The Self-Made Tapestry Phần 8 pdf

The researchers set out to construct a simple theoretical model of this process. To do so, they had to establish criteria for when avalanches started and stopped. These criteria are well explored for piles of grains, and you can see them for yourself by tipping up bowls of granulated sugar and long-grain rice until they undergo avalanches. First, smooth the surfaces of the materials so that they are both horizontal. Then slowly tilt the bowls until a layer of grains shears off and runs out in an avalanche. There is a critical angle, called the angle of maximum stability (θ m ), at which sliding takes place. Moreover, when the avalanche is over, the slope of the grains in the bowl will have decreased to a value for which it is stable. This is called the angle of repose (θ r ), and the slope always relaxes to this same angle (Fig. 8.10). Both of these avalanche angles depend on the grain shapeyou'll find that θ m for rice is larger than that for sugar, whereas granulated sugar, caster sugar and couscous (all with roughly spherical grains) all have a similar angle within the accuracy of this kitchen-table demonstration. You can see the same thing by letting a steady trickle of sugar pass through a hole in a bag so that it forms a heap on the table top. The heap grows steeper and steeper until eventually there is a miniature landslide. Thereafter, you'll find that, however much more sugar you add, the slope of the pile stays more or less constant as it grows, with little landslides making sure that this is so. The angle of the steady slope is the angle of repose. Fig. 8.11 The stratification that takes place when mixed grains are poured can be mimicked in a simple theoretical model in which the two grains have different shapes: square and rectangular. The model assumes that as the are poured, the grains stack up into colums, with all of the rectangular grains upright (a). Although this is a highly artificial assumption, it reporduces the effect of different grain shapes, which is the cause of the stratification. The angle of maximum stability θ m is such that the difference in height between one column and the next cannot exceed three times the width of the square grains; and the angle of repose θ r is equivalent to a height difference of two (b). If a new grain added to the top of the slope creates a slope greater than m , it tumbles from column to column until it finds a stable position (c). But if the grain has to go all the way to the foot of the pile (as in c), this implies that the slope is equal to θ m everywhere. The pile then undergoes a landslide to reduce the slope everywhere to θ r or less (d). (After: Makse et al. 1997.) It was quite by chance that Hernán Makse had decided to conduct his initial experiments with sand Page 208 and sugar, which have slightly different shapes and therefore slightly different angles of maximum stability and reposedifferent sizes alone wouldn't have given the stratification. So in the model that he and his colleagues developed, they tried crudely to mimic this difference in shape. They considered two types of grain: small square ones and larger rectangular ones. These were assumed to drop onto the pile so as to stack in columns (Fig. 8.11). This model seems highly artificialthe experimental grains are clearly not squares and rectangles, nor do they stack up in regular vertical columns. But it's only a rough first shot, aimed at capturing the essentials of the process. The heap was assigned characteristic angles of repose and maximum stability. When a grain drops onto the pile to create a local slope greater than θ m , it tumbles down from column to column until it finds a position for which the slope is less than or equal to θ m . But if a grain tumbles all the way to the bottom, which means that the slope everywhere is already equal to θ m then a landslide is considered to occur in the model: all the grains tumble, starting at the bottom, until the slope everywhere is reduced to the angle of repose θ r . Fig. 8.12 The model outlined above generates the same kind of stratification and segregation as seen experimentally. (Image: Hernán Makse, Schlumberger-Doll Research, Ridgefield, Connecticut.) Because the large grains are 'taller' and so more readily introduce a local slope greater than θ m , they tumble more readilyjust as in the experiments (remember that the large grains are less easily trapped on the slope). This accounts for the segregation of grains, with the larger ones at the bottom. The researchers found that all experiments showed this segregation when the grains were of different sizes. Stratificationstriped layersrequires something more, however. They found that this happened experimentally when the two types of grain not only have different sizes but also different angles of repose; that of the smaller grains being less steep than that of the larger grains. Because the particles in the model were not just of different sizes but also of different shapes, the model captures this feature of the experiments too. So when played out on the computer, it is able to produce piles that are both segregated and stratified (Fig. 8.12). The simple model, therefore, does a fair jobbut it may neglect some important factors such as dynamical effects of grain collisions. These may explain, for example, why the pouring rate is also critical to obtaining good stratification. Roll out the barrel Fig. 8.13 Avalances of grains in a rotating drum will mix different grains that are initially divided into two segments (a). As the drum turns, there is a succession of avalanches each time the slope exceeds the angle of maximum stability, transposing the dark wedges to the white wedges (b, c, d). If the drum is less than half-full (b), the wedges overlap, and the two types of grain eventually become fully mixed. If the durn is exactly half-full (c), the wedges do not overlap, so mixing takes place only within individual wedges. When it is more than half-full (d), there is a central core in which avalanches never take place, so this circular region never gets mixed. As bricklayers know well, an easy but generally effective way to mix two substances is to place them inside a rotating drum, like a cement mixer. But when the substances are powders, don't expect the obvious. This became clear to Julio Ottino and co-workers at Page 209 Northeastern University in Illinois when they tried to mix two types of salt, identical except for being dyed different colours, in this way (Fig. 8.13a). If the drum rotates slowly enough, the layer of granular material remains stationary until the drum tips it past its angle of repose, whereupon the top layer slides in an avalanche (Fig. 8.13b). This abruptly transports a wedge of grains from the top to the bottom of the slope. The drum meanwhile continues to rotate until another wedge slides. Fig. 8.14 14 The unmixed core is clearly visible in experiments. (Photo: Julio Ottino, Northwestern University, IIIinois.) Each time a wedge slides, the grains within it get scrambled (because they are identical apart from colour). So if the grains are initially divided into two compartments separated by a vertical boundary (Fig. 8.13a), they become gradually intermixed by avalanches. But are grains also transported between wedges? They are if the drum is less than half-full, because then successive wedges intersect one another (Fig. 8.13b). But when it is exactly half-full the wedges no longer overlap (Fig. 8.13c), and mixing occurs only within individual wedges. If the drum is more than half-full, something strange occurs. There is a region around the outer part of the drum where avalanches and mixing take place, but in the central region is a core of material that never slides (Fig. 8.13d). The initially segregated grains in this core therefore stay segregated even after the drum has rotated many times. This, Ottino and colleagues observed, leaves a central pristine region of rotating, unmixed grains, while the region outside becomes gradually mixed (Fig. 8.14). In theory, you could spin this cement mixer for ever without disturbing the core. Even if you start with a well-mixed concoction of different grains, it won't necessarily stay that way in a tumbling barrel. Engineers have known since 1939, when the effect was observed by Y. Oyama in Japan, that this process can cause grains to segregate out in a series of bands when the barrel is a long cylindrical tube (Fig. 8.15a). This happens if the grains have different angles of reposefor example, tiny glass balls (with θ r of 30°) will separate from sand (θ r of 36°). And in a rotating tube with a periodic change in width, so that it has a series of 'bellies' connected by a series of 'necks' grains will separate according to their size even if the angle of repose is the same for both (Fig. 8.15b). In the case shown here, small glass balls segregate into the bellies while large balls gather in the necks. Crucially, segregation in both uniform and bulging cylinders requires that the grains only partially fill the tube, so that there is a free surface across which grains can roll in a constant landslide. Fig. 8.15 (a)Grains of different shapes (and thus angles of repose) will segregate into bands when rotated inside a cylindrical tube. Here the dark bands are sand, and the light bands are glass balls. (b) In a tube with an undulating cross-section, a difference in size alone is enough to separate grains, which segregate into the necks and bellies (Photo: Joel Stavans, Weizmann Institute of Science, Rehovot.) Where I grew up on the Isle of Wight in the south of England, there is a place called Alum Bay that is famous for its multicoloured sands. Tourists are invited to fill glass cylindersmodels of the Needles lighthousewith stripes of these sands by carefully adding each colour in sequence. I very much doubt they would believe you if you were to suggest that they might get much the same result by mixing up all the sands and then rolling the tube! Page 210 Joel Stavans from the Weizmann Institute of Science in Israel and co-workers proposed, when they investigated this phenomenon in 1994, that it might be put to good use as a way of separating different kinds of grains in a mixture. It certainly beats doing the job by hand. Stavans and colleagues suggested that the banding results from the complex interplay between two properties of the grains: their different angles of repose and the differences in their frictional interactions with the edge of the tube. They built these differences into a model of the tumbling process, and found it predicted that the well-mixed state of the grains was unstable: small, chance variations in the relative amounts of the two grains would be amplified such that the imbalance would grow. A tiny excess of sand in one region, for instance, would enhance itself until that region contained only sand and no glass balls. But you might think that this would give rise to bands of random width, whereas the experiments showed bands whose widths were all more or less the same within a narrow range (Fig. 8.15a). In other words, a certain preferred length scale appears spontaneously in the pattern. The researchers pointed out that this is analogous to a phenomenon called spinodal decomposition, which takes place when a mixture of fluids is suddenly made immiscible (for example, by cooling the mixture to a temperature at which the two fluids separate). Spinodal decomposition takes place in quenched mixtures of molten metals: the two metals separate, as they freeze, into blobs of more or less uniform sizes. This too is driven by random fluctuations in concentrations of the two substances, which conspire to select a certain length scale. For the tube with bulges (Fig. 8.15b), the two types of glass balls have the same angle of repose, yet segregation still occurs. This is because the variation in width along the tube imposes a change in slope on the free surface of the tumbling grains, and the large balls roll down the slope more readily than the small ones, whichas we saw earlierare more easily trapped by bumps on the surface. Although it's not obvious without looking at the profile of the free surface, downhill carries the larger balls into the necks if the tube is more than half full but into the bellies if it is less than half full. Thus, shaking, tumbling or even simple pouring of granular media can cause a mixture of different grains to mix, unmix or form striking patterns. At present there is no general theory that allows us to predict which of these will take place for a given system: again, you don't know until you try it. Organized avalanches So the slope of a granular heap, fed with fresh grains from above, remains at the angle of repose. This is a dynamically stable shape, because it is maintained in the face of a constant throughput of energy and matter (grains flow onto and off the heap). It is, in fact, a non-equilibrium 'dissipative' structure (see p. 255). In the past decade, it has become clear that in this everyday structure, familiar to millers and quarrymen everywhere, there is an unguessed complexity. The mass and profile of such piles are subject to continual fluctuations, as landslides remove grains from their slopes. Studies of the way in which sand piles maintain their shape while executing these fluctuations have led to a whole new field of research in nonequilibrium science and pattern formation. In 1987, physicists Per Bak, Chao Tang and Kurt Wiesenfeld at Brookhaven National Laboratory on Long Island, New York, devised a model to describe the dynamics of a heaped pile of sand. They were led to do so not by the kind of practical concerns that would motivate an engineer, but because this system provided an easy-to-visualize model for studying a rather recondite question about the electronic behaviour of solids. In essence, their hypothetical model described a pile of sand grains with a well- defined angle of repose, to which new grains were gradually added. In the simplest version of the model, the sand pile was twodimensional and was shored up against a vertical wall at one end. (This is like the experimental system of Makse and colleagues described above, with the plastic plates so close together that the pile is only one gram thickand with the important provision that the friction between the walls and the grains is ignored.) Grains were added to this pile one by one at random points. The pile builds up unevenly, so that its slope varies from place to place (Fig. 8.16a). But nowhere can the slope exceed the angle of maximum stability, the critical slope above which an avalanche takes place. If a single additional particle tips the slope locally over this critical value, a landslide is induced which washes down the 'hillside' and reduces the slope everywhere to a belowcritical value (Fig. 8.16b). But here is the curious thing: in their model, Bak and colleagues found that a single grain can induce a land-slide of any magnitude. It might set only a few grains tumbling, or it might bring about a catastrophic sloughing of the entire pile. There is no way of telling which it will be. Page 211 Fig. 8.16 The slope of a granular heap varies locally from place to place (a). In this pile of mustard seeds, small variations in slope can be seen superimposed on a constant average gradient. When the slope approaches the angle of maximum stability, the addition of a single seed can trigger an avalanche (b). This avalanche can involve any number of grains, from just a few to the entire slope. Notice that only grains within the first few layers move (here the grains in motion are blurred). (Photos: Sidney Nagel, University of Chicago). [...]... sand from the air and provides more wind shelter for the grains on the leeward side But in doing so, the dune removes the sand from the wind and so suppresses the formation of other dunes in the vicinity The balance between these two processes establishes a constant mean distance between dunes which depends on wind speed, sand grain size (and thus their mobility in the wind) and so forth The essence... at the end of it all, the researchers concluded that the behaviour of these granular piles depended on the kind of rice that they used: specifically, on whether it was long-grain or short-grain To physicists, these varieties differ not in terms of whether they are to be used for risotto or rice pudding but according to their so-called aspect ratio: the ratio of length to width Long-grain rice has the. .. surface (Fig 8. 24a) They finally Page 219 attained a triangular shape, and at the same time they began to migrate across the surface in the direction of the windjust as they do in real deserts Remember that this migration is not in any sense a result of the ripples being 'blown' by the wind; instead, the solemn procession is the indirect effect of individual grain impacts followed by saltation Fig 8. 24 Self-organized... different regions of the pattern elements For sand ripples, the coarsest grains appear preferentially at the crests, with a thin veneer also coating the stoss face For large dunes the reverse is commonly the case: the finest grains collect at the crests, and the coarsest in the troughs When there is net deposition, so that ripples are gradually laid down on top of each other, the result is a series... material coating the face of the stoss slope (Fig 8. 27a), like the ripples found in the real world: the stoss slope receives more impacts, and so gets more coarsened, than other regions The crests of the ripples were particularly enriched in coarse grains (as seen in nature), which the researchers explained as follows The larger grains make smaller hops, because they are more massive and are therefore ejected... are therefore ejected from the impact splash with lower velocities By means of these little jumps the large grains gradually make their way up the stoss slope and jump just over the crest, into a sheltered region just at the top of the lee slope within the impact shadow Here they remain, protected from impacts, while further coarse material gradually climbs on top of them The smaller grains, meanwhile,... in their model sand pile, they were consequently hugely excited Here was a relatively simple model system, for which they knew all of the ingredients (because they had mixed them up themselves), that might offer some clues about the origin of the puzzling 1/ f behaviour This kind of scaling law is the consequence of abrupt avalanche-like events happening on all scales irrespective of the size of the. .. activity, in the length of streams in river networks, and in the fossil record of fluctuations in the abundance of life on Earth through the geological pastto name just a few examples There is clearly no link between the physical mechanisms that control these phenomena, but nonetheless it seems that they may show essentially the same statistical behaviour There appears to be something universal about the probabilities... exerted by a body falls off as the inverse of the square of the distance The power law relating avalanche frequency to avalanche size in the model of Bak and colleagues falls off rather less sharply than this: the frequency (or probability, if you like) of an avalanche falls off as the inverse of its size (Fig 8. 17) Conversely, the size is proportional to the inverse of the frequency of occurrence, denoted... losing the most crucial features of a system by throwing out the detailed specifics and focusing only on the statistics On the other hand, who would otherwise have guessed at this 'hidden' kinship between the nine o'clock news and a baroque concerto? The main point I wish to make is that both of these sound signals are clearly distinct from random noise The latter is called white noise (Fig 8. 18b), and . (c). But if the grain has to go all the way to the foot of the pile (as in c), this implies that the slope is equal to θ m everywhere. The pile then undergoes a landslide to reduce the slope. occur in the model: all the grains tumble, starting at the bottom, until the slope everywhere is reduced to the angle of repose θ r . Fig. 8. 12 The model outlined above generates the same kind. according to their size even if the angle of repose is the same for both (Fig. 8. 15b). In the case shown here, small glass balls segregate into the bellies while large balls gather in the necks.