In Verne's story a group of explorers survive a catastrophe in the Arctic by means of an ice lens. The ship's captain had set out to reach the North Pole, but a mutiny had left him and a few supporters with a broken and sinking ship and few supplies. Their only chance to survive was to reach another ship that had been abandoned when it became locked in the ice. The group gathered wood from their broken ship, packed up their meager food supplies and the flint and steel for making fire and set out.

Because their food was running out, they stalked and killed a polar bear, but as they were doing so their fire went out and the flint and steel were lost. Although they had enough food for the remainder of the trip, they were threatened with freezing to death if they could not restart the fire.

They were saved by optics. Noting the bright sunshine, a physician in the group recalled how a fire can be started with a lens. Lacking a glass lens, he decided to make a lens of ice. He chose a clear section of ice and hacked out a piece about a foot in diameter. Shaping it roughly with a hatchet, he whittled and smoothed it with his knife and then polished the surface with his bare hands "until he had obtained as transparent a lens as if it had been made of magnificent crystal." The ice lens was positioned above some tinder, focusing the rays of the sun. Within a few seconds the tinder had caught fire. "The stove was soon roaring, and it was not many minutes before the savory odor of broiled bear-steak" had roused the men.

What the physician had made was a convex lens. Since it was a large one, it intercepted a goodly amount of sunlight. The concentration of the light on the tinder placed at the focal point of the lens quickly delivered enough energy to kindle a fire.

I told this story on one of my regular contributions to Quirks and Quarks, a radio show about science on the Canadian Broadcasting Corporation. I also described how one can fashion a convex lens of ice in a simpler way with curved watch glasses. (A watch glass is the kind of glass that covers the face of a watch. Glasses of similar shape but larger size can be found among the standard supplies of a chemistry laboratory.) Water is poured into two watch glasses and allowed to freeze. Then the flat sides of the two pieces of ice are placed together to form the lens.

Wheeler was a listener who decided to improve on my idea. He made a lens from the ice in a cattle trough. He started with a section that was about an inch thick and free of bubbles. To shape the piece he put it in the lid of a milk pasteurizer, which is curved like a lid for a saucepan. Then he floated the lid (with the piece of ice) in a sink of warm water. The heat melted the bottom of the ice until it fitted the inside of the lid. He occasionally rotated the ice to increase the symmetry of its bottom surface. Next he turned the piece over to shape the other side.

Wheeler now had a convex lens of ice. To test its focusing power (and to mimic the physician in the Verne story) he held the lens by its edge and focused the light of the sun onto a newspaper: "The paper. quickly singed, smoked and burst into flame." Wheeler tells me the lens worked best when the air temperature was above freezing. At lower temperatures drops of water condensed on the lens, distorting its surface and interfering with its focusing of light.

Wheeler made a smaller lens in the concave base of a pressurized container. When he placed the lens in front of his 35-millimeter camera, it functioned as a telephoto lens with a focal length of about 80 millimeters. Wheeler tested it outdoors on a cold day with snow on the ground. The lens was too fast to photograph snow scenes, and so he stopped it down to f9 with a cardboard iris. The iris was mounted between a collar of black plastic tubing and a conventional rubber lens shade. The purpose of the collar was to eliminate any light that did not pass through the lens.

Initially Wheeler held the lens in place with his gloved hand. Later he found that a cardboard mount was more convenient. The lens was clamped to it with three cardboard flanges stapled to the card. On the front of the card he put a cardboard shade. Once the lens was mounted in its holder the card and the lens were held against the rubber lens shade and adjusted for position. Wheeler told me an ice lens will work well for 15 minutes and will serve even when the air temperature is 16 degrees Celsius (about 60 degrees Fahrenheit).

Wheeler has now made many lenses with a variety of molds. In some cases he begins with liquid water that he freezes. He says the clearest ice results when the freezing begins at the top. To make this kind of ice he insulates the sides and the base of his mold.

If you make an ice lens with a pair of watch glasses, experiment with placing the two pieces of ice together to complete the lens. The pieces should have good contact. Try rubbing them together or stroking them with a finger so that the contact surfaces are wet and smooth. When they freeze together, the contact should be firm.

By choosing appropriate molds you can fashion both convex and concave lenses or combinations of the two. The index of refraction for ice is less than that for glass, so that an ice lens does not bend the rays of the sun as much as a glass lens of the same shape. Therefore an ice lens (either convex or concave) has a longer focal length than a similar lens made of glass.

Middle Eastern coffee is a strong drink brewed from sugar and finely ground coffee. The sugar is needed to offset the bitter taste of the coffee, but I think it also is a factor in the brewing. A mixture of ground coffee, sugar and water is heated in an ibrik, a container that tapers slightly toward the top and has a long handle. The coffee is rapidly brought to a boil and then poured into a demitasse, which is half the size of a standard coffee cup. What is poured is a mixture of liquid, foam and grains of coffee. The foam is highly regarded in Middle Eastern countries. In fact, some people say it is an insult to serve a guest a demitasse of coffee without foam.

One must let the coffee remain in the demitasse for several minutes before it is tasted. The grains of coffee settle slowly to the bottom. If the brew is sampled too soon, the liquid is unpleasantly gritty. After the grains have settled the liquid is sipped slowly until nothing remains but the bottom layer of coffee grains, which is about the consistency of a sludge.

I have spent hours meditating on the brewing and sedimentation of Middle Eastern coffee as I prepare demitasse after

demitasse of the drink. It seems to be about the only coffee drink in which grains of the coffee remain in the liquid after the brewing process. It is also noteworthy in several other respects. Why must the coffee be finely ground? (When you buy coffee for the drink, you should ask for an extra-fine grind.) Why is sugar heated with the coffee rather than added later as it usually is with other coffee drinks? Why does the liquid suddenly foam up the sides of the ibrik as the coffee is being brewed? How long must one let the coffee stand in a demitasse to ensure an ungritty sip?

I buy finely ground coffee roasted by the Viennese or the French method. Both coffees are. darker than the others commonly sold in the U.S., meaning that they have been roasted longer and have lost more of the volatile substances in them. Nevertheless, I doubt that the type of coffee alters anything but the taste of the drink.

I prepare my coffee in a tin ibrik large enough to fill two demitasses, each of which holds about two fluid ounces (roughly 60 milliliters). Usually, however, I make only a single demitasse at a time, otherwise the foam forms so fast 11 that it cascades onto the stove. Hence I add a bit less water to the ibrik than is needed to fill one demitasse. I add two teaspoons of coffee grounds and from one teaspoon to four teaspoons of sugar to the water. (One teaspoon is equivalent to five milliliters.)

Some people mix the ingredients before brewing the drink, but I usually do not. I put the ibrik on the heating coil of my electric stove with the control high. Soon the mixture begins to rumble and click. After several minutes the noise suddenly stops and the foaming starts. The foam rapidly builds upward to the top of the ibrik. The transition is so fast that I sometimes fail to remove the ibrik from the heating coil before the foam overflows. If I pour the coffee slowly into the demitasse, some of the foam remains in the cup, but most of it collapses.

In order to look inside the mixture while it was brewing I replaced my ibrik with a 1 80-milliliter beaker made of Pyrex. Again I added the proper amount of water and coffee for one demitasse. The coffee grounds floated and very slowly became wet. Apparently they are held together with enough electric force to keep them in a clump. Water seeps between the grains only slowly. Although the grains are denser than water and so should sink. the clumps contain enough trapped air to make them float.

I put two teaspoons of sugar onto the coffee. Most of the sugar sank, sometimes tipping over a clump of coffee as it did so. When I placed the beaker on the heating coil, the sugar and coffee layers were well separated by a layer of relatively clear water.

Even if I stir the mixture before heating it, the same three layers appear. In cold water the sugar dissolves slowly and so settles to the bottom after being stirred because it is denser than water. Even with stirring, the coffee grains still carry enough adsorbed air to make most of them float to the surface. The only difference when the mixture is stirred is that some of the grains end up in the water layer.

Soon after the heat was turned on bubbles of air appeared at several sites along the bottom of the sugar layer and began oscillating rapidly. The air in the bubbles had been adsorbed onto the sugar grains or had been dissolved in the water between the grains. Once bubble nucleation starts at a given place more air is added to the site from the surrounding water and sugar. The bubbles oscillate because when they expand upward into cooler liquid (concentrated sugar water in this case), they suddenly collapse. Soon afterward, however, they are reheated and reexpand. The rapid and chaotic oscillation of the air bubbles generates the clicks and rumbles characteristic of the early stage in heating a fluid.

Bubbles of air and water vapor eventually break free and stream upward through the sugar layer. When they penetrate the top of the layer and enter the clear water, their motion is arrested. The water is cooler than the sugar layer and makes the bubbles collapse.

By shining a flashlight into the beaker I can spot threads of sugar water projected upward from the collapse of each bubble. The threads are visible because their index of refraction differs from that of the water. The threads twist and turn before they fall back to the sugar layer. Sometimes small droplets of sugar (or concentrated sugar water) are left in the wake of the threads.

Soon the entire sugar layer becomes turbulent. Whereas until then bubbles conveyed heat upward to the layer of water, convection streams now seem to dominate. The variation in the index of refraction makes some of these streams visible. With further heating the turbulent sugar layer moves upward into the water layer. Some of the turbulence or one of the bubbles from the top of the sugar layer occasionally reaches the bottom of the coffee, but otherwise the coffee seems to be unaffected by what is taking place below it.

As the turbulent layer of sugar nears the coffee the grains begin to glisten with tiny air bubbles, none larger than a millimeter and most much smaller. Just as the sugar layer reaches the coffee the grains are stirred around and yield a great many of the tiny bubbles. They form the foam, which builds up rapidly. Within a few seconds the foam is three times as high as the initial mixture.

As soon as the grains are stirred around the rumbles and clicks stop. The noise originated in the formation of large bubbles along the bottom of the container. Once the grains reach the region they provide plenty of nucleating sites for smaller bubbles. The large, noisy bubbles disappear.

I employ a long pair of tongs to grasp the hot beaker. If I hold the beaker above the heating coil, I can control the rate of production of the foam: the rate slows as I move the beaker away from the coil. The bubbles are small at first, but as they are pushed up the beaker by newly made bubbles they merge and give rise to larger bubbles. The larger bubbles are too fragile to survive when the coffee is poured. The foam some people like to have on their Middle Eastern coffee consists of the tiny bubbles that were adhering to grains of the coffee when the foaming started. When the coffee is poured into a demitasse, some of the tiny bubbles break free of the grains and form a raft that may hold its shape for quite a while.

I do not always get such a raft on my coffee. When I do, it can survive for hours if the demitasse is left undisturbed. One reason for the stability of the bubbles is that the sugar in the coffee increases the viscosity of the fluid. When a bubble raft has formed, the fluid in the walls between the pockets of gas drains because of gravity. The walls get thinner until the bubbles burst or merge. The more viscous the fluid is, the slower it drains. The fluid in the walls of the bubbles on my coffee drains very slowly.

To change the conditions I made the coffee without the sugar. Again the coffee grains floated in clumps on top of the water. When the beaker was placed on the coil, the water began to heat rapidly and with much noise. Air bubbles appeared and oscillated. Then bubbles of water vapor formed and escaped upward to beat against the bottom of the coffee layer.

Soon streams of these bubbles broke the clumps of coffee. The grains quickly dispersed throughout the volume of the fluid. Although the clumps lost some of their air and thus their buoyancy, the grains still bore adsorbed air. They were stirred down from the top surface only because of the strong convection currents in the beaker. Small bubbles formed and then the foaming began. A11 these things happened faster than they did when I had sugar in the mixture.

I tried several other preparations. In one I heated only sugar (two teaspoons) and water. As before, the sugar sank to the bottom and was heated first. Viscous bubbles and turbulent convection appeared again. When the mixture was at full rolling boil, bubbles appeared on the top, but with this preparation there was no rapid foaming. The bubbles were relatively large (several millimeters across) and did not last long.

After a few minutes the fluid was transparent, indicating that the sugar had dissolved fully and was uniformly mixed. I allowed the mixture to cool. Even at room temperature all the sugar remained dissolved. Clearly the amount of sugar normally added to the drink does not exceed the saturation limit of sugar in hot water.

Next I boiled the usual amount of water in the beaker but omitted the sugar. After the rolling boil began I added a single teaspoon of coffee. Almost immediately the grains were swirled throughout the volume of water and tiny bubbles developed on them. The foaming was so fast that some of the mixture spilled onto the stove before I could remove the beaker from the heat.

In another preparation I replaced the sugar with an equal amount of honey. I poured the honey into the beaker first, then the water and finally the coffee. When the honey began to heat up, it formed tiny geysers that rose into the layer of water. They were visible only when I shined a flashlight into the beaker, making the difference in the index of refraction between the geysers and the water apparent. If a geyser pinches off from the main layer of honey, it falls back to that layer because honey is denser than water. Often the stream loops or releases drops of honey. Some of the drops remain stationary, apparently because they. have been diluted enough to acquire about the same density as that of water.

At this point in my investigation I could answer some of my questions about the brewing of Middle Eastern coffee. The principle of the brewing is to cook the grains fast. If they are overcooked, the drink tastes too bitter or acidic. It is then like regular coffee that has been left simmering in the pot for a long time. The grains for Middle Eastern coffee are finely ground so that they can be rapidly heated and cooked.

The foam is almost a side effect. The grains are so small that they provide plenty of surface area for adsorbing air. The tiny pockets of air are sites for the nucleation of bubbles. When the coffee and the surrounding water get warm, air dissolved in the water comes out of solution at the nucleation sites, forming bubbles just barely large enough to be seen.

When the hot liquid layer reaches the grains, water vapor expands the bubbles. They pinch off and break free from ,the nucleating sites and then form the !t foam of large bubbles that climbs the ibrik. Many of the nucleating sites on the grains remain active because they retain part of the air or vapor after a bubble has left. Moreover, they are now stirred around in the water near the hot bottom of the container. Hence they continue to generate bubbles as long as the surrounding fluid is hot enough to provide more water vapor.

Sugar is added before brewing for several reasons. Heating ensures that the sugar is fully dissolved by the time the coffee is drunk. If the sugar were added after the coffee had been poured into a demitasse, you would have to stir the sugar into the drink to dissolve it. This action would cause the coffee grains to be suspended for much longer. The coffee would be cold before they had settled out enough to provide a drink that was not gritty.

The sugar also facilitates brewing. The viscous layer of sugar at the bottom of the ibrik holds heat better than water. If the layer consisted of water only, bubbles of water vapor would transfer heat to the coffee layer early in the brewing procedure. With the sugar in place the bubbles and their heat transfer are retarded. Only later does the hot and turbulent layer of sugar water reach the coffee and begin to heat it. The period of heating and cooking of the coffee is sudden and short. It begins when the hot liquid reaches the coffee, it ends in less than a minute when the foam threatens to overflow the container.

Finally, the sugar. is beneficial to the brewing process because it helps to stabilize-the foam of tiny bubbles. Without the sugar the foam is less likely to survive being poured into the demitasse.

Although the brewing process is fast, it must not be given too little time. When I make enough coffee for two demitasses simultaneously in my ibrik, I often wind up with gritty coffee. No matter how long I wait large grains fail to settle. The reason is that they still carry enough adsorbed air to remain buoyant. They may even remain clumped together. Since the ibrik has too much fluid, I must take it off the stove prematurely to keep foam from overflowing. As a result the time from wher1 the hot, turbulent sugar layer reaches the coffee to when the coffee is poured is too short for all the adsorbed air to be eliminated. From experience I have learned to prepare only one demitasse of coffee at a time. If it is necessary to make a larger amount in one preparation, I must hold the ibrik off the heat source to control the foam and to cook the coffee long enough.

My next set of questions has to do with what is going on in the demitasse as the grains of coffee sink to the bottom. Each grain sinks at a speed that is approximately constant. Since the sedimentation rate of suspended particles is an important datum in both science and technology, much work has been done to develop a theory of how fast particles settle in water.

If the particles are smaller than about .06 millimeter in diameter, they fall according to Stokes's law, named after the British physicist and mathematician G. G. Stokes. The speed of-a small particle depends on two factors: the effective weight of the particle and the drag applied to it by the fluid. The effective weight is the difference between the particle's true weight and the buoyancy imparted to it by the surrounding fluid. This difference is usually described in terms of the density of the particle and the fluid. Since a coffee grain is denser than water (and denser than a mixture of sugar and water), the buoyancy imparted to it is too weak to keep it from falling. Its effective weight is a downward force smaller than its true weight.

Countering this force is an upward force generated by drag. According to Stokes's law, the drag depends on the diameter of the particle, the speed of the particle and the viscosity of the fluid. When the particle begins to descend, its speed is small and so is the drag. The particle steadily accelerates and soon is moving fast enough for the upward drag force to match the downward force of the particle's effective weight. Thereafter the particle no longer accelerates but it continues to fall at a constant speed. This terminal speed depends on the size of the particle. Smaller particles have lower terminal speeds and therefore take longer to settle.

The value of the terminal speed also depends on the viscosity of the fluid. Since the sugar increases the viscosity of the coffee, all the particles have lower terminal speeds than they would have without the sugar.

Unfortunately for drinkers of Middle Eastern coffee, Stokes's law applies only to spherical particles. The coffee grains small enough to be affected by the law are irregular in shape and therefore encounter a drag force that is difficult to calculate. Nevertheless, the overall dependence on particle size generally holds: smaller grains fall slower.

Many of the grains in Middle Eastern coffee are too large to fall according to Stokes's law, which predicts a drag force that depends on the first power of the particle's speed through the fluid. The larger particles get a drag force that depends on the square of the speed. In addition the equation for the force includes an experimentally determined drag coefficient, which takes into account the effect of the particle's irregular shape.

Another factor is that if there are many particles falling through a certain volume of fluid, they change the situation in two ways. They increase the effective density of the fluid, thereby decreasing the effective weight of a particle and its settling rate. The particles also interact as they pass one another on the way down. In spite of this added complication the same rule usually holds: smaller particles-fall slower.

To demonstrate the rule I poured a small amount of freshly brewed Middle Eastern coffee into a tall, transparent jar containing water. I had left the water undisturbed-for an hour so that it was still. I added freshly brewed coffee because I wanted grains that had lost most of their adsorbed air. Unbrewed grains would have tended to float.

With the aid of a flashlight I monitored the descent of the particles. Large ones sank quickly to the bottom at a speed of about one centimeter per second. Smaller grains were much slower. Some of the smallest ones appeared to take 10 minutes or more to reach the bottom.

The settling rate of particles in a demitasse is somewhat different from what I saw in my demonstration. The number of particles per unit of volume is large enough to make it likely that the particles interact as they fall. More important, the fluid is hot enough to create circulation cells. Hot fluid from near the bottom rises to the top, cools by evaporation and then descends. The circulation forms what are called Benard cells, after the French scientist Henri Benard. In some fluids and under certain heating conditions the cells create a stable geometric pattern on the top surface.

In my coffee the pattern is chaotic and constantly changing. It is best seen when the coffee lacks foam, since foam not only covers the surface but also alters the evaporation rate. I rig a light that reflects off the surface, revealing that over the region of ascending hot-fluid tiny drops are suspended just above the surface. They condense from the water vapor when it cools in the air just above the surface. No drops are visible over the regions of descending fluid. Since the drops reflect light, the regions of ascending fluid look slightly white while the regions of descending fluid are the dark color of the coffee. Benard cells in a cup of coffee with no grains are merely a novelty. In Middle Eastern coffee the circulation in Benard cells delays the settling of the grains.

The result of all this physics is that one would do well to strike a balance when drinking Middle Eastern coffee waiting long enough. for the larger particles of coffee to settle to the bottom of the demitasse and for the Benard circulation to decrease but not so long that the drink gets unpleasantly cool. The foam on the coffee makes the compromise easier by insulating the coffee with the vapor trapped in the bubbles, thereby decreasing the cooling rate.

Bibliography

OBSERVATIONS OF AN EARLY MORNING CUP OF COFFEE. Vincent J. Schaefer in American Scientist, Vol. 59, No. 5 September-October, 1971.

GRAIN SHAPE EFFECTS ON SETTLING RATES. P. D. Komar and C. E. Reimers in Journal of Geology, Vol. 86, No. 2, pages 193-209; March, 1978.

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