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 Page 61 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 [...]... 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... doing this: they perfume themselves The cells emit a chemical compound, called a chemoattractant, into the medium around them, much as animals emit pheromones to attract mates Other bacteria can sense how much of this chemical signal is coming their way from different directions, and they start to move in the direction where the concentration of the chemical rises most rapidlyin other words, they wriggle... ring of cells, which cluster into spots behind the advancing front when they attract one another through chemotactic signalling (Fig 3. 28a) The spot patterns become Page 74 Fig 3. 29 The patterns in E coli are formed as cells migrate outwards in a 'swarm ring' The ring breaks up into a series of spots, some of which remain immobile while other cells advance and reform the swarm ring (Photos: Elena Budrene,... measure of the local voltage The researchers saw spiral waves of electrical activity whose centres meandered over the surface of the heart (Fig 3. 23a) The electrocardiograms associated with this behaviour showed the uncoordinated oscillations characteristic of VF (Fig 3. 23b) Fig 3. 23 A spiral wave developing in a rabbit heart, traced out by monitoring voltage-dependent dyes (a), and the associated electrocardiogram... 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... 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. .. 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... equations that were known to describe the basic properties of heart activity In their simulations, the spirals were very clear (Fig 3. 24a and Plate 7), and when the model was adjusted to allow the tips of these waves to meander, the simulated electro-cardiograms were very similar to those seen in the real sheep hearts (Fig 3. 24b) How might spiral waves arise out of the regular travelling waves found in... 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... involved in a feedback loop that regulates the formation of more cAMP In the latter case, cAMP outside the cell interacts with another protein molecule in the cell membrane in such a way that the protein is stimulated into influencing the catalytic activity of the adenylate cyclase enzyme In this way, cAMP produced by the enzyme can enhance the rate at which further cAMP is formed This autocatalytic behaviour . 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 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 Page 61 Fig. 3. 10 The oscillations in the. 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