1 Grains, Seeds, and Powders

To see a World in a Grain of Sand

And a Heaven in a Wild Flower,

Hold Infinity in the palm of your hand

And Eternity in an hour.

William Blake, “Auguries of innocence”

Highly diverse in shape and origin, grains of matter occupy a key position in our environment. Granular matter, our object of study in this work, represents the result of a vast number of grains combining in a pile, just as a crowd represents many people coming together to form a group. The phenomenon we will be examining in the following pages is a physical reality, and countless examples abound all around us, on a human scale. This opening chapter is devoted, above all, to the basic unit of matter, the tiny object that, in analogy to the seed of plants (granum in Latin), is called a grain.

A Close Look

Many Powers of Ten

To understand granular matter as a whole, we’ll start by examining a single grain. The field to be explored is vast indeed. All states of matter admit description as an assemblage of particles of highly variable dimensions, even though, in empirical terms, certain properties of organization recur on very different scales: the atoms of a crystal stack up just like a regular pile of oranges! At the most fundamental level, we have atoms—the elementary constituents of matter. Nuclear physicists view atoms as a collection of elementary particles. At the other end of the spectrum, a meteorite may serve as an example of a very large grain. The difference between a grain of so-called colloids*1 (for instance, the particles suspended in milk that give it a whitish color), a powder that weighs less than a millionth of a gram, and a meteorite involves more than twenty-four powers of ten!

In this book, we’ll be limiting ourselves to grains of intermediate size; hereby—as the boxed text that follows explains—thermal agitation* generally proves to be negligible. As we will see, this is the case for powders and sand on the beach, just a hundred micrometers in diameter. It can be difficult to appreciate that, after water, grains like this represent the most frequently used material on our planet.

In the course of this chapter, we will define what scientists mean by a “grain,” which, together with many other grains, makes a “pile” or a “packing.” This initial characterization will be retained throughout the book. Here, the only properties of a single grain to receive our attention are those relevant to later chapters.

Of course, if we were discussing how to use these grains in chemical reactions, or how to account for their mechanical or electrical properties, further distinctions would be necessary. This information will be introduced at a relevant juncture, in keeping with the subject explored. For the moment, however, a geometrical description of the individual grain will suffice.

Tiny Grains

1 Words and terms followed by an asterisk are explained in the glossary at the end of the book.

A crystal may be viewed as an ensemble of atomic grains that are well ordered and fixed in place. A liquid also consists of atoms or molecules in a compact state, but they are disordered and mobile. Finally, a gas comprises the same elements in dispersed and diluted form; they can be pictured as a mass of nanoscopic* bumper cars. Grains just a few angstroms (1 Å = 10^–10 m) in diameter are constantly colliding—a manifestation of the gas’s absolute temperature. Such movement can overlap with mean continuous flows on a larger scale, which belong to the realm of fluid mechanics.

At a scale slightly larger than that of atoms, one finds small particles, or micrograins, dispersed in liquid. One example is the aforementioned microdroplets suspended in milk that diffract light and make it look whitish. These particles retain a certain thermal agitation that decreases as their mass increases. A continuum extends from objects the size of a molecule to those the size of these colloids, which are ten to a thousand times larger than an atom; thermal agitation declines in proportion to growing size. Programs of research and applications in this domain—which is that of the nanosciences—have recently charted major developments on the basis of new possibilities of preparation and manipulation; their study must be reserved for a separate work. Here, we will generally limit ourselves to objects bigger than a few microns, for which thermal motion is insignificant. The objects in question are grains large enough to be felt “in the palm of your hand.”

Thermal Agitation

In 1827, the botanist Robert Brown first described the continuous and disordered movement exhibited by small grains of pollen suspended in water, thereby renewing an observation made by Lucretius some two thousand years earlier! This motion is a manifestation of thermal agitation affecting inert particles of matter in permanent collision with each other. The root mean square of these particles’ velocity is related to their mass and to absolute

temperature by , where is the Boltzmann constant (which

is universal) and “≈” means “of the order of.” When applied to the molecules of a gas, the formula yields a velocity of several hundred meters per second, depending on the value of . This velocity is visible under a microscope for colloidal particles a few thousand angstroms in diameter, with a mass a million times larger. In contrast, it is completely negligible for grains of matter larger than a few microns (which we mainly will be discussing in this book). The thermal agitation of particles that settle in a fluid limits their accumulation: on the surface of the Moon, for instance, it is sufficiently large that gravity cannot hold oxygen near the surface. In general, the density of Brownian particles decreases rapidly (exponentially) the higher the altitude and the weaker the gravity. Using this law when conducting research based on Albert Einstein’s theoretical work, Jean Perrin (figure 1.1) was able to determine the Avogadro constant (the number of gas molecules contained in a volume of 22.4 liters). He received the Nobel Prize for the experiment.

The jagged lines on figure 1.1 might have been produced by the random motion of bacteria propelled by their flagella. In such a case, it does not make much sense to define an equivalent temperature for characterizing the agitation of “biological objects” that are active (in the sense of the bumper cars mentioned earlier). This book considers only passive objects without an internal motor. Like pucks in a game of air hockey, macroscopic grains can remain in a constant state of agitation if they are constantly shaken; all the same, they’re still passive objects!

Large Grains

We have eliminated excessively small grains from our field of study. At the other end of the spectrum, are there objects too large to examine? Does a planetary object count as a grain? The rings of Saturn, which are readily visible through a small telescope, are composed of a very thin layer of ice particles and dust the size of what we find on Earth. Determining an outer limit is tricky. The meteorites—“space stones”—that reach our planet have sizes, shapes, and chemical compositions that might be mistaken for those of “Earth stones.” Is this so surprising, given that our rocks have resulted from extended periods of disintegration and reconstitution here on Earth? Indeed, our planet itself emerged as an agglomeration of meteorites made up of grains with different chemical constitutions, perhaps in the wake of a supernova’s explosion. Certain elements (gold and heavy metals, for instance) probably have such an origin; at any rate, they could not have been synthesized in the stars themselves. The elements stuck to each other and separated through sedimentation, with heavy ones like iron gathering at the core and light silicates forming at the crust. Although this process has slowed down significantly, several tons of these grains fall to Earth every year. The smallest aren’t more than a gram; but the huge rock that made Meteor Crater in Winslow, Arizona, weighed roughly three hundred thousand tons! Today, the physics of granular media also concerns planetologists; we will return to this point.

Grains That Can Be Held in the Palm of One’s Hand

William Blake invites the reader to contemplate a whole “world in a grain of sand.” Following these inspiring words, we will now look at one grain, just big enough to be felt, whose shape and surface state may be readily discerned through a magnifying glass or microscope. The “infinity” that opens up is the voyage over the Earth that a single grain can make across geological time up to the present day. Our grain, which is too large to be subject to thermal motion, reacts to gravity and external forces. Every step we take leads through history, and we tread transformations of the mineral and animal worlds underfoot. Without even realizing it, we’re kicking up the documents and archives housed on the Earth’s surface.

A professor and his students—grain collectors, or arenophiles—created a museum of sand on at a sea resort in Château d’Olonne, Planète sable (Sand Planet). There, for universal admiration, the rich array of samples they collected or were given is on display; the young curators share their aesthetic sensibility with visitors by means of lenses, which activate the “memory” stored in the grains (figure 1.2). Taking a close-up look resembles the way a geologist observes a stone when performing research: seeking clues, magnifying glass in hand. A Sherlock Holmes of petrography, the late Maurice Mattauer was a professor at the University of Montpellier and author of a remarkable book, Ce que disent les pierres (What Stones Have to Say). There was nothing like going on one of his geological excursions through the Cevennes mountain range he loved. Just clean the surface of a pebble, feel its heft, and turn it around (without much attention to the grains immediately surrounding it—it’s been traveling for so long …), and a whole history, genealogy, and vanished epoch came back to life. A little spit and polish is all it takes to attenuate the surface imperfections diffusing the light, and all the contrasts and internal patterns of unprepossessing stones emerge. Now, the merest pebble is fit for a jeweler’s display case. One discerns material structure due to heterogeneous composition as well as the deformation* experienced as resulting from a tectonic journey: stripes, furrows, and deposits. As we proceed, we shouldn’t forget the humble pebbles on the way, which also have a story to tell (figure 1.3a).

We can take the same approach to a single grain. Which one to choose from a handful gathered on the beach, which all have different properties and points of origin? What’s the common feature shared by that slightly translucid grain of white sand and the one next to it, entirely black, both of which may have traveled thousands of miles to wind up here? The first one is quartz, perhaps the result of sandstone* breaking down. It’s more angular and might have come from a crystal inside a granitic rock. A grain of quartz is so hard that the back-and-forth of particles rubbing together as ocean waves move barely softens its forms. Yet again, the shape bears a record of its origins. The black particle, which is denser, might be a fragment of basaltic lava—the kind that abounds on the beaches of volcanic islands like Hawaii. This kind of grain is quite different and humidity breaks it down rapidly.

We can also find colored grains, very bright ones, whose irregular shape directly reveals their origin: they’re elements of broken, worn, and carbonated shells with a chemistry that has partially dissolved in the sea (which is already rich in carbonates). A grain like this may also represent the result of a rock that cracked because of ice or lightning. The protracted flow of a glacier may have chiseled it and ground it down (figure 1.3b). Successive shocks might have fragmented it before it was carried in a river or borne through the air in a process called “saltation,” causing it to land at various locations before settling in a dune; sometimes, it will have traveled at an altitude of thousands of meters over thousands of kilometers—like the sands of the Sahara that make scattered red stains on Nordic glaciers. Perhaps this grain of sand has been “recycled”; originally part of a rock that broke apart, then subjected to abrasion by air and water, it may have sedimented once already and formed part of a block of sandstone, only to be sent on its way again before reaching us after a “second birth.” This “geological cycle” represents the very motor of changes to the Earth’s surface (figure 1.4). Investigation proceeds by looking at the marks that record the grain’s travels; if the stone is big enough, the deformations will be plain.

This history permits one to draw up an “identity card” for grains. For quite some time, it was standard practice among soil engineers to use the geological origins of grains when referring to them. This science—pedology*—was long the province of naturalists and chemists like Justus von Liebig (who is better known, today, for having invented a certain bouillon cube, or for having quarreled with Louis Pasteur). Only in the course of the twentieth century were methods devised for characterizing grains in mechanical and physical terms. After World War II, large-scale projects (dams, tunnels, and so on) contributed to these quantitative approaches and to developing models to account for soil behavior. The methods in question rely on two components: at an elementary level, grains must be characterized; on a larger scale, it’s a matter of measuring the texture and deformation of a mass of grains. Sixty years ago, a civil engineer, Pierre Dantu, was the first to observe the heterogeneous way forces are transmitted by means of photoelasticity, which we will examine in chapter 7. On the basis of these two factors, soil engineering could be progressively standardized and, in more recent times, automated.

Here, we won’t go much further into the details a geologist could provide. Discussion will be limited to how physicists and engineers identify grains in terms of size, shape, surface state, mass, density, and interactions. These parameters correspond to the origin and history of the deformations that particles bear. Thus, a roundish specimen attests to journeys on wind and in water; successive collisions between one grain and others gradually efface irregularities. In deserts, winds carry the sand around for so long that dunes consist of practically nothing but rounded particles of uniform size. But on relatively large grains—starting with the fractures on pebbles—deformations can be traced back to the effects of tectonic or seismic movements.

Grains also belong to the world of vegetation. During the Devonian Period, for reasons that remain poorly understood, the first seed-bearing plants developed. So many plant groups have been discovered that their rapid development (which then continued during the Carboniferous Period) has come to be known as the “Devonian Explosion.” The common ancestor of plants is algae; vegetation had great difficulty doing without water and needed to reproduce in aqueous media, often in a very short period of time. Under these conditions, the first true seeds appeared, containing an embryo, or germ, that could hibernate (that is, enter a state of dormancy) when conditions proved unfavorable. Seeds’ ability to undergo intensive dehydration, which makes them resistant to predation and attacks by microorganisms and fungi, represents a key asset. Flowering plants, which appeared during the Cretaceous Period about 130 million years ago, made seeds that could remain inactive for quite some time. Lupin seeds, for instance, have sprouted after spending 10,000 years in the frozen soil of northern Canada. Nature is quite an inventor! Like a ship carried to and fro by currents and waves, the embryo is protected by an outer casing that resists shocks, predators, and changes in humidity and temperature. Some seeds have even developed strategies for covering more ground—for instance, the tufts of dandelions or maple samaras (which children often call “helicopters”). This natural use of hydrodynamics has inspired many researchers and designers. Still now, nature continues to show the way to those who wish to benefit from millions of years of improvement and optimization.

Counting Grains

In The Adventure of Numbers, the mathematician Gilles Godefroy tells the tale of Scottish warriors who, when setting out for battle, had the custom of placing a pebble on a pile. Those who made it back alive each removed a stone. By counting the remaining pebbles, they knew how many of their companions had fallen on the field. After all, calculus means “little stone”! To determine the approximate number of grains in a pile, one needs only weigh the whole and figure out the average weight of a single grain (assuming that the grains are reasonably close in size). The result can yield mind-boggling quantities. In The Sand Reckoner, the ancient Greek mathematician Archimedes set out to determine how many grains of sand it would take to fill the universe. According to the physicist David Louapre, there are nearly as many grains of sand on the Earth as there are stars in the universe; the numbers are big enough to make your head spin! On a more modest scale, one can content oneself with counting particles by passing them, one by one, in front of a beam of light, which they temporarily obscure—like people walking by a ticket window. It should be noted, however, that this method only works if the concentration of particles is low enough for our optical “doorman” to count them out individually. There are also other methods—optical ones, as a rule—for obtaining fuller information about an ensemble of grains. In particular, the way that light passes through a medium containing grains in diluted suspension tells us about their average density and size. When the grains reach one-tenth of the light’s wavelength, the light is diffused, again, as in the case of milk or of a cloud. This is called Rayleigh scattering; it derives from the fact that the density and, in consequence, the index of refraction of a colloidal liquid fluctuates in space over time. Shining a beam of white light through a glass of dilute milk powder in water will reveal diffusion that produces a reddish glow on the optical axis and a bluish one transversally, because the different wavelengths of the source are sensitive to the scale of these fluctuations. This experiment simulates the diffuse radiation of the sun’s light in the atmosphere. With particles of a bigger size, it’s important to bear in mind that they cast a shadow when light is transmitted. This is called Mie scattering. Here, the wavelength does not play a major role, and the scattering does not affect color. A prime example is the effect of little particles of water in white clouds overhead.

Measuring Grains

In order to understand the organization of granular matter, quantitative evaluation of certain constituent elements is necessary. We need to establish the grain’s identity card. Although characterizing a grain in geometrical terms can sometimes require elaborate instruments, the principle of the process itself isn’t complicated. In general, one proceeds by taking a sample of the population; as when dealing with a human being, weight, height, and girth are informative. Does the grain have a smooth surface? Is it bulky? Is its profile jagged? Whereas a round grain may often be characterized in one dimension, two or three parameters are necessary for grains that have the shape of a needle or a chip; on occasion, more parameters will be necessary if the grain is irregular. Since a complete inventory of factors would take up too much space, we’ll discuss two basic principles to observe when measuring.

The Perspectives of Physics and Engineering

For physicists, obtaining data efficiently is a cardinal principle. Introducing new parameters always means enlisting new instruments, new diagnostic tools, for recording information and calculating averages. The second principle, which is closely related to the first, is managing data effectively. In an ideal case, a reduced and well-defined amount of information will closely approach the real facts by omitting characteristics that qualify as secondary. What is relevant for one problem might not bear on another one. For instance, whether it’s polished or rough, bright or dull, any ball will fall at the same speed in a viscous liquid; that said, optical observation of the same process requires different tools. Engineers need to take into consideration all the details at work in a system to reach a precise quantitative assessment. They need to know grains and the assemblages they form as fully as possible, drawing up catalogs and general rules of their properties. Such charts are easy to manage using databases, and engineering now incorporates many parameters calculated by computer. On one occasion, when exploring the mechanics of threads of fiberglass insulation, we encountered an engineering program that incorporated fourteen parameters—which still didn’t account for all the material’s mechanical properties! That said, mathematicians can trace a line describing an elephant’s profile by setting just five parameters; sometimes, a model can try to incorporate too many factors.

This, in schematic terms, is the difference between the approaches taken by physicists and engineers. Physicists look for hidden order by expressing the most general physical laws they can. Symmetry and scale can tell us a lot about grains’ properties. In this book, we’ll be trying to combine the insights afforded by both physics and engineering.

Instead of inventorying the geometrical properties of grains in detail, each problem we discuss will consider only what is absolutely necessary. The simplest model we’ll be using has a single parameter—what could be better than that? Called the hard sphere model,* or, more colloquially, the “billiard ball model,” it works with undeformable spheres that all have the same diameter. The only parameter that admits adjustment is the number of particles per volume unit (or surface unit in the case of balls on a flat billiard table). Because of its universality, this model is appealing, but it won’t do for many problems: try going down a slope made up of steel balls instead of rocks!

With this spherical grain as a reference point, it’s possible to describe any other grain’s deviation from regularity in terms of elongation,* expressed as the ratio between its largest and smallest dimensions. Devices (granulometers) that capture simultaneous images of falling grains from different angles enable us to gauge this aspect of shape. Deviation may also be calculated by describing the smallest sphere enclosing the whole grain as well as the largest sphere completely included in the grain. For an almost-spherical grain, the ratio between the radii will differ little from unity.

Grain Surface

The smoothness or, conversely, roughness of a grain and its angularity are key parameters for understanding how it will interact with neighboring particles in a pile, and what their relative motion will be. Anyone who has ever been in a large, dense crowd and tried to move in the midst of all those people will easily imagine the jostling, resistance, and shocks at play between grains.

Surface condition bears witness to the voyage that grains or stones have made. In one particular case, this fact interests historians directly: prehistoric objects used as tools tell a story to anyone who knows how to listen. The uninitiated will confuse a sharpened flint (a rock made up of disordered aggregates of quartz and siliceous cement of chalcedony) and a stone that has been broken accidentally. However, the experts can reconstruct its preparation and use by reproducing the gestures performed by an ancestor long ago with tools (likely a wooden or bone hammer, or a rock); the process involves looking for the waste left in the course of fabrication, the surfaces of fracture, touch-ups, and alterations, as well as imperfections left behind after actual, practical use of them. In this way, a piece of flint can inform us about our forebears, who from the Paleolithic Period up to the Middles Ages used stones as weapons, tools, and ornaments (figure 2.1). Today, it’s even possible to determine an object’s age by studying how light elements such as fluorine, which initially were contained within the rock, have spread to the surface over time.

Examining the surface, then, is essential for understanding the nature of grains. An exemplary case is the uniformly rough surface that sometimes results when a material is fractured. Whereas splitting a crystal yields flat planes on an atomic level, breaking a hetereogeneous material generally yields rough surfaces at varied scales, on which the properties of surrounding areas depend; in this context, as we will discuss further on, the term fractals* offers a key point of reference. Instruments have been developed for studying surface conditions and measuring roughness. They incorporate sensors directed at the surface of the material, and their vertical movement provides a record of it. The most remarkable device based on this principle is the atomic force microscope, an early version of which (the scanning tunneling microscope) earned the Nobel Prize for Gerd Binnig and Heinrich Rohrer, two Swiss physicists working for IBM; atomic force microscopy* (AFM) allows one to detect, and even move, atoms and molecules that, magnified, look like a layer of billiard balls (see figure 3.2).

It is also possible to characterize mean surface condition indirectly. A more significant quantity of atoms can be adsorbed on an uneven surface than one that is smooth. At an extreme end of the spectrum, a surface with crevices on all scales displays highly irregular geometry of the fractal type. Such a surface can adsorb a monoatomic layer of gas. When the surface is rough, the smaller the grains are, the more they will manage to work their way into small cracks. (Thus, a fishing boat can find more mooring space available than an ocean liner!) Take, for example, a pile of powder consisting of spherical grains one micron in diameter. If one evaluates the total surface area geometrically, the result is 1,000 square meters for one liter of grains. The ratio is a thousand times greater for the same volume of grains of active carbon, or the zeolite powder used in industrial adsorbents for catalytic converters. This is due to the fact that these grains possess an extremely rough surface with many internal voids. The roughness of a grain governs its accessibility: increasing surface at a constant volume increases efficiency; the filters in kitchen exhaust hoods operate on this principle. In batteries, finely divided electrodes make it possible to increase the contact area with the electrolyte and, as such, improve overall efficiency.

Another technique for measuring roughness involves illuminating the surface with light at the wavelength λ: the light is split in all directions, that is, diffused, by surface irregularities. The distribution of luminous intensity as a function of the scattering angle provides quantitative information on the surface condition. Today, intense light can be produced by large instruments called synchrotron* light sources (for instance, the National Synchrotron Lab at Brookhaven, or the European Synchotron Radiation Facility in Grenoble), which emit eletromagnetic radiation ranging from the short wavelengths of X-rays to wavelengths in the visible spectrum. Modifying a wavelength is a bit like adjusting a gauge to determine the size of imperfections on the surface under examination. The latter may display smooth characteristics when illuminated by light at a broad wavelength, and rough ones when exposed to rays at the small values of .

In the following, such analysis will enable us to characterize contacts between grains, which may be described metaphorically as a meeting between two individuals: permanent or temporary; rough or smooth, hard or soft. Often, a third factor comes into play (water, a lubricant, or a glue).

Chapter 2 continues to explore the nature of grains by considering how they are made. That said, describing single grains and the contacts between them is not enough to understand the properties displayed by numerous grains in immediate contact with several others. We must be wary of a “reductionist” approach: knowing what happens on a small scale is not enough to account for what occurs on a larger scale. A packing of granular matter is much more than the sum of its parts!