MANY POLYMER SHEETS, INCLUDING some common plastic food wraps, behave peculiarly when they are stretched. They neither snap like a thread nor expand like a rubber band. Instead they first strongly resist the stretching and then suddenly yield by narrowing in thickness or in width (perpendicular to the direction of stretch), or in both dimensions. The narrowing is called necking or cold drawing. (The second term refers to a similar narrowing that a hot metal rod undergoes when it is drawn.)

Test a length of stretchable plastic sheet by pulling on the two ends. At first you need to pull hard to get any movement at all, but then the plastic abruptly stretches and necks. Once it has necked, the plastic becomes easier to stretch, and the narrowing begins to travel toward the ends. After the necking has spread appreciably, though, you must again pull hard, and finally the plastic rips somewhere in the narrowed region.

Before I explain the mechanics of necking, let me mention a rather puzzling effect of the phenomenon. Some polymer sheets are transparent but murky. To read a printed page through one, you have to hold the sheet close to the page; as you move the sheet farther from the page, the print soon becomes too obscure to read. The murkiness is caused primarily by the scattering of light by the molecules of the sheet. The greatest distance at which you can read through a sheet is one measure of how strongly the light is scattered.

Now, if you stretch the sheet and it necks in thickness, the light passes through less material-fewer molecules-and the net scatter should be less; you should be able to hold the sheet farther from the page and still be able to read the print. As logical as the argument seems, clearly it does not hold true in the situation pictured below. To make the photograph, I stretched and necked half of a polymer strip, laid the strip flat on a microscope slide and then, with modeling clay, propped up the slide over a printed page at an angle, so that the distance between the page and the strip varied. In the section of the unnecked region farthest from the page the print is blurred but readable, whereas in the necked region the print blurs out completely even where the strip is much closer to the page. The reason is subtle, as will appear below.

A polymer is a large molecule built up by the repetition of some basic chemical unit called a monomer. The many different polymers encountered in daily life present a wide range of chemical structures and consequently of mechanical and optical properties. From the subset that displays necking, I chose two examples to study. One is polyethylene, most commonly found in the kitchen in the form of storage bags or sheets-often self-sticking- for wrapping foods. The monomer of polyethylene is a simple array of two carbon atoms and four hydrogens. In a sheet of polyethylene, regions where the long polymer molecules form tiny crystals are separated by amorphous regions that lack any organization. The other polymer sheet I chose to study is Parafilm, which is ubiquitous in biological and chemical laboratories, where it serves to seal off beakers and other containers. Parafilm is a mixture of wax and polyethylene.

Necking is in large part a consequence of the orientation of polymer molecules. Consider a polymer sheet that is necking in thickness, and imagine that you can see the molecules in the sheet. Initially they might be organized on a small scale-partially crystallized or somehow aligned by the manufacturing process-or they may have completely random orientations. As you pull, the sheet is able to stretch only if the molecules can be made to turn or shift to line up with the direction of the pull and thereby accommodate the increase in length. At first their chemical bonds resist the reorientation, and the sheet stretches only slightly. But once the pull reaches some critical value, the molecules in the weakest region of the sheet surrender, break their weaker bonds, slip over one another and move into alignment [right]. The sheet grudgingly yields by sacrificing thickness for length in that region. The sheet does gain strength, though, in the sense that the reoriented molecules have shown their weakest connections.

If you continue to pull, you align more of the molecules in the necked region and further diminish the thickness. Once many of the molecules are aligned, the thickness is at its narrowest, and then the bonds among the molecules are too strong to allow any further yield. If you keep pulling, the molecules in the "shoulders" of the necked region are the next to submit, which spreads the necking along the sheet in the direction of your pull. When the full sheet is necked, the "hardened" nature of the molecular interconnections throughout the sheet requires that you once again pull strongly on the sheet to stretch it still more, and soon the sheet snaps rather than give in much further.

To prepare Parafilm to be necked, I cut a rectangular strip with scissors, laid the strip out flat on a table and then tacked each end down with sturdy packaging tape. Next, with ruler and pen, I marked off lines that ran across the width of the strip and were spaced two millimeters apart. Then I transferred the strip to a laboratory jack, pressing each taped end firmly to a face of the jack. (You could substitute a household vise.) The strip extended vertically between the jack faces; it was straight but was under no tension.

I recorded the vertical length and the left-to-right width of the strip. To measure its thickness with a micrometer, I first backed off the instrument's mobile prong enough to allow the strip to slip through the prongs and then gradually tightened the prongs while also gently moving the micrometer around. When the prongs were close enough to catch on the strip, I backed them off just enough to eliminate the catch and recorded the micrometer setting. I repeated the measurement several times and averaged the results. The strip was approximately .14 millimeter thick.

Now I began to turn the jack screw, moving the two faces of the jack apart and thereby stretching the strip. The turning was difficult at first, but when I had stretched the strip by about 9 percent, it suddenly yielded: it necked in thickness within a narrow band that ran across its width. I found that inked lines in the necked region were separated by an additional 25 percent; the lines elsewhere showed no extra separation.

As I continued to turn the jack screw, the necking inched toward the two faces of the jack, stretching apart lines along the way. The progress was not uniform: after each turn of the screw, the shoulders of the necked region were marked with "islands" of unstretched material that were visibly different from the stretched material surrounding them. With each turn of the screw I measured the length, width and thickness of the strip and the separation of lines at various spots along the strip. When I had stretched the strip by about 90 percent, all the lines on the strip displayed additional separation.

When the stretch reached 150 percent (that is, when the strip had been stretched to two and a half times its original length), the lines in the initially necked region were as much as nine millimeters apart (which corresponds to a stretch of 350 percent) and the thickness was only .064 millimeter. Elsewhere the lines were 3.5 millimeters apart (the material between them had been stretched by 75 percent) and the thickness had been somewhat reduced. The stretching and thinning were obviously spreading along the strip. With one more turn of the screw, the strip began to tear in the necked region, probably because of a nick I had left along the side of the strip with the scissors. (I could have sealed the tear with a small patch of packaging tape but had not done so.)

The stretching of the strip can be followed with a graph in which "conventional stress" is plotted against "strain". The strain, which is the same as the extent of stretch, is the ratio of the change in the length of the strip to its initial, unstretched length. For example, when the length of the strip is doubled, the stretch is 100 percent and the strain is 1.0. The conventional stress is the ratio of the pull on the strip to the area of a cross section of the strip along the edge of a face of the jack [see Figure 6]. That area did not change in the course of the experiment, and so any change in the conventional stress reflected the change in the strength of the pull.

Because I could not measure the pull, I have graphed only my subjective observations. As I first turned the screw, the stress climbed and the strip stretched marginally. When the strain reached .09, the strip suddenly necked and lengthened, thereby relieving the pull by the jack and decreasing the stress on the strip. As I turned the screw more, increasing the strain, more of the strip necked and the stress decreased even further. If I had been able to neck the full strip without its tearing because of an accidental nick (I came close with several samples), the stress would have begun to increase again as strongly linked molecules resisted further stretching. Eventually the stress would have snapped the strip like a thread.

Before necking appears, the conventional stress applies to the entire strip and also to any given section. After necking, however, the stress in the necked region is larger than the conventional stress because the cross-sectional area is reduced there. Such local stress is called the true stress; it is the ratio of the pull on a region to the region's cross-sectional area. The strain in the region is said to be the local strain.

In the illustration at the right above, true stress is plotted against local strain before and after necking appears. Prior to necking, every section of the strip undergoes the same stress and strain, and the point representing those values climbs up into the broken part of the curve. Just as the strip necks, point A, representing unnecked regions, slides back down the curve and point B, representing the necked region, climbs higher on the curve: the necked region is under more stress and strain than the unnecked regions. The line directly connecting A and B indicates stress and strain through the shoulders.

I repeated the stretching experiments with strips of Glad Cling Wrap, a clear polyethylene sheet. Like Parafilm, a strip of the food wrap strongly resisted the initial stretching and then suddenly necked. Unlike Parafilm, however, it necked in its left-to-right width rather than in thickness. The narrowing began in one place and then gradually spread up and down the strip, giving it a distorted hourglass shape. Just before the strip ripped, it was stretched by 180 percent and the initial width of three centimeters had a been reduced to .9 centimeter at the narrowest region. Inked lines there indicated a local strain of about 4.0.

I now return to the matter of reading print through a polymer sheet. The matter was discussed in 1973 by David Miller of the Beth Israel Hospital and Harvard University and George B. Benedek of the Massachusetts Institute of Technology in their book Intraocular Light Scattering. They called the effect "the nude in shower phenomenon." (I am grateful to Craig F. Bohren of Pennsylvania State University for pointing out the reference.)

To see how the nickname for the effect got its start, imagine that you watch a bather through a rigid shower enclosure or curtain made of textured plastic. If the person is close to the plastic, body features are readily visible, but if the person is farther from the plastic, the features are too murky to distinguish. A similar dependence of visibility on distance can be observed with a strip of common transparent plastic tape. Hold the tape (adhesive side up to prevent sticking!) just above this page, and then gradually move it upward. As its distance from the page increases, the words begin to blur and become unreadable.

To understand the blurring, consider a dot on the page. When light travels from the dot up through the tape, it scatters from the tape's molecules [see drawing at left in Figure 7]. The scattering spreads each original ray into a bundle of rays forming a narrow cone centered on the direction of the original ray. The cone can be characterized by its half-angle: the angle between the most severely scattered rays in the cone and the direction of the original ray.

Suppose that the tape is just above the dot and that you look down on it from a distance of at least 40 centimeters with one eye closed. Your open eye then intercepts rays that are approximately perpendicular to the tape and that have been scattered by molecules along, or adjacent to, the direct line to the dot. Rays that scatter from molecules farther out miss your eye and do not contribute to your vision. To perceive the source of the intercepted rays, your brain automatically extrapolates them back to the paper, where they appear to have originated from a small spot centered on the actual dot. The spot will not be as crisp as the dot, but it is still recognizable. Its radius is approximately equal to the product of the dot-to-tape distance and the tangent of the half-angle of the scatter cones.

Now lift the tape a few millimeters. As is shown in the drawing at the right in the illustration, the half-angle of each scatter cone is unchanged, but the extra dot-to-tape distance spreads the light on the tape, and so you intercept rays from a somewhat larger region of the tape around the direct line between the dot and your eye. As you extrapolate the rays backward, they appear to originate from a somewhat larger spot on the page. The spot is now less distinct; it is harder to recognize it as being a dot.

Suppose there is a second dot next to the first one. When the tape is near the dots, the perceived spots are small and sharp enough to be distinguished as being separate. As you lift the tape and the spots widen, they eventually overlap too much to be distinguished. So it goes with the nude behind the textured plastic: when the figure is close to the plastic, details are distinct and recognizable, but when the figure is farther away, the details blur out.

While experimenting with Parafilm, I noticed that it obscured the details of a printed page seen through it, just as tape does. I assumed that the extent of blurring must depend on the thickness of the sheet. For a certain thickness, each ray from a given detail must pass through a certain number of molecules and thus be spread into a cone with a certain half-angle, and that half-angle should determine how far from the page I can hold the sheet and still make out the detail. Presumably, if I thinned the sheet by necking it, the light would be scattered by fewer molecules and be spread into a narrower cone, and I should then be able to distinguish a detail with the sheet farther away from it.

To test the idea, I placed a small section of unstretched Parafilm over the hole in a mechanical-drawing template and positioned the hole over three small dots I had penciled on a sheet of paper. The dots formed a corner and were separated vertically and horizontally by a millimeter. I lifted the template while looking at the dots through the Parafilm. When the Parafilm was about two centimeters from the paper, the dots blurred together. I repeated the observation a number of times and averaged the "blur-out" heights.

I next substituted a small section of Parafilm that had been necked with the jack. To my surprise, I found that the dots blurred out when the section was only half a centimeter above them. Moreover, the two dots aligned with the section's direction of stretch blurred out about a millimeter lower than the two dots oriented perpendicular to that direction. (Of course, that discrepancy could have been caused by an unequal separation between the dots in the horizontal and vertical directions, but the discrepancy persisted even when I turned the paper around my line of sight by 90 degrees.) Although my observations were certainly crude, they indicate that when the Parafilm necks, it scatters light in cones that are roughly four times broader than the cones created when the sheet is unstretched.

I think the increase in scattering results from the alignment of molecules that is brought about by the necking. When light travels through a transparent material in which the molecules are randomly oriented or in which their organization is on a scale much smaller than the wavelength of light, each molecule is said to scatter the light independently of the other molecules. In such a case, any light that is scattered out of the forward direction is likely to be canceled by light scattered in the same direction by another molecule. (That is, each crest of one wave falls on a valley of the other wave; the waves interfere destructively.) The half-angle of a scatter cone is thereby kept small.

If, instead, there is some ordered arrangement of the molecules, and if the spacing associated with the arrangement is about the size of the wavelength of light, the molecules scatter not independently but in an organized way. When two light waves are then scattered in the same direction, the crests of one wave may not fall precisely on the valleys of the second wave; the cancellation of light scattered to the side is less complete, and the half-angle of each scatter cone is broader. Apparently, when I aligned the molecules by necking the Parafilm, I organized the scattering, widened the scatter cones and increased the blurring by the sheet.

Bibliography

X-RAY DIFFRACTION STUDIES OF THE STRETCHING AND RELAXING OF POLYETHYLENE. Alexander Brown in Journal of Applied Physics, Vol. 20, No. 6, pages 552-558; June, 1949.

THE NECKING AND COLD-DRAWING OF RIGID PLASTICS. P. 1. Vincent in Polymer, Vol. 1, No. 1, pages 7-19; 1960.

ON THE EXTENSION OF THE NECK OF POLYMER SPECIMENS UNDER TENSION. G. I. Barenblatt in Journal of Applied Mathematics and Mechanics, Vol. 28, No. 6, pages 1264-1276; 1964.

INTRAOCULAR LIGHT SCATTERING: THEORY AND CLINICAL APPLICATION. David Miller and George Benedek. Charles C. Thomas, Publisher, 1973.

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