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 (θm), 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 (θr), 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 θm 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 θm 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 θr 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 θm everywhere. The pile then undergoes a landslide

to reduce the slope everywhere to θr 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

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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 θm, it tumbles down from column to column until it finds a

position for which the slope is less than or equal to θm. But if a grain tumbles all the way to the bottom,

which means that the slope everywhere is already equal to θm 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 θr.

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 θm, 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

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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.