BLOWING UP A COMMON rubber balloon is child's play, but the inflation raises several challenging questions. Why is the inflation usually difficult at first and easier once the balloon is partially filled? (You might expect the inflation to get progressively more difficult as you stretch the rubber surface.) If you are inflating a cylindrical balloon, why does the inflation begin at one region and then travel down the length of the balloon? Suppose that you take two identical balloons, inflate them to different sizes and connect them with a tube. Will air flow from one balloon to the other so that they change in size, and if so, will they finally reach the same size?

To answer these questions, I must first explain a simpler object, a spherical soap bubble. The size of a bubble depends on a competition of two forces. One force derives from the internal air pressure, which attempts to expand the bubble. (Actually, the force is caused by the difference between the internal pressure and the smaller atmospheric pressure, but here I shall let "internal pressure" stand for the difference.) Countering the expansion is the surface tension of the soap film. The tension arises from the electrical attractions between molecules on and near the interior and exterior surfaces of the film. The tension attempts to contract the bubble.

When a bubble is blown, the competing forces automatically adjust the bubble's size and pressure until the forces are matched. In that state of equilibrium, the pressure is proportional to the curvature and inversely proportional to the radius of the bubble. Notice that the result means the air in a smaller bubble is under more pressure than the air in a larger one.

To understand why the pressure is less when the radius is larger, consider a cross section through a small patch on the bubble [see Figure 1]. The edges of the patch are pulled outward along the surface by adjacent regions of the film. Each pull can be broken up into a component that is tangential to the center of the patch and a perpendicular component. The tangential components are matched in strength and pull on the patch in opposite directions, and so they cancel each other. The perpendicular components are also matched in strength, but since they both point radially inward, they add. When the bubble is in equilibrium, the air pressure on the patch balances the inward pull from the radial components.

The pull on each side of the patch depends only on the surface tension, which does not vary as the bubble is inflated. How much of the pull is radial, however, depends on the curvature of the surface. Because the air pressure opposes the radial components, the pressure also depends on the curvature. When the bubble is small and tightly curved, the radial components are large, and so is the pressure. If the bubble is inflated to some larger size, the surface becomes less curved, and the radial components and the pressure become smaller.

In principle you can test the result by inflating a soap bubble. Dip a drinking straw into a bowl of water mixed with dish detergent and then blow into the other end of the straw to start a bubble. As the bubble grows, the air pressure you must produce decreases because the pressure in the bubble decreases. Of course, the expansion is so easy, even initially, that you may not notice any difference.

Because the surface tension in a soap bubble does not vary during inflation, the resistance to further stretching of the surface does not vary either, and the internal pressure is set only by the curvature. The rubber membrane of a balloon is different. It is elastic: the more it is stretched, the more it resists further stretching. The resistance can again be attributed to surface tension, but the tension does not come from a simple electrical attraction between molecules. Rather it comes from a cross-linked network of flexible, long-chain molecules. When the membrane is stretched, the interlinked molecules resist, trying to contract the membrane to its original size. The more the membrane is stretched, the harder the network resists.

The elastic property of a rubber balloon greatly complicates any study of how the internal pressure varies when the balloon is inflated. As with a soap bubble, the curvature still matters because it determines how much of the pull on the sides of a patch is radial. But with the balloon, the pull itself changes in size because of the elasticity. As the balloon is inflated, the decrease in the curvature tends to reduce the pressure while the increase in the stretching tends to raise it.

To see which tendency wins out during the inflation, one needs a model of how the rubber in a balloon stretches. In 1986 Graham Read of the Open University in Milton Keynes, England, suggested that the stretching can be likened to the expansion of a spring. A spring resists stretching with a force that is proportional to the degree of elongation. For example, if you stretch a spring to twice its previous length, the force resisting you is twice as large as before, and so you must pull twice as hard to maintain the elongation or to increase it somewhat.

With his model Read computed how the pressure inside a spherical balloon varies with the radius of the balloon. The results are illustrated in the graph at the top left. (Recall that the pressure is actually the difference between the interior and exterior.) According to the computation, as the balloon expands, the pressure first increases in a region labeled A, peaks and then decreases in a region labeled B. In A the variation in the pressure is dominated by the rubber's resistance to stretching. In B the variation is determined by the decrease in the curvature. The peak is where the curvature takes control and the balloon begins to act like a soap bubble.

To follow the curve, imagine that you inflate a balloon. The balloon is initially flat, and your first puff serves make it spherical without stretching the rubber. Let R be the radius of the balloon just then. The pressure and radius are represented by the lowest point of the curve in A.

When you blow in another puff of air, the point representing the balloon on the graph is driven up the curve: the balloon expands and the pressure increases until the balloon reaches an equilibrium state. Because the rubber's resistance to stretching dominates the balloon's behavior in A, the pressure of successive puffs must increase to expand the balloon further, to new equilibrium states. Read found that the peak pressure for an equilibrium state is reached when the radius is 2R. When the radius is even larger (in B), each additional puff of air leaves the balloon in an equilibrium state with a lower pressure, and so the inflation becomes progressively easier. As you continue to blow into the balloon, you still must exceed the equilibrium pressure, but the equilibrium pressure gradually wanes.

According to Read's analysis, the pressure falls until the balloon is so large that it has no curvature. At that ideal limit, the tension would be quite large, but the pull on the sides of any patch on the balloon would have no radial, inward component. Of course, a real balloon would burst well before it reached such a limit.

Another study of a balloon's inflation was published in 1978 by David R. Merritt and Frederick Weinhaus, both then at the University of Santa Clara. They employed a more complicated model of how the rubber of a balloon stretches, but their graph of the pressure versus radius is similar to Read's [see Figure 3]. Again the pressure increases to a peak and then falls. They calculated that the peak pressure occurs at a radius of about 1.38R. In addition, they noticed that if a balloon is inflated to a radius of several times R, the pressure begins to climb again, in a region I call C The increase in pressure indicates that when the rubber is highly stretched, the resistance to further stretching increases faster than the curvature decreases.

A related study of inflation was published in 1973 by H. D. Crane of the Stanford Research Institute. Crane considered how a membrane would inflate if it were fastened over the opening of a tube in which the air pressure could be controlled. This arrangement differs considerably from the previous examples. When you blow into a bubble or balloon, you deliver a puff of air under a pressure that is higher than the existing pressure, but you do not maintain your pressure for more than a moment. In Crane's arrangement a regulator attached to a source of compressed air maintains the pressure continuously.

Suppose that a soap film spans the opening on the tube. Recall from the examples with soap bubbles that air pressure inside such a film is proportional to the curvature. Before the regulator is opened, the film is flat-it has no curvature. As the pressure is increased, the film bulges outward from the tube and the curvature increases in response. When the radius of the film matches the radius of the tube, the film is hemispherical. The curvature is then at a maximum, and so the pressure that the film can withstand reaches a peak. If the pressure is increased a little further, the film must expand outward, but the expansion reduces the curvature and hence the film's resistance to pressure. The film is no longer stable. Instead it expands uncontrollably until it bursts.

The soap film's behavior is charted in the left-hand illustration in Figure 5, where the pressure is plotted against the volume contained by the film. The graph resembles earlier graphs, although the horizontal axis shows volume instead of radius. The film's controlled expansion into a hemisphere takes place in A. The rapid expansion at the peak pressure is represented by the broken line that extends horizontally from the peak. The curve in B shows the pressure that could be balanced by the curvature when the film is larger than a hemisphere. The difference between the curve and the broken horizontal line is the pressure that drives the film outward during its uncontrolled expansion.

The analysis is similar when a rubber membrane is substituted for the soap film on the end of the tube. Now, however, the graph of pressure versus volume has a region C in which the curve climbs a second time. C reflects the elasticity of the membrane; it is where the resistance to stretching begins to increase faster than the curvature decreases, thereby driving up the pressure.

Again imagine what happens when the pressure in the tube is gradually increased. A point representing the membrane first climbs through A until it reaches the peak. If the pressure is increased another notch, the membrane suddenly expands because the curvature cannot withstand the pressure. Like the soap film, the membrane expands dramatically. Just before the peak pressure is reached, the membrane swells only moderately from the tube. When the regulator is opened another notch, the membrane suddenly expands to several times its previous size. Yet the expansion does not continue uncontrollably, because the membrane can reach a new equilibrium state in C.

Next imagine what happens if the air pressure is slowly decreased. The point representing the membrane gradually slides down the curve until it reaches the lowest point-the bottom of the valley between B and C If the pressure is then decreased slightly more, the curvature is too much for the pressure, and the membrane suddenly contracts to a new equilibrium state in A. The broken horizontal line extending to the left from the valley represents the contraction.

Crane suggested that a similar sudden contraction may be seen when an inflated balloon is released and allowed to rocket around the room. Suppose the balloon is initially inflated into region C As air is discharged, the pressure drops, and the point representing the balloon on a graph falls to the valley between regions B and C There the curvature of the balloon overwhelms the remaining pressure and suddenly ejects the rest of the air, which gives the balloon a final spurt.

How do these considerations of size and pressure apply to cylindrical balloons? In 1984 E. Chater and J. W Hutchinson published a study of how a cylindrical balloon expands. As with a spherical balloon, blowing up a long balloon is hard at first, and then it becomes easier. In contrast to the inflation of a spherical balloon, however, the expansion is at first restricted to one spot, usually at the end nearest the opening.

I will follow the graph of pressure versus volume that was developed by Chater and Hutchinson [see Figure 7], but I will interpret the mechanics of the inflation somewhat differently than they did. As the pressure is driven up through A on the graph, the balloon expands radially in a short bulge. The bulge initially behaves like a spherical balloon: once the pressure peaks, the decreasing curvature around the circumference of the bulge allows the pressure to decrease through region B. When stretching of the rubber in the bulge becomes extreme and the pressure is forced to rise a second time, however, something different happens. Just as the pressure begins to increase again, the bulge starts to extend.

Consider what happens when one more puff of air is blown into the balloon. The balloon needs to expand somewhere. The existing bulge resists further expansion because of the extreme stretching of the rubber there. Most of the uninflated part of the balloon also resists inflation because it lies in region A, dominated by the rubber's elasticity. Yet the uninflated part adjacent to the bulge is actually easy to inflate because of the membrane's curvature. Although the membrane is convex around the circumference of the bulge, it is concave for a short distance where the bulge meets the uninflated part. Components of the tension along the concave shape point radially outward, and so they actually help to expand the surface when more air is blown into the balloon. As a result, the front of the bulge travels along the balloon. Until the front reaches the end of the balloon, the pressure stays constant.

If you stop the inflation before it is complete, you can manually collapse a spot in the inflated region. Squeeze the spot and shove the resulting smaller bulges off toward either end of the balloon. Twisting the balloon where you collapsed it also serves to drive the bulges apart. When you release the collapsed spot, it stays collapsed. Just like the uninflated part at the far end of the balloon, it cannot expand again unless you blow more air into the balloon. (Some party entertainers collapse several partially inflated balloons so that they can be wrapped around one another to form novel shapes.)

I now return to the question about balloons that are connected by a tube. Again soap bubbles can serve as a simple model. Suppose two bubbles of different sizes are connected by a tube with a valve that is closed at first so that air cannot flow between the bubbles. What happens when the valve is opened? Because the smaller bubble is under more pressure than the larger one, the smaller bubble collapses as its air flows into the larger bubble, which expands. The smaller the first bubble becomes, the harder it will push its air into the larger one because the pressure imbalance grows.

If the original bubbles are identical in size, they can remain so in principle, but their equilibrium states are unstable. If either bubble experiences the slightest chance disturbance, one will collapse and the other will expand.

The demonstration of interconnected bubbles was published in 1902 by Charles V. Boys, the British scientist who is remembered for his popularization of science. He did not comment on the final shape of the smaller bubble. I wondered about the final shape after reading about Crane's work with a soap film on a tube. The pressure in the smaller bubble does not increase steadily as it shrinks: instead it reaches a peak when the bubble contracts to a hemisphere and decreases thereafter, as the curvature decreases. In the meantime the pressure in the expanding larger bubble must also be decreasing since its curvature is also decreasing. Depending on the initial conditions, the bubbles must reach some final states of equilibrium in which their pressures match. The smaller bubble would then be smaller than a hemisphere, but not flat.

What happens when the bubbles are replaced with rubber balloons that are inflated to different sizes? The answer was given in 1978 by Weinhaus and William A. Barker, then his colleague, and independently in 1986 by Read. The answer depends on the initial states of the balloons when the valve between them is opened. If both balloons are in region A of their pressure-versus-radius graphs, the larger balloon will transfer air to the smaller balloon until the balloons are the same size and under the same pressure.

If the amount of air is greater than a certain critical value, however, the balloons will come to a common pressure but will have different sizes. One will be in A, the other in B. The critical amount of air is the amount needed to inflate both balloons to the pressure peak between A and B. One balloon could begin in A with a small amount of air, provided that the other balloon begins in B with enough air to put the total amount of air over the critical value, or both balloons could begin in B. On the other hand, if one balloon begins in A and the other in B but the total volume of air is below the critical value, then both balloons will end up in A with the same size and pressure.

Bibliography

SWITCHING PROPERTIES IN BUBBLES, BALLOONS, CAPILLARIES AND ALVEOLI. H. D. Crane in Journal of Biomechanics, Vol. 6, No.4, pages 411-422; 1973.

ON THE EQUILIBRIUM STATES OF INTER-CONNECTED BUBBLES OR BALLOONS. E Weinhaus and W. Barker in American Journal of Physics, Vol.46, No.10, pages 978-982; October, 1978.

THE PRESSURE CURVE FOR A RUBBER BALLOON. David R. Merritt and Frederick Weinhaus in American Journal of Physics, Vol. 46, No. 10, pages 976-977; October, 1978.

ON THE PROPAGATION OF BULGES AND BUCKLES. E. Chater and J. W. Hutchinson in Journal of Applied Mechanics, Vol. 51, No. 2, pages 269-277; June, 1984.

AN APPLICATION OF ELEMENTARY CALCULUS TO BALLOONS. Graham Read in European Journal of Physics, Vol.7, pages 236-241; 1986.

The Society for Amateur Scientists (SAS) is a nonprofit research and educational organization dedicated to helping people enrich their lives by following their passion to take part in scientific adventures of all kinds.