Time to relax: the concept of fractal time ties together the stretchiness of silk and the brittleness of polymers.

Time to Relax

The rubber in a pair of boots, retrieved after a long sojourn in an attic, shows its age in an annoying way. No longer as flexible as it once was, the material -- an elastomeric polymer -- readily cracks and falls apart. Under the same conditions, many other plastics suffer a similar fate.

One cause is chemical. Sunlight or oxygen can initiate chemical reactions that alter the material's properties. But deterioration occurs even when a material is kept in the dark or away from oxygen. The material gradually becomes denser amd more brittle, losing its toughness and impact resistance.

The explanation lies in the way "defects" within amorphous, or noncrystalline, materials reorganize themselves over long periods of time. When expressed in terms of a relatively new concept known as fractal time, the same mathematical model used to describe polymer aging also applies to the stretching of glass or silk fibers, the recovery of glassy materials after a stress has been removed and a wide range of other phenomena in amorphous materials.

"It's an issue of practical importance," says John T. Bendler of the General electric Research and Development Center in Schenectady, N.Y. Slow aging processes, both environmental and physical, control the lifetime of a great variety of manufactured products, from electronic devices to optical fibers and advanced composite materials. The new theory of how such processes occur suggests novel techniques for toughening ceramics and for designing polymers having particular characteristics.

Pull on a glass fiber, then let go. The glass first stretches, then shrinks. Apply a strong electric field to a polymer, then turn it off. Areas of positive and negative charge in the polymer line up with the field, then drift out of alignment. In each case, the material endures a stress, then recovers or "relaxes" when the stress is removed.

Relaxation in a crystalline material typically proceeds at an exponential pace. That type of relaxation follows the same pattern as the decay of a radioactive isotope. Such a process is characterized by a certain time, known in the case of radioactive decay as the half-life.

In contrast, physicists have discovered that an amorphous solid takes longer to relax than would be expected if the relaxation simply followed an exponential decay. No characteristics time can be defined for such an extended relaxation process.

"Normally, one finds that relaxation is clustered around a certain time," Bendler says. "It takes either a second, a day or a week. What we see in amorphous systems is that some parts relax very quickly, say, in seconds. But other pieces relax on a time scale of minutes, still others at days or weeks. If we wait long enough -- even years -- we still see things happening. There is no definable time scale."

This type of behavior goes by the name of "stretched exponential relaxation." It fits a wide range of relaxation processes in disordered systems, including the way many polymers, glasses and ceramics respond to stresses caused by changes in pressure and temperature, and the imposition of electric and magnetic fields.

Because so many different systems behave in such a strikingly similar fashion, physicists, in their search for an explanation, have concentrated on what these systems share. They find what's important is not the details of a material's atomic and molecular structure but rather its state of disorder.

An amorphous material's constituent atoms or molecules lie in random positions rather than at well-defined sites, as happens in an orderly crystal lattice. Moreover, just as crystal structures are rarely perfect and contain dislocations and imperfections, amorphous materials also contain "defects," in which bonds between atoms or molecules may be strained, distorted or displaced. For example, such defects occur during glass formation because molecules find they have too little time during cooling to orient themselves into their proper positions to form a closely packed crystal structure. Inevitably, glasses end up containing "vacancies," or low-density regions.

In 1984, Michael F. Shlesinger of the Office of Naval Research in Arlington, Va., and Elliott W. Montroll of the University of Maryland in College Park proposed that migration, or diffusion, of mobile defects could account for stretched exponential relaxation in the case of an amorphous material relaxing after the application of an electric field. They suggested that defects, in order to move, must overcome different energy barriers scattered throughout the material. Whereas small barriers are easy to hurdle, larger ones significant reduce mobility. Consequently, relaxation, which occurs through the movement of defects, stretches out over a long period of time.

"In the early stages of relaxation, those defects that are near low barriers don't have any trouble," Bendler says. "There's enough thermal energy, so they can jump and cause relaxation. Others, faced with moderate barriers, are not so well off. It takes them longer to get moving." Thus, a random distribution of energy barriers implies a wide range of relaxation times, leading to the stretched exponential relaxation observed for amorphous materials.

Mathematically, the situation relates close to the problem of determining the length of a fractal, a geometric object in which the same pattern is repeated on ever smaller scales. Magnifying a fractal by any amount reveals a miniature version of the larger form. Finer and finer scales show more and more detail and lead to greater and greater estimates of total length.

For example, measuring the length of a fractal coastline leads to different answer depending on the scale used. On a world globe the eastern coast of the United States looks like a fairly smooth line roughly 3,000 miles long. The same coast drawn on an atlas page showing only the United States looks much more ragged. Adding in the lengths of capes and bays extends the coast's length to 5,000 or so miles. Piecing together detailed navigational charts to create a giant coastal map reveals an incredibly complex curve perhaps 12,000 miles long. Each change in scale reveals a new array of features to include in the measurement.

Similarly, the relaxation processes in amorphous materials have no characteristic time scale, They occur on all time scales, an idea embodied in the concept of fractal time. Each shift in time scale -- from seconds to minutes to days to years -- adds new features to include in a relaxation measurement.

"What we have in amorphous materials is every time scale, just as we have every distance scale in the coastline problem," Bendler says. "It isn't as picturesque to think of infinitely many time scales as it is to think of patters within patterns on different length scales, but the analogy is exact."

To support this theoretical picture, researchers have discovered that in polymer relaxation, some phenomena occur within picoseconds while other effects aren't apparent for years. "It's an astonishing array of time scales," Bendler says. "That makes it tricky experimentally because it's hard to measure things over so many orders of magnitude in time."

Shlesinger and Bendler have now applied the theory to key aspects of glass formation and to various types of relaxation phenomena in polymers. "One is now able to go from microscopic motions to macroscopic behavior," Shlesinger says. "And the theory can provide ideas on ways to modify in a useful manner the properties of industrially important polymeric materials."

Bendler and his colleagues used the theory to explain the properties of Lexan, a tough polycarbonate resin used for making bulletproof windows for limousines. Defect diffusion turns out as a good model for how the material responds to stresses and how it ages.

A chunk of Lexan consists of an irregular, three-dimensional network of long polymer molecules, each a twisted chain thousands of atoms long, with a precisely defined, repeating pattern of atoms. Experiments indicate that cooling the polycarbonate freezes in a small population of high-energy "kinks" in the molecular chains. The kinks occur within segments where a ring of carbon and hydrogen atoms called a phenylene group meets a combination of carbon and oxygen atoms called a carbonate.

It's the movement of these kinks in a fractal-time process along the polycarbonate chains that leads to relaxation. They reorganize the molecular backbone and effectively absorb mechanical energy, such as the impact of a bullet or a sledgehammer. They are also responsible for aging. As energy-absorbing kinks reach chain ends, the material gradually becomes more brittle and weaker. With that insight, researchers now believe they might possibly slow aging by modifying chain ends.

The fractal-time, or defect-diffusion, model also helps explain the stretching of silk and glass threads. In 1835, German physicist Wilhelm E. Weber noticed that attaching a weight to a long thread causes it to stretch to a certain length immediately. But that instantaneous elongation is unexpectedly followed by a gradual further lengthening that depends on how long the weight is applied.

The reason for such behavior lies in the fractal-time movement of defects within the materials. Silk is a complicated natural polymer, occurring in a variety of different amorphous and crystalline forms. Under an applied load, the material tries to rearrange itself to redistribute and minimize stresses. Under those conditions, silk molecules relax by unwinding and changing the hydrogen bonding along their backbones. In a glass fiber, the mobile defects correspond to imperfections in the distorted, tetrahedral network of oxygen and silicon atoms.

"It's these mobile units of structure that cause something to happen under a load," Bendler says. "The material isn't just sitting there. There's a mechanical reorganization."

Although ceramicists, engineers and artisans have long recognized the peculiar behavior of glasses, polymers and ceramics and have taken these properties into account when working with materials, until recently researchers made little progress in understanding relaxation phenomena because the mathematics used to describe such processes seemed so complicated and difficult, Bendler says. Now, the new concepts of mobile defects and fractal-time motion appear to provide a self-consistent picture of viscoelastic and thermodynamic behavior in supercooled liquids and glassy solids.

"One of the chief merits of the theory is that it's so simple mathematically," Bendler says. "We're able to use defect-diffusion mathematics -- the mathematics of intermittent pausing -- to model the kind of behavior displayed by almost all amorphous materials. It gives us a nice, satisfying picture."