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DRY sand might seem an uninteresting material. There are huge quantities of it on most beaches, it varies little from one bucketful to another and has no obvious high-technology applications. And, sand has always found unexciting uses, such as in ballast or as an abrasive, mixed into concrete, or dumped into children’s sandpits. But some physicists find sand fascinating because its behaviour is often complex, intriguing and unpredictable.

One of the reasons why physicists like sand is that it provides a good model for other powders. Many foods, for example, are in the form of particles or granules. Manufacturers need to understand how these particles behave during processing, which is one reason why many experimental and theoretical physicists are trying to understand the intricate dynamics of sand grains.

Sand can behave in many different ways. Grains blown by the wind form large, well-defined dunes that are constantly in motion. Sand continually pounded by breaking waves develops a clear gradation of particle sizes which seems to contradict what you would expect to result from the effects of gravity. Sometimes, sand may appear to flow freely like a liquid and at other times it may support itself, or support an applied stress, like a solid. The hourglass acts as an example of both kinds of behaviour. For hundreds of years, people have exploited the regular trickling flow of sand through a constriction in its glass to make a simple, accurate clock, but the sand pouring from the hole onto a level base forms a pile, with sloping sides, rather than a pool with a flat surface.

Although engineers have developed sophisticated descriptions of the way that sand behaves in bulk, we still do not have a detailed knowledge of how the behaviour of individual particles influences the way it flows, for example. This knowledge is necessary if we want to apply what we know about the physics of sand to powder technology and materials research. In particular, we might want to predict how granules and powders behave at the limits of the normal working conditions in a manufacturing process. Often, it is these limits that are the most important when it comes to designing safe and efficient processing plants.

Farmers, for instance, frequently store grain in large metal hoppers, or silos. Unlike silos for liquids, grain silos occasionally collapse spontaneously. We do not know exactly why this happens but it is probably because the flow of the grain becomes unstable. Simple fluid dynamics applied to sand flow does not predict the instability. Nevertheless, silo manufacturers must include some details of the physics of particle flows in their design criteria.

Nowhere is the study of particulate materials more important than in food processing – Britain’s largest manufacturing industry. Many foods consist of dry particles at some stage. The average supermarket trolley contains packets and jars of powders and granules, from traditional ingredients, such as flour, sugar and salt, to freeze-dried coffee, potato and gravy. Soap, talc and medicines often come in the form of powder or tablets. These products are processed in enormous quantities so that even modest improvements in the design of process equipment, the flow of the powder and the way it is mixed can give a manufacturer a commercial edge over competitors in the industry.

Even more important is that food companies have to consider ways of handling and controlling the flow of particles hygienically so as to safeguard public health. Ideally, a plant should have an efficient flow regime of pipes, hoppers, mixing vessels and so on, with no stagnant areas, and the machinery should be accessible and easy to clean.

Particulate foods, such as cornflakes, granulated sugar and flour, are obviously all very different in the shapes, sizes and texture of their constituent particles or granules, and they need very different specifications during manufacture. Rapid advances in milling technology and in the computer software controlling the processes have allowed manufacturers to lay down more precise specifications for these materials, even though tolerances are very small. This means that food technologists can make a wide range of different products from the same raw material. Think of the varieties of different cereals and snacks that you can buy, all made from the same ingredients – wheat, oats and corn.

In order to set the specifications for milling machines, manufacturers need to be able to classify powders according to their properties and how they are made. This is extremely difficult: it is unrealistic to come up with a description including the positions, shapes and orientations of all the particles. At the same time, just stating the density and distribution of the size of particles does not convey the richness of the materials. Why are some powders dusty, others crumbly and some almost impenetrable? How do we describe such attributes quantitatively? A team led by Sam Edwards at the Cavendish Laboratory in Cambridge has been working on a theory that will help to answer these questions. The researchers use the physics of gases and liquids, which are basically random collections of atoms or molecules, to model powders which are random collections of particles.

The Cambridge team considered three important qualities of particulate solids. First, powders are made up of extremely large numbers of similar components. An average teaspoon of sugar crystals, for example, may contain between a thousand and a billion particles depending on whether the sugar is plain granulated, castor or icing sugar. Secondly, when you pour, shake or compress a powder, the particles do not move independently. They all jostle each other, readjusting their positions at the same time.

Thirdly, it is clear that what happens when a powder is poured, shaken or compressed is reproducible. That is not to say that the properties of a powder remain constant. If you pour coffee granules into a jar and immediately measure how dense it is, the density will be slightly lower than if you had first tapped the side of the container a few times. It is only after many taps that the coffee’s density becomes constant. This final measurement is called the bulk density. If we perform the same sequence of operations, pour, tap, measure, and so on, with a different sample of the same coffee granules, we will reproduce both the same intermediate measurements and the same bulk density.

These three observations prompted the physicists to compare the grains in a powder with the atoms in a dense fluid. Atoms in liquids move around in a random way but they are not independent. They also jostle each other.

Despite these similarities, there are some important differences between liquids and powders. In liquids, molecules are attracted to each other by weak forces known as van der Waal’s forces. One moving molecule soon transfers some of its thermal, or kinetic, energy to other molecules around it. This means that the thermal energy fluctuates as the atoms move about into different configurations, but eventually the liquid settles down to a thermal equilibrium. The volume of the liquid does not change. In contrast, in a powder, the particles do not move around so easily. If we shake the powder so that it forms a completely new configuration of particles, it may have a different overall density (and therefore a different volume per particle) but there will be little exchange of thermal energy between grains. So physicists use the volume per grain, rather than the energy, as a variable, when describing different powder configurations.

Disorder in a jar of coffee

For any value of the volume per particle, there are many possible arrangements of grains in a powder. Imagine jars of coffee stacked on supermarket shelves. Each jar contains granules at the same density because they have been packed and transported using the same method, usually on a production line. But it is highly unlikely that any two jars contain exactly the same arrangements of the individual grains. This freedom that you find with the large number of equivalent arrangements of powder grains provides a measure of disorder called entropy.

Liquids and gases also have an entropy associated with their disordered configurations of molecules. In this case, the entropy, along with the energy and the temperature, are the fundamental quantities used to describe the thermodynamic properties of fluids. The theoreticians in Cambridge have developed a similar thermodynamic scheme to describe the large scale properties of powders. They identified entropy, the volume and a particulate analogue of the temperature, a property they called the ‘compactivity’, as the variables that control the way the powder behaves. In a liquid, the temperature measures the level of the heat energy in the system, whereas, in a powder, the compactivity measures how inefficiently it is packed.

Preliminary results show that compactivity is very important. As it varies from zero to infinity, the powder grains adopt configurations that progress from the most compact to the least compact arrangements provided that the powder remains as a stable set of grains in contact. In model systems, the range of the compactivity may divide into domains that correspond to different kinds of packing of the particles. Evaluating the compactivity of real powders is clearly an important step towards describing the textural properties that are otherwise so difficult to express in quantitative terms.

Progress in understanding the physics of powders has been helped by the recent advances in computer modelling. This alliance between theory and computer simulation had worked well in describing the physics of the liquid state. It was not until the 1960s, when computer techniques really developed, that physicists produced good statistical theories of liquids. But programs for modelling the behaviour of powders have to include additional features. Physicists usually consider molecules as billiard balls when describing their thermodynamic properties. When molecules collide, they conserve their total kinetic energy. Powders often have particles with irregular shapes, and collisions between them do not conserve the kinetic energy. Because of the large number of unknown parameters needed to describe particle collisions in powders, physicists need vast simulations, involving hundreds of particles, that take hundreds of hours of computer time, to assess how the material behaves overall.

Computers can successfully model the flow of powders down chutes, along conveyors and out of hoppers. This is difficult to monitor in real situations because reliable instruments also obstruct the powder flow. The simulations can not only test theory quantitatively but also provide striking graphic displays of what happens at the microscopic level . During processing, powders often get shaken about by vibrations. The vibrations might be part of a stirring or sieving process or they might come from pipes and containers not properly insulated from surrounding machinery. Vibrations may cause small groups of particles to jiggle about so that they temporarily become more loosely packed, resulting in a series of changes of positions of particles in localised areas in the powder. At this level, the effects of shaking are complex and disorganised. In contrast, vibrations often cause a powder to move about in a highly organised way. If you place lycopodium powder, for example, on a vibrating metal plate, it spontaneously runs into small heaps. How the movement of the powder changes from a disorganised to an ordered kind of behaviour is interesting to physicists, not only because it represents ‘order out of chaos’, but also because it is important commercially for maintaining and improving process control.

One of the most fascinating effects of constant shaking is the separation of particles into different sizes. Even when a sample contains only a small spread of particle sizes, the largest ones rise to the top of a container during repeated shaking. Check your muesli packet. Are the raisins, nuts and large flakes on the top with the crumbs on the bottom?

Here segregation causes problems of presentation for the manufacturer when packaging the cereal. It can also reduce the quality and performance of blended powder mixtures for soups and drinks packed for dispensing in automatic vending machines. In fact, two separation processes are happening simultaneously. In the first, small particles filter through the network of pores created by the packing of larger particles. This effect is very rapid and shaking makes it only slightly worse. In the second, vibrations cause particles with comparable sizes to segregate.

We at the Institute of Food Research in Norwich and Anthony Rosato and colleagues at Carnegie Mellon University in Pittsburgh have performed computer simulations of continuous shaking that show the segregation clearly and reveal the underlying local process. During the periods when shaking loosens the packing, individual small particles can move into voids beneath large particles and so prevent them from returning to their previous positions. It is far less probable that several small particles will move together so as to create a void that can be occupied by a single large particle. The net effect is that the smaller particles occupy the lower positions during the active part of the shaking process and then become trapped there when the grains fix into a new arrangement.

The simulations show that the driving force on the particles is roughly proportional to their size and that for asymmetrical particles only their widest dimension is important. The rate of energy put in from shaking controls the time that the particles spend loosely packed and, therefore, how quickly the particles segregate. Below a certain energy, no separation happens. These simulations suggest ways of ensuring that particle separate out less when powder mixtures vibrate. They should also allow us to analyse particular problems of powders in a quantitative way.

Physicists can also use the simulations to measure how voids are distributed in each new, static powder configuration generated by shaking. Large voids tend to be trapped below the large particles. If many voids appear together, the particles above them, supported only by each other and at the ends, can form a bridge or arch that has a large empty volume trapped beneath it. A little more shaking can cause these structures to collapse, producing an unstable flow of particles.

Convection in powders

One remarkable dynamical property that has long intrigued physicists is ‘convective grain flow’. Here, a horizontal layer of rigid particles on a vertically vibrating bed spontaneously forms a heap with sloping sides. Michael Faraday observed this phenomenon as long ago as 1831. He described the dynamics of the particles to a meeting of the Royal Society – ‘the particles of the heap rise up at the centre, overflow, fall down upon all sides, and disappear at the bottom apparently proceeding inwards’. More than 150 years later, people are still hotly disputing the origin of this motion. If you vibrate salt grains at a frequency of 50 hertz, this slope may be inclined at 20Degree to the horizontal. You can see the convective motion described by Faraday by using a few contrasting tracer particles. But it is clearly not convective in the sense used to describe the thermal rolls formed in heated layers of fluid.

Recently, Stephan Fauve and colleagues in Lyons, France, have observed vibrating piles of glass beads with a stroboscope. While it was being shaken, the pile developed regions with different packing structures. There was a region where the beads moved in unison, as if they were part of a solid, underneath a region where the beads jiggled about independently as if part of a flowing liquid. During a single shake up and down the boundary between these regions moved, causing the particles in the body of the heap to flow inwards and upwards. When particles reached the surface at the top of the pile they fell down the slope in a series of avalanches until they reached the bottom, where they started to move inwards again. It is not clear whether air surrounding the powder, acting as a fluid, affects the way this internal boundary moves and the extent to which energy dissipated by collisions between particles is important. The ‘avalanches’, which can be triggered without any shaking simply by pouring more sand into an existing pile or by tilting its base, last for a well-defined time, and are separated by almost regular intervals. Moreover, most avalanches involve the whole free surface of the pile. This behaviour arises because, as more sand is added, the pile temporarily sustains slopes slightly steeper than the preferred slope. Avalanches reduce the slope to this preferred value when the steepening goes too far.

There is clearly much more work to be done on the dynamics of powders. The variability of the granules, their interactions with each other and with their container, and the diversity of the external conditions increase the complexity of the problem and make theory and experiment difficult. Computer simulations are, however, an ideal tool for predicting how granular solids behave because they allow theorists to model a wide variety of particle characteristics and external influences. In the food industry, where quality, presentation and, above all, hygiene have to be considered alongside profit, simulation techniques have much to offer.

Clearly, physicists still have plenty to learn from simple patterns in the sand.

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Modelling the textural properties of mixtures

ONE of the most long-standing and celebrated problems in theoretical physics is how to evaluate the way regular objects such as discs and spheres pack. If we fill a large box with billiard balls, so that they pack randomly, how much of the total volume will the balls occupy? What are the sizes of any irregularities and how are these affected by including some marbles of a different size? Although categorical answers do not exist, we need a thorough knowledge of these problems if we want to understand the packing of cubic sugar grains, irregular coffee granules or wheat flakes. Computer simulations provide the best source of this information. The random packing efficiencies of discs and spheres of the same diameter obtained in computer simulation experiments are approximately 82 per cent and 64 per cent respectively.

At the Institute of Food Research at Norwich, we in the theory and computational science group have used this kind of computer model to study what happens when you mix different particles randomly to form a granular material such as muesli. In this way, the ‘textural’ properties of the material are related to the components of the mixture and to the way they are put together. We can also picture the shape and the geometry of the voids that can form, which allow fluids to trickle through the powder and which may harbour contaminants.

Further insights come from these simulations when we study the networks formed from the particle-particle contacts in the ‘close-packed’ material. These in dicate the kind of local ordering that is maintained by the grains and which is important in determining the mechanical properties of a powder. We can then transfer this expertise to other, more everyday packing problems.

Gary Barker and Malcolm Grimson work in the theory and computational science group at the AFRC Institute of Food Research in Norwich.