium, oscillations are impossible. Far from it, a steady flow of free energy can hold up patterns in time and space. Ilya Prigogine called these dissipative structures and won the 1977 Nobel Prize in Chemistry for the theory. A sealed BZ dish ticks for an hour or so, then runs out of fuel and goes quiet.
Elsewhere
Any medium that can fire, rest and recover, with a way for neighbours to trigger each other, supports these waves. The pictures below are illustrations drawn in the style each field uses to image them.
Voltage-sensitive dye
Spiral waves of electrical activity drive some arrhythmias. In thin slices of ventricle they rotate about every 0.18 s.
Dark-field
Starving Dictyostelium cells relay pulses of cyclic AMP. The waves form targets and spirals that lead thousands of cells into one mound.
Calcium dye
Calcium waves inside a single Xenopus egg form spirals that annihilate when they collide, reported by Lechleiter, Clapham and colleagues in 1991.
Photoemission microscope
Carbon monoxide burning on platinum forms spirals tens of micrometres across, imaged by Gerhard Ertl’s group in 1990. Ertl won the 2007 Nobel Prize in Chemistry.
Model notes
u and x are scaled HBrO2, v and z the oxidised catalyst (the colour you see), y bromide, and φ bromide produced by light in the ruthenium-catalysed reaction. Bromide is fast enough to sit at its steady state, which reduces the Field–Noyes model to two variables.
The dish is a 512 × 512 grid inside a circular wall with no flux. The stiff HBrO2 kinetics are stepped with a backward-Euler Newton solve on the GPU, diffusion with an isotropic nine-point stencil, and the catalyst relaxes exactly. Spiral tips are found by measuring how far the (log u, v) phase winds around each grid square, then refined with a bilinear root-find.
The flask uses rate constants measured for the cerium-catalysed reaction; ferroin behaves similarly. Simplifications: the dish uses q = 0.002, larger than the textbook value, a common choice that keeps the equations tractable; real dishes also have convection, CO2 bubbles and oxygen effects, which are left out.
| Quantity | Dish | Flask |
|---|---|---|
| q | 0.002 | 7.6×10−5 |
| ε | 0.02 (lab: 0.02–0.09) | 9.9×10−3 |
| f | 1.4 | 1.0 (slider) |
| φ | 0.01 | – |
| Δx, Δt | 0.4, 0.005 | –, 5×10−4 |
| time unit τ0 = 1/(kc[MA]) | ≈ 50 s | |
| length unit √(Dτ0), D = 2×10−5 cm²/s | ≈ 0.32 mm | |
| dish diameter | ≈ 6.2 cm | |
Sources