SAND CASTLES AND MUD PIES are made for fun but present intriguing questions. Although sand and mud consist of particles of much the same composition, they have very different properties. What forces hold sand castles and mud pies together? Why is it that sand castles disintegrate when they dry and mud pies do not? Why do only the mud pies shrink when they dry? How can water, which normally functions as a lubricant, increase the cohesiveness of sand and clay? The answers to these questions are found in the electric interactions between the water and the particles of sand or clay.

Sand consists of fairly large, roughly spherical grains that remain closely packed even when they are wet. Their close packing is indicated by the fact that sand does not shrink when it dries. A wet grain of sand has positive and negative ions at its surface, usually grouped in pairs with one member a little farther from the surface than the other. Which ion is the outer one varies over the surface. Both orientations contribute an electric field to the space surrounding the grain. When the outer ion is positive, the electric-field vector points away from the surface. When the outer ion is negative, the vector points toward the surface.

Although the electric field surrounding the sand grain should average out to zero, it does not because of a polarization of charge in the negative ions on the surface. On one side of a negative ion there is a positive ion a little farther from the surface; on the other side there is no charge. Since the negative ion has more negative charge than positive charge, its negatively charged electron cloud shifts slightly away from its positively charged nucleus toward the neighboring positive ion. This slight separation of positive and negative charges, called an electric dipole, has a relatively weak electric field away from the surface of the sand grain. A positive ion on the surface has a deficiency of negative charge, undergoes no separation of its charges and has the electric field of a point source of charge instead of the weaker dipole field. Hence the electric field around a sand grain is primarily due to the positive ions on the grain's surface.

Wet sand is firm because of an electric interaction between these positive ions and the water between the grains. According to a slightly dated model of the interaction the water provides negative ions (OH- that adhere to the grains and electrically shield the sites of positive charge. Positive ions (H₃O+) remain in the water that is farther from the surface. The attraction between the positive ions in the water and the negative ions adhering to the surface of the grains makes it harder for one grain to slide over another. Although this model does explain the firmness of wet sand, it has a flaw: the water is unlikely to be ionized so that it can provide the required OH- and H₃O+ ions.

No current model of wet sand is complete, but one modern interpretation of the electric interaction does not require that the water be ionized. The electric field generated by positively charged sites on the surface of the sand grain itself partially depletes the positive charge of the water adjacent to the surface. This depletion is not ionization, because the charges within a water molecule are not totally removed from each other. The electric field simply shifts the average positions of the hydrogen nuclei (protons) in the molecules that are farther from the surface. The layer next to the surface is left relatively negative; the water farther from the surface is relatively positive. The transition from relatively negative to relatively positive is gradual, this state of affairs is described as a diffuse double layer of charge.

In the older ion model the strength of the wet sand is attributed to the attraction of the oppositely charged ions in the water. In the diffuse-double-layer model the strength arises from the fact that the water tends to be immobilized. Since the fields generated by positively charged sites at the surface of a sand grain redistribute the average positions of the hydrogen protons in the water, the water's ability to flow is reduced. The sand is firm because the increased viscosity of the water hinders the sliding of one grain with respect to another.

The immobilization of the water diminishes with distance from the surface of a grain. When wet sand flows to the degree that it cannot be used for building a sand castle, it does so because of the large amount of water between the grains. Although the relatively negative water surrounding a grain still attracts the relatively positive water farther from the grain, the grains are too much separated by water. Some of the water between the grains is not close enough to any one grain to participate strongly in the electric interaction. This less viscous water flows, allowing the adjacent grains (and their halos of relatively charged water) to slide over each other.

It is a common notion that sand castles are held together by the force exerted by the surface tension of the water in the sand. Water molecules are said to be polar because they have a permanent electric dipole. These dipoles attract each other. At a surface between air and water the mutual attraction of neighboring water molecules puts the surface in a virtual state of tension. When the surface is curved, the mutual attraction provides a collective force on it.

On the surface of a sand castle the collective force does help to firm up the sand. The sand is just as firm, however, within the sand castle, where no air spaces remain. Although the water molecules there are loosely attracted to each other by their electric-dipole fields, the attraction is too weak to account for the sand's firmness and resistance to flow. Water molecules in a glass of water are attracted to each other but the water is not firm.

When a sand castle dries, it disintegrates because the amount of relatively charged water in it has decreased. During the transition the charges on the surface of the sand grains rearrange themselves. These ions form a double layer of charge, called the Helmholtz double layer, with a positive ion slightly inside the surface below a negative ion outside. The electric field generated by this positioning of charges is weak except directly at the surface. Wherever two dry sand grains touch they repel each other because both surfaces have an outer layer of negative ions. Hence dry sand grains do not adhere.

The cohesion of sand is destroyed if some of the water is replaced with glycerol. The molecules of glycerol electrically shield the sites of positive charge on the sand grains, reducing the electric field in the water. With weaker fields there is less ordering of the water layers, and with no immobilization of the water the sand does not hold together.

You can demonstrate the effect of glycerol easily. First pack a small container with wet sand so that the sand is reasonably firm. Over the container place a flat plate. Quickly invert the plate and the container without allowing the sand to spill out. With the plate on the floor pull slowly upward on the container so that a mound of sand remains on the plate. Test the strength of the mound by shaking or tapping the plate.

Now repeat the experiment with glycerol added to the water mixed with the sand. When you have the assembly inverted, again pull upward on the container. The chances are that the mound left behind is noticeably weaker than the mound without the glycerol. A shake or a tap will probably cause it to collapse. The more glycerol you add to the sand, the less stable the mound is. The result is somewhat surprising. A viscous fluid (glycerol) is poured onto a very viscous mixture (wet sand), and the result is a decrease in viscosity.

Wet clay differs from wet sand in several subtle ways. Its particles are considerably smaller than sand grains and are platelike, with one dimension much smaller than the other two. In wet clay the particles are loosely packed and well separated by water. One indication of the looseness is the fact that clay shrinks as it dries.

On the surface of a clay particle positively charged sites are exposed, just as they are on a sand grain. According to the older ion model of wet clay, OH- ions adhere to these sites, leaving H3O+ ions in the water between the clay particles. The strength of the wet clay is due to the attraction between the two sets of ions. In the more modern model of wet clay the water is said to be immobilized by the electric fields set up by the positively charged sites on the clay particles. As with the model of wet sand, the immobilized water has a high viscosity. The wet clay is firm because the clay particles cannot flow over each other. If the clay is very wet, it is not firm. The water away from the particles is less viscous and can flow.

Drying clay differs from drying sand in an important respect. Whereas the charges on a sand grain can shift and give rise to a Helmholtz double layer on the surface of the grain, the charges on a clay particle cannot. The clay particle is too thin in one dimension for double layers to form on the opposite sides of that dimension. The innermost layers of charge (which would be of the same sign) would repel each other too strongly to be stable.

When wet clay dries, the fact that a Helmholtz double layer cannot form requires that the particles retain their diffuse double layer and that the resulting viscous water remain between them. The clay structure continues to be firm. In fact, its rigidity increases as the particles move closer to one another, further increasing the immobilization of the water. As the structure shrinks and more contact points develop between the particles, the particles bond together by sharing ions. They also resist movement because of the remaining viscous water. Other forces, all electric, also help to bond them together.

Sand grains are closely packed. Clay particles are loosely packed. Their packing differs because of the difference in their shapes and sizes. Suppose a sand grain falls and comes in contact with a stationary grain. Does the fallen grain remain attached to the other grain or does it roll or slide over the other grain and continue its downward motion? At the region where the grains come in contact the attractive forces are strong enough to hold them together. The gravitational pull on the fallen grain, however, puts a torque on it that tends to rotate the grain about its point of contact with the other grain. Some of the attractive forces create a torque opposing the rotation. Because the grain is relatively heavy the torque due to gravity overwhelms the torque due to the attractive forces, and the grain rotates. Continued motion of this kind finally brings the grain downward into the other grains until it is closely packed.

When a clay particle falls onto a collection of other clay particles, it is subject to attractive forces on encountering a

stationary particle. Gravity continues to pull it downward, creating a torque that tends to rotate it about the point of contact. The particle is so light, however, that the torque created by gravity cannot accomplish the rotation. Hence the particle remains in the orientation it had when it first collided with the stationary particle. When more particles fall, they too go no farther once they land. As a result the packing of the particles is loose.

When wet or dry sand is compressed, it takes a different form only if the sand grains slide relative to one another. Suppose a pressure is applied to wet sand but not enough to cause relative motion of the grains. The contact area of two grains that are pushed against each other increases, and so does the extent of the electric forces holding them together. At the center of the contact area the additional pressure on the grains irreversibly deforms their surfaces. The energy expended in the deformation cannot be reclaimed when the pressure on the grains is removed. Mast of the energy of the compression of the grains, however, goes into deforming the region surrounding the center of the contact area, which is not irreversible. That energy is stored as elastic potential energy, much as energy is stored in a compressed spring.

When the pressure on the grains is removed, the elasticity of the grains pushes them apart. The additional electric forces that are developed during the time of increased contact between the grains are relatively weak and are easily broken by the rebound of the grains. Hence after the pressure is removed the grains return to their earlier form.

Something different happens when wet clay is compressed. Consider the loose collection of clay particles shown in Figure 5. The applied force bends one of the platelike particles so that its previously free end is now touching another particle. At the point of contact attractive forces bond the two particles together. When the applied force is removed, the particles remain bonded because the bonding forces are strong enough to hold the particles in their bent configuration. When you push with your fingertips against wet clay, you compress the ; structure. It does not rebound when you stop pushing. Such material is said to be plastic. Wet clay is plastic and wet sand is not.

The best type of sand for building a sand castle is sand with a little clay mixed into it. Choose sand near the water so that it is already wet enough. Sand right at the edge of the water is probably too wet to be firm. Besides, any sand castle built there would soon be destroyed by waves or the tide.

You can build sand castles in several ways. Wet sand can be dumped carefully from a container so that the shape of the container's interior is preserved. An interesting container leads to interesting designs. The sand can also be shaped by packing it between two lengths of board. Long castle walls are easily constructed in this way. A more difficult technique is sculpture. Heap up a large mound of sand and then, starting at the top, whittle away. Another technique mimics the growth of stalagmites in caves. From a clenched fist filled with wet sand allow a small stream to dribble out. The stream creates a thin spire that resembles the steeple of a futuristic cathedral.

If your castle is to last throughout a hot day, you should periodically spray it in order to preserve its firmness. The added water seeps between the grains. There the electric fields from the grains shift the average positions of the hydrogen protons in the water and so immobilize the water. The increased viscosity of the water ensures the structure's stability until the next spraying is needed.

Although sand does not change in volume as it dries, it does expand when it is suddenly stressed, a kind of behavior termed dilatancy. You may have noticed such expansion when you walked across a stretch of beach just wetted by a wave. The sand looks wet before you step on it but dries just as you do so. Only after a while does the footprint look wet again.

Wet clay does not dry under a footstep because its particles are loosely packed. Any pressure on the clay moves the particles closer together, decreasing the clay's volume. Dilatancy is characteristic of closely packed particles. When a sudden pressure is applied to the particles, forcing them to move, they can do so only if their average distance from one another increases. With wet sand pressed down by the sole of a foot this expansion amounts to the surface of the sand's rising. The water is left behind and takes a few minutes to seep to the surface.

The dilatancy of sand also figures in an avalanche of dry sand. A simple demonstration exhibits the important features of an avalanche. Gradually tilt a trough of sand until it flows. The avalanche begins when the tilt of the surface exceeds a certain angle called the dynamic friction angle. The sliding sand decreases the tilt to an angle, called the friction angle, at which the surface is stable. If you again tilt the trough, the surface again exceeds the dynamic friction angle and a new avalanche begins.

At any angle less than the friction angle the sand grains on the slope are stable because they are closely packed. Only if the tilt of the slope reaches the dynamic friction angle can the grains slide. To do so they first must rise, so that they can slide or roll over the grains under them. The entire overburden of the sand enters the state of dilatancy and slides down the slope.

Many granular substances slide in much the same way. Snow is a well-known example. In addition to sand I investigated sugar (both granulated and powdered), salt, cornstarch and cocoa powder. I also studied the avalanching of dry pinto beans, oranges and apples. Each had its characteristic friction angle and dynamic friction angle.

With sand the difference between the two angles is about five degrees. When I spray the sand with water, the grains no longer flow down the tilted surface. Instead the entire bulk of sand in the trough. slides across the bottom of the trough once the trough is sufficiently tilted. With pinto beans the difference between the two angles is about four degrees. Cornstarch and cocoa powder are so cohesive that with them I have trouble making the measurement.

Many powders are cohesive: their particles stick to each other and resist flowing. Some, such as face powders, are meant to be cohesive so that they stay put. Others, such as salt and cocoa powder, are a nuisance when they are cohesive. The cohesion in a powder is due to a variety of electric forces.

One such force, the electrostatic, is due to charges on the surface of adjacent particles. The strongest of the electrostatic forces is the Coulomb force, which attracts particles that have the same sign of charge and repels particles that have the opposite sign. Weaker is the attractive force that develops between a charged particle and a neutral (uncharged) atom or molecule. The electric field of the charged particle induces a separation of charge inside the atom or molecule. For example, if the particle is negative, the positive constituent of the neutral atom or molecule moves slightly toward the particle and the negative constituent moves slightly away. Although the atom or molecule is still neutral, it now has the field of an electric dipole and is attracted to the charged particle.

Some molecules are permanent electric dipoles and thus attract each other when they are neighbors; the negative end of the one pulls toward the positive end of the other. The van der Waals force is a similar type of electric force, but it is due to the quantum nature of atoms. Although an atom might be electrically neutral and so not be a permanent dipole, it is nonetheless a dipole at any one instant. In the simple Bohr model of an atom the electron is considered to be in orbit around the nucleus. Therefore at any given instant it is on one side of the nucleus and is hence separated from the positive charge. This instantaneous dipole can induce a dipole in a neighboring neutral atom. Once the two dipole fields are established they pull the atoms together.

The van der Waals force attracts almost any two surfaces that are in contact. Everything in the world is not stuck to everything else only because Coulomb repulsion usually keeps surfaces from adhering. Many common powders are cohesive because of the van der Waals force between the particles.

If the particles in such powders touch, the frictional forces between them oppose the sliding of the particles relative to each other. If there is water between the particles, their cohesion can be due to its immobilization by the charged sites on the particles. Surface tension can also play a role if there is air between the particles or if the outside of a cake of the powder is held together by water.

The cohesion of a powder cannot be predicted from a detailed knowledge of the structure of its particles except perhaps in a very general way. For example, there is probably more friction between irregularly shaped particles, but only if they happen to touch each other. The cohesiveness of a powder is often reduced when the particles are ground into smoother and more regular shapes. The cohesiveness can also be altered if the powder is dried so as to diminish the role of the water between particles. The compression of a powder usually increases its cohesion by increasing the strength of the van der Waals force or the attractive electrostatic forces or by adding to the number of particles touching and interlocking. In general the smaller the grains of a powder are, the more cohesive the powder is. Suppose you prepare a powder whose particles are nearly uniform in size. The powder is placed in a hopper whose bottom has a circular opening to allow the powder to flow out. To measure the rate of flow of the powder you measure the time it takes to empty the hopper by a particular amount.

When the particles are very small, their rate of flow from the hopper is either very low or zero. They encounter large cohesive forces that restrict their motion relative to one another. If progressively larger particles are substituted, the rate of flow increases until it peaks and then decreases. When the particle size is roughly a fifth the size of the escape hole, the flow stops At this point the particles have locked onto one another, building across the opening something like a dry-stone wall.

Some powders form crumbs that are held together by water between particles. The form of the water layers between the grains falls into one of three categories. If the particles of the powder were originally quite dry, the water lies between the regions of contact of the particles in what is termed a pendular formation. Other regions between the particles are filled with air. When more water is present, the water-filled regions are larger but are still separated by air gaps. This formation is called funicular. If the space between the particles is filled entirely with water, the formation is said to be capillary. The detailed nature of the forces holding together a crumb of particles is not well understood. The adherence is often attributed to the surface tension of the water, and that model may suffice for the pendular and funicular formations. In any event the immobilization of water is important in all three formations.

Powder can cake even if most of it is dry. You may have noticed that when water is added to a powder for hot chocolate, the powder is often so caked that it resists dissolving. Such a cake is dry on the inside but wet on the outside. When a large mound of hot-chocolate powder floats on the surface of the water, the powder along the edge of the water gradually collapses the perimeter of the mound, sending part of the powder avalanching into the water.

Powder cakes when the water between the particles on the outside of the cake is so viscous (owing to immobilization) that additional water cannot easily seep into the interior. Attractive forces between the particles may also play a role. Eventually enough water seeps between the particles so that their electric fields can no longer immobilize all the water or strongly attract each other. Then the cake falls apart. Still, some particles remain clumped long after the main body of the powder has dissolved and disappeared.

Wet cornstarch is similar to many other wet powders except that its viscosity depends on the stress put on the mixture. It is an example of a non-Newtonian fluid, a class of fluids whose properties I discussed in this department for November, 1978. Cornstarch is thixotropic, which is to say that its viscosity increases when the stress on the fluid is increased.

I dissolved a small amount of cornstarch, rolled it into a ball and then examined it periodically as it dried. At first the firmness of the ball depended on the stress placed on the mixture. When I pressed the ball, it was very firm and looked dry. When I released the pressure, the ball immediately flowed and looked wet. As water evaporated from the mixture the ball became progressively firmer until its surface was quite rigid to the touch. The structure was weak to any twisting force but strong to any pressure on its surface.

Because cornstarch is composed of very small particles its behavior might be modeled on the electric interactions in wet clay. I am not certain why the mixture is thixotropic. The easiest explanation is that the increase in viscosity is linked to the immobilization of the water between the particles. When I press the ball of cornstarch, I move particles of starch closer to one another. The water between them becomes less mobile because it is now closer to one or more particles. That region of the ball thus becomes much more viscous than it was before I applied the pressure.

Most of the cohesive powders are also adhesive and cling to my finger or a kitchen utensil. Which of the many types of electric force are important in adhesion may be difficult to determine. The van der Waals force usually plays a role, but some of the electrostatic forces probably participate also. If the particles in the powder are wet and fairly large, the surface tension of water may adequately explain the adhesion.

When a fine powder is even slightly damp from the humidity in the room, part of its attraction to your finger is probably due to the immobilization of the water between the finger and the individual particles of powder. On the surface of a particle there is a layer of relatively negative water. Somewhat farther from the particle is relatively positive water. When this system is brought close to a surface such as a finger, the electric field from the relatively positive water forces the surface of the finger to be negative. The particle clings to the finger because of the Coulomb attraction between adjacent layers of charge. The surface of the particle is positive. The water adjacent to the particle is relatively negative. The water farther from the particle is relatively positive. The surface of the finger is negative.

This kind of adhesion is strong, however, only if the particles are in the size range of a few microns. Particles much larger are probably too heavy to cling to a finger. Particles much smaller do not cling either, but that is not because of their weight. Smaller particles are able to generate thicker halos of immobilized water because they are not as good at generating a Helmholtz double layer of charge on their surface. Since they cannot be as close to a finger as the somewhat larger particles with their thinner halos, they fall off. If the adhesion to your finger were not so selective, life would be miserable, particularly on humid days. Everything you touched would stick to your fingers because the water between them and the particles was immobilized.

Bibliography

ATOMISTIC APPROACH TO THE RHEOLOGY OF SAND-WATER AND OF CLAY-WATER MIXTURES. W A. Weyl and W. C. Ormsby in Rheology Theory and Applications, Vol. 3 edited by Frederick R. Eirich. Academic Press, 1960

THE COHESIVENESS OF POWDERS. N. Pilpel in Endeavour, Vol. 28, No. 104, pages 73-76; May, 1969.

THE PHYSICS AND MECHANICS OF SOIL. Ronald F. Scott in Contemporary Physics, Vol. 10, No. 5, pages 449-472; September, 1969.

CRUMB FORMATION. N. Pilpel in Endeavour Vol. 30, No. 110, pages 77-81, May, 1971.

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