THE TENDENCY OF A THREADLIKE liquid to draw much of itself into beads has intrigued many investigators. The beads can be found on sticky strung-out liquids such as the circular strands of certain cobwebs and a bit of saliva drawn out between the thumb and forefinger. C. V. Boys, author of the classic Soap Bubbles and the Forces Which Mould Them, demonstrated another form in a lecture at the London Institution many years ago. He dipped a straw into castor oil and coated a quartz fiber thinly with the oil, which quickly pulled itself into beads less than.O1 millimeter in diameter.

Lord Rayleigh examined beading in a set of papers published in 1879 and 1892. His work was based on earlier experiments done by the Belgian physicist Joseph A. F. Plateau, who is still renowned for his pioneering study of thin films. When a liquid is drawn into a thread, surface tension minimizes the surface area by pulling the thread into a cylindrical shape. At the same time, however, waves playing along the thread work to distort it into a periodic series of narrow and wide regions. The waves are unavoidable, arising from the motion when the thread is drawn out, from any slight shaking of the mount and from motion of the air.

Plateau demonstrated that the wavelength of a wave determines whether the thread resumes a cylindrical shape after the wave has passed. If the wavelength is less than the circumference of the thread, the distortion produced by the wave increases the overall surface area. The area of the narrowed regions decreases but that of the widened regions increases much more. As soon as the wave has passed, surface tension pulls the thread back into a cylinder to minimize the surface area.

If the wavelength is greater than the circumference, the distortion decreases the overall surface area. The area of the narrowed regions decreases more than that of the widened regions increases. Then surface tension enhances the distortion, pulling the widened regions into beads and the narrowed regions into a thread thinner than the original one. The spacing of the beads is equal to the wavelength of the wave. A weak thread may rupture, freeing the beads. If a thread is strong, additional waves may create a second set of smaller beads mixed in with the first set-all of them strung together by a thread that may now be virtually imperceptible.

Rayleigh found that the initial liquid thread is most unstable when a passing wave has a certain critical wavelength. A wave of that length generates beads faster than waves of other lengths and therefore dominates the formation of beads. If the viscosity of the liquid can be disregarded, the critical wavelength is approximately one and a half times the circumference of the thread. Waves of about that size leave a number of beads separated by a distance roughly equal to the initial circumference of the thread. When the viscosity is high, as in a thread of molten glass, the critical wavelength is much longer. Beads are fewer, and they are farther apart.

A commoner example of a strung-out liquid is a soap film suspended between two mounts. Jurjen K. van Deen of The Hague has studied several of the stability limits in such an arrangement. (I described some of his equipment in this department last month.) For his experiments on stability he prepared two rings, each of them 50 millimeters in diameter, by soldering copper wire. The rings are mounted horizontally, one above the other, on a stand.

To form a bridging film van Deen blows a soap bubble between the rings and makes sure that it anchors on them. Since the solution gradually drains along the film, fresh solution is provided through a tube near the top of the bubble. An extension of the wire forming the top ring fits loosely into the tube, guiding the solution to the bubble. As liquid drains to the bottom of the bubble, it tends to collect in a drop that makes the bubble oscillate when it detaches. Van Deen eliminates the oscillations by placing a length of copper wire just below the bubble. Because of the wire, the drop detaches gently.

The bubble forms a sphere because surface tension pulls inward, tending to minimize the surface area. Air pressure inside the bubble counters the inward pull. (Actually the cause is the excess of the inner air pressure over the atmospheric pressure.) The bubble is stable when its air pressure is equal to 4S/r, where S is the film's surface tension and r is the radius of the bubble.

In one set of experiments van Deen reduces the bubble's air pressure with a vacuum cleaner, which sucks air out by means of a tube stuck through the top ring. A clamp on the tube controls the rate at which air is removed. As the air pressure drops, the bubble begins to change shape. When the pressure reaches 2S/r, the sides of the bubble form a cylinder. This shape requires less pressure than a sphere because the net curvature of the cylinder is less than that of a sphere. (A cylinder does not curve along its length.)

A cylindrical bubble is peculiar in that its existence depends on the ratio of L, the separation between the rings, to d, the diameter of the rings. If the ratio exceeds , the bubble becomes unstable and collapses to form two smaller bubbles, one on each ring. This limit on stability is less surprising if it is recast in the following form: a cylindrical bubble can exist if L is less than the circumference of the cylinder.

In this form the limit of stability is related to the beading on a liquid thread. Remember that the waves responsible for beading have a wavelength longer than the circumference of the thread. A cylindrical bubble also develops waves from chance disturbances, but their wavelength cannot be longer than L. Hence if L is kept smaller than the circumference of the cylinder, the waves are too short to make the cylinder collapse. Every time a wave passes along the cylinder the overall surface area increases. Surface tension immediately pulls the bubble back into a cylinder. If L exceeds the circumference, the waves can be at a wavelength that exceeds the circumference. Such a wave decreases the overall surface area of the bubble. Surface tension tends to distort the cylinder. As a result the center of the bubble collapses, leaving a spherical bubble on each ring.

In another experiment van Deen anchors a spherical bubble to the rings and pricks the two sections enclosed by them. The ruptures open the bubble to the atmosphere, whereupon the internal pressure drops to zero. This change causes the curvature of the remaining soap film to become zero also. How can the film stretch between two rings without curvature?

The explanation is that the bubble forms a catenoid: a surface generated when a catenary curve is rotated about an axis. (A catenary curve is formed when a stringlike object is suspended between two fixed points.) The catenoid's surface constitutes the minimum area for a film stretched between two rings when there is no pressure inside the structure. The surface also has a net curvature of zero. Although the surface is in fact curved, any point on it is part of a concave curvature in one direction and a convex curvature in a perpendicular direction, so that the net curvature is zero.

This configuration also sets a limit on the film's stability that involves the ratio of L to d The catenary shape can exist only if L/d is less than .663. If the limit is exceeded, the waist of the catenoid suddenly collapses, leaving a flat film on each ring. The reason is that when L/d exceeds .663, the flat films have a combined area smaller than the area of the catenoid. (Actually the combined area of the flat films is smaller than that of the catenoid when L/d exceeds .528. The catenoid can be stable between .528 and .663, however, because its surface area would have to increase momentarily if it could transform to two flat films. Unless the waves on a soap film are vigorous, surface tension causes the catenoid to re-form after the passage of each wave. It fails to h do so when L/d exceeds .663. Then even a feeble wave gives rise to an immediate collapse.)

Van Deen discovered that maintaining a stable catenoid with L/d greater than .5 was difficult. The catenoid nearly always collapsed because of waves from various disturbances. If you experiment with a catenoid film, isolate it from vibrations and air currents.

In the early 1970's M. A. Erle, R. D. Gillette and Derek C. Dyson of Rice University also studied the stable shapes of soap films, but in their arrangement the film formed a bridge between two coaxial, parallel disks. The workers dealt with films that were symmetrical around the central axis and with symmetrical disturbances to the films. In their experiments the film was anchored to protruding rims on the disks. Air was removed from or added to the film through a small hole in one of the disks. When the film was inflated, it eventually reached a limit of stability, a condition in which its sides bulged outward. When more air was pumped in, the film broke free.

When air was removed from the film, the sides moved inward until it reached another limit of stability. Then the center collapsed, leaving a hemispherical film on each disk. The shape of the film just before collapse again depends on L/d The limiting shape is an inward bulge when the ratio is less than and an outward bulge when it is greater than .

Suppose that L/d is less than . Visualize the change in the film's shape as air is removed. Begin with the film at the limiting shape of an outward bulge. As the volume of air in the film decreases, the structure shrinks and its curvature increases, raising the inner air pressure. For any given inner volume of air the film has a unique shape determined by the interplay of the net curvature and the air pressure. When more air is taken out, the curvature begins to decrease and the air pressure drops. When the curvature reaches zero, the side of the film is a catenary.

This configuration is called a type1 catenary. A type-2 catenary can also have a net curvature of zero, and so be compatible with an air pressure of zero inside a film. The mathematical functions for both catenaries are similar, but the type-2 catenary has a narrower waist and a larger surface area. When the film is on rings, surface tension prevents the formation of a type-2 catenary; instead it decreases the surface area by pulling the film into a type-1 catenary.

A type-2 catenary is possible when the soap film is mounted on disks and the volume of air in the film is reduced. Beginning with the film in a type-1 catenary, imagine the change in its shape when more air is removed. Its sides begin to move inward as the inner pressure drops below the external air pressure, and the net curvature is outward. After a minimum air pressure is reached the pressure begins to increase until it again reaches zero and the net curvature is also zero. The curve of the film between the disks is then a type-2 catenary.

Erle, Gillette and Dyson found that a film with a type-2 catenary is possible if L/d is between .47 and .663. At any value less than the lower limit the film always collapses into two hemispherical shapes before the internal pressure reaches zero for a second time. If L/d is, say, .5 and the soap solution is long-lasting, the film can survive as a type-2 catenary for more than an hour.

The research team also investigated the stability of greatly inflated films. Theoretically, if the film is symmetrical with respect to the central axis and if only symmetrical disturbances are considered, the film can bulge outward quite significantly before it becomes unstable. The calculated limit is reached when the surface of the bridge is perpendicular to the back of each disk.

Recently M. R. Russo and Paul H. Steen of Cornell University considerably reduced this theoretical limit by examining asymmetrical disturbances. They prepared a silicone-oil bridge that stretched between two brass plates. The bridge was surrounded by a mixture of n-propanol and water that had the same density as the oil. The match of densities effectively removed the distortion normally caused by gravity because the buoyancy on the bridge was equal to the weight of the bridge.

The liquids were put in a Plexiglas cell to eliminate evaporation and consequent changes in density. The oil was added to or removed from the bridge by means of a flexible tube inserted through a small hole in one of the plates. A syringe at the other end of the tube facilitated the change in the volume of the oil.

Beginning with a cylindrical oil bridge, Russo and Steen decreased the distance between the plates by turning a micrometer drive. Because the volume of oil remained constant, the bridge billowed outward. Each plate's beveled edge and coating of paraffin wax helped to anchor the oil on the plate. The bridge remained symmetrical until its surface at the plates was tangent to the plane of the plates. When the distance between the plates was further decreased, increasing the bridge's outward bulge, the bridge suddenly became distorted on one side. The buckling reduced the surface area of the bridge.

Cylindrical bridges of liquid have been the subject of study by J. M. Haynes of the University of Bristol, who considered their stability as their length increases. He knew that they are unstable when L/d exceeds . The bridges break up into hemispheres on the plates to which they are anchored. Haynes also discovered that they show surprising stability at smaller ratios. When L/d exceeds 9/4, the surface area of a cylindrical liquid bridge is actually more than the total surface area of the hemispheres. If the ratio is less than , however, the bridge does not collapse to form hemispheres because to do so it would momentarily have to increase its surface area. That barrier disappears when L/d reaches .

Several readers have commented on what I said in June about the Minotaur Cube. They found two solutions instead of one. The first letters to arrive were from John Stewart and Jim Rostirolla of Bellevue Community College in Bellevue, Wash., Leonard Gordon of Chico, Calif., and Michael Keller of Ellicott City, Md. Keller found the first solution "by hand" and then was surprised when his computer turned up the second one. The solutions have identical plays for pieces A and B. The problem as I gave it was to employ the six pieces of the puzzle-three quadcubes and three pentacubes-to form a cube with three units on a side.

Bibliography

STABILITY OF FLUID INTERFACES OF REVOLUTION BETWEEN EQUAL SOLID CIRCULAR PLATES. R. D. Gillette and D. C. Dyson in The Chemical Engineering Journal, Vol. 2, No. 1, pages 44-54; January, 1971.

STABILITY AND OSCILLATIONS OF A SOAP FILM: AN ANALYTIC TREATMENT. Loyal Durand in American Journal of Physics, Vol. 49, No. 4, pages 334343; April, 1981.

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