The Self-Made Tapestry Phần 5 pps

Page 119 highest) to the value in the bulk of the oil. If we think of a model analogy in which the pressure is equivalent to the height of a hill and the motion of the air bubble is equivalent to the motion of a ball, the ball accelerates more rapidly down the hill the steeper it isin other words, it is the gradient that determines the rate of advance. Saffman and Taylor pointed out that the gradient in pressure around a bulge at the air/oil interface gets steeper as the bulge gets sharper. This sets up a self-amplifying process in which a small initial bulge begins to move faster than the interface to either side. The sharper and longer the finger gets, the steeper the pressure gradient at its tip and so the more rapidly it grows (Fig. 5.14). Fig. 5.14 The Saffman-Taylor instability. As a bulge develops at the advancing fluid front, the pressure gradient at the bulge tip is enhanced and so the tip advances more rapidly. (Contours of constant pressureisobars, like those in weather mapsare shown as dashed lines.) This amplifies small bulges into sharp fingers. Compare this to the growth instability in DLA (Fig. 5.8). This instability is called the SaffmanTaylor instability. In 1984, Australian physicist Lincoln Paterson pointed out that the equations that describe it are analogous to those that underlie the DLA instability described by Witten and Sander. So it is entirely to be expected that viscous fingering and DLA produce the same kind of fractal branching networks. Both are examples of so-called Laplacian growth, which can be described by a set of equations derived from the work of the eighteenth-century French scientist Pierre Laplace. Within these deceptively simple equations are the ingredients for growth instabilities that lead to branching. But tenuous fractal patterns directly comparable to those of DLA occur in viscous fingering only under rather unusual conditions. More commonly one sees a subtlely altered kind of branching structure: the basic pattern or 'backbone' of the network has a comparable, disorderly form, but the branches themselves are fat fingers, not wispy tendrils (Fig. 5.15; compare 5.12). And under some conditions the bubbles cease to have the ragged DLA-like form at all, and instead advance in broad fingers that split at their tips (Fig. 5.4a). This sort of branching pattern is called the dense-branching morphology, and is more or less space-filling (twodimensional) rather than fractal. Why then, if the same tip-growth instability operates in both viscous fingering and DLA, do different patterns result? All viscous fingering patterns differ from that of DLA in at least one important respectthey have a characteristic length scale, defined by the average width of the fingers. This length scale is most clearly apparent at relatively low injection pressures, when the air bubble's boundary advances quite slowly. Then one sees just a few fat fingers that split as they grow (Fig. 5.16). There is a kind of regularity in this so-called tip-splitting patternthe fingers seem to define a more or less periodic undulation around the perimeter of the bubble with a characteristic wavelength. But a length scale is apparent in the widths of the fingers even for more irregular patterns formed at higher growth rates (for example, Fig. 5.15). For the self-similar DLA cluster (Fig. 5.7), on the other hand, there is no characteristic sizeit looks the same on all scales. Fig. 5.15 Viscous fingering has a characteristic length scale, which determines the minimum width of the branches. So the fingers are fatter than the fine filaments of DLA clusters. (Image: Yves Couder, Ecole Normale Supérieure, Paris.) Eshel Ben-Jacob of Tel Aviv University explained the reason for these differences in the mid-1980s: between the air bubble and the surrounding viscous fluid there is Page 120 an interface with a surface tension. As I explained in Chapter 2, the presence of a surface tension means that an interface has an energetic cost. Surface tension encourages surfaces to minimize their area. Clearly, a DLA cluster is highly profligate with surface areathe cluster is about as indiscriminate with the extent of its perimeter as you can imagine. This is because there is effectively no surface tension built into the theoretical DLA modelthere is no penalty incurred if new surface is introduced by sprouting a thin branch. In viscous fingering, on the other hand, there will always be a surface tension (provided that the two fluids do not mix), and so there would be a crippling cost in energy in forming the kind of highly crenelated interface found in DLA. The fat fingers represent a compromise between the Saffman-Taylor instability, which favours the growth of branches on all length scales, and the smoothing effect of surface tension, which washes out bulges smaller than a certain limit. To a first approximation, you could say that the characteristic wave-length of viscous fingering is set by the point at which the advantage in growth rate of ever narrower branches is counterbalanced by their cost in surface energy. Fig. 5.16 At low injection pressures, the length scale of viscous fingering is quite large, and the advancing bubble front then has a kind of undulating shape with a well-defined wavelength. The relation between DLA and viscous fingering is made very apparent when DLA growth is conducted in a system where a surface tension is built in. The surface tension has the effect of expanding the cluster's branches into fat fingers (Fig. 5.17). Ben-Jacob showed that the generic branching pattern in such cases is the dense-branching morphology. Conversely, a wispy DLA-like 'bubble' can be produced experimentally in the HeleShaw cell by using fluids whose interface has a very low surface tension. Fig. 5.17 When surface tension is included in the DLA model, it generates fat, tip-splitting branches like those in viscous fingering. Here the bands depict the cluster at different stages of its growth. (Image: Paul Meakin and Tamás Vicsek.) Physicists Johann Nittmann and Gene Stanley have shown that, somewhat surprisingly, the fat branches of viscous fingering can be generated instead of the tenuous DLA morphology even in a system with no surface tension. They formulated a DLA-type model in which they could vary the amount of 'noise' (that is, of randomizing influences) in the system. In their model the perimeter of the cluster can grow only after a particle has impinged on it a certain number of times (in pure DLA just one collision is enough). This reduces the tendency for new branches to sprout at the slightest fluctuation. Nittmann and Stanley found that, when the noise is very low, the model generates fat branching patterns (Fig. 5.18a), which mutate smoothly to the DLA-type structure as the noise is increased (Fig. 5.18b,c). This suggests that one way to impose a DLA-like pattern on viscous fingering in a Hele-Shaw cell is to introduce a randomizing influence (that is, to make the system more 'noisy'). A simple way of doing this is to score grooves at random into one of the cell plates until it is criss-crossed by a dense network of disorderly linesthis was how the pattern shown earlier in Fig. 5.4c was obtained. The lesson here is that noise or randomness can influence a growth pattern in pronounced ways. Page 121 Fig. 5.18 Dense-branching patterns appear in DLA growth even in the absence of surface tension, when the effect of noise in the system is reduced by reducing the sticking probability of the impinging particles (a). As the noise is increased (from a to c), the branches contract into the fine tendrils of the DLA-type pattern. Again, contours denote different stages of the growth process. Note that, despite their differing appearance, all of the patterns here have a fractal dimension of about 1.7. (Images: Gene Stanley, Boston University.) The six-petalled flowers Just as random noise can jumble up branching growth, so can an underlying symmetry have the opposite effect of introducing order. Take another look at Fig. 5.4b, which is a viscous-fingering pattern formed in a HeleShaw cell in which one plate has been scored with a regular hexagonal lattice of grooves. The sixfold symmetry of the underlying medium shows up clearly in the pattern, whose branching form is reminiscent of a snowflake. The beautiful, symmetric complexity of snowflakes (which share such hexagonal symmetry) has captivated scientists for centuries. Their hexagonal character was apparently known to the Chinese almost two millennia before Western natural philosophers became aware of it. Around 135 BC Han Ying wrote with astonishing perception that 'Flowers of plants and trees are generally five-pointed, but those of snow, which are called ying, are always six-pointed'. (About five-pointed flowers we have heard already in the previous chapter.) Yet as late as 1555, the Scandinavian bishop Olaus Magnus could be found claiming that snowflakes display a variety of shapes, including those of crescents, arrows, nails and bells. The Englishman Thomas Hariot seems to have been the first in the West to note the six-pointed shape, in 1591; but it was not until 1611 that this fact became common knowledge, when Johannes Kepler wrote a treatise entitled De niva sexangula ('On the Six-cornered Snowflake'). Herein Kepler pondered over the mysterious origin of this shape. Although lacking the theoretical Page 122 Fig. 5.19 Snowflakes are symmetrical branching patterns of infinite variety. (Photos: from Bentley and Humphreys 1962, kindly provided by Gene Stanley.) tools and concepts needed to make much impact on the problem, Kepler did have the remarkable insight that the hexagonal symmetry must result from the packing together of constituent particles on a regular lattice. The symmetry, he said, was a consequence of their 'Patterns of contact: for instance, square in a plane, cubic in a solid'. At a time when atoms and molecules were barely conceived of, this was truly a leap of inspired imagination. Modern techniques for analysing crystal structures have now shown us that water molecules do indeed pack together on a regular lattice that, looked at from certain directions, has sixfold symmetry (which is to say that it looks the same when rotated through a sixth of a full revolution). Astonishingly, we can see in this an echo of ancient Chinese wisdom about the cosmic schemes of nature: the number six was associated with water (then seen as one of the fundamental elements), and the scholar T'ang Chin wrote 'Since Six is the true number of Water, when water congeals into flowers they must be six-pointed'. Everyone now believes that the hexagonal symmetry of snowflakes is a manifestation of this deep- seated symmetry in the crystal structure, just as the cubic shape of table-salt crystals reflects the cubic packing of its constituent ions. But that is only a small part of the problemby analogy with other crystals, we might then expect ice crystals to be dense polyhedra with hexagonal facets, whereas instead we find these flat, highly branched and infinitely varied natural sculptures (Fig. 5.19). Just how varied they are becomes evident from a glance through Snow Crystals by amateur photographer William Bentley and his colleague W.J. Humphreys. This astonishing book documents thousands of snapshots of snow crystals captured and photographed by the authors shortly after the turn of the century. A book of the same title published in 1954 by Japanese physicist Ukichiro Nakaya adds about 800 more snapshots to the family album, each one an individual. Nowhere in these two books will you find two identical snowflakes. From where does nature obtain this ability to turn out endless variations on a theme? There is still no complete, universally accepted answer to that question. Indeed, in 1987 Johann Nittmann and Gene Stanley began a paper on snowflake patterns by confessing that 'There is no answer to even the simplest of questions that one can pose about snowflake growth, such as why the six arms are roughly identical in length and why the overall pattern of each Page 123 arm resembles the five others'. Nor, they added, are we quite sure why snowflakes are (mostly) flat. But although ice seems to be unique in forming these highly symmetrical flakes, regularly branched crystals analogous to a single snowflake arm may be seen in many other solidifying materials, including metals crystallizing from a melt (Fig. 5.20a), salts precipitating from supersaturated solution, and electrodeposits (Fig. 5.20b). These structures, known as dendrites, are generally formed when solidification is rapidthat is, far from equilibrium. For metals freezing from their melt, for instance, rapid solidification can be induced by cooling the molten metal suddenly to far below its freezing point. Slow growth of crystals close to equilibrium gives instead compact, facetted shapes. (I should point out that these dendrites are not the same as the mineral dendrites mentioned at the start of the chapter, which instead have a more random DLA-like structureunfortunately researchers in different fields have been rather inconsistent with the 'tree' metaphor.) [...]... Figs 5. 8 and 5. 14 If a bulge develops by chance (that is, because of the random fluctuationsnoisein the system) on an otherwise flat solidification front, the temperature gradient becomes steeper around the bulge than elsewhere, because the contours of constant temperature get pressed closer together (Fig 5. 22) So the bulge grows more rapidly than the rest of the frontand the sharper it gets, the steeper... Because the bacteria could not penetrate through the gel, the colony could grow only by pushing back the gel at its boundary The more agar they added to the growth medium, the harder the gel wasit could vary in consistency from jelly-like to rubbery And the harder the gel, the harder it became for the colony to expand The researchers observed fractal, DLA-like colonies under conditions where the gel... be proportional to the inverse of the body mass raised to the power 1/4 The metabolic rate of individual cells in an organismthe rate at which they consume energyfollows the same mathematical law In other words, big organisms have a slower metabolism What's more, the total metabolic ratethe net rate of energy consumption of the whole organismvaries as the 1/3 power of body mass And the cross-sectional... advancing edge of the solid This in turn depends on how steeply the temperature drops from that of the liquid close to the solidification front to that of the liquid further awaythe steeper the gradient in temperature, the faster heat flows down it (It may seem odd that the liquid close to the freezing front is actually warmer than that further away, but this is simply because the front is where the latent... indeed they are not the same; but our eyes are fooled into seeing more symmetry than there really is by the uniformity of the branching angles As the model is modified to make the depths of the 'fjords' more accessible, the snowflakes become denser (b, ccompare Fig 5. 19) (Images: Gene Stanley, Boston University.) Page 127 diverging at the same (60°) angle and because the envelopes traced out by the tips... century, by identifying the following principles: 1 When the central stem forks into two branches with equal width, they both make the same angle with the original stem 2 If one branch of the fork is of lesser width than the other (so that it can be regarded as a side branch, the wider one being a continuation of the main stem), then the thinner branch diverges at a larger angle than the thicker 3 Side... extent can these ideas help us to understand biological form? Fig 5. 24 Branching patterns, like that shown here in electrodeposition, can undergo abrupt changes in shape as the growth conditions are varied Here the change took place as the electric-field strength (given by the voltage drop between the edge of the cluster and the edge of the triangular cell, divided by the distance between them) exceeded... substance called agar They injected a few bacteria into the centre of the dish, added some of the nutrients needed for growth, and let nature take its course By varying the conditions under which growth occurred, they found that they could obtain colonies with very different shapes They looked at the effect of changing just one of two variablesthe concentration of nutrient and the hardness of the gelwhile keeping... the interface between the solid and the melt can advance in a whole family of parabolic shapes: all possible parabolas are allowed, on the condition that the thinner they become, the more rapidly they advance (Fig 5. 21) So thin, needle-like tips should shoot rapidly through the melt, while fatter bulges make their way forward at a more ponderous pace Fig 5. 21 A simple analysis of the solidification of... harder: there Fig 5. 32 The morphology of the bacterial colony depends on the conditions under which it grows: the amount of nutrient, and the hardness of the gel medium (a) 'Eden'-like growth (b) The dense-branching growth mode (c) Compact, non-branching growth (Photos: Mitsugu Matsushita.) Page 1 35 is a certain degree of hardness beyond which the bacteria simply cannot move Under a microscope, they could . closer together (Fig. 5. 22). So the bulge grows more rapidly than the rest of the frontand the sharper it gets, the steeper the gradient and so the more rapidly it grows. The situation is mathematically. rates (for example, Fig. 5. 15) . For the self-similar DLA cluster (Fig. 5. 7), on the other hand, there is no characteristic sizeit looks the same on all scales. Fig. 5. 15 Viscous fingering has. Here the change took place as the electric-field strength (given by the voltage drop between the edge of the cluster and the edge of the triangular cell, divided by the distance between them)