Time to Relax II

Time to Relax I

An amorphous material's constituent atoms or molecules lie in random positions rather than at well-defined sites as in an orderly crystal lattice.

In a crystalline material, atoms or molecules sit in an orderly array (left). In amorphous solids, the pattern is more irregular (right).

Moreover, just as crystal structures are rarely perfect and contain dislocations, vacancies, and other 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 the cooling process to orient themselves into their proper positions to form a closely packed crystal structure. Inevitably, glasses end up containing low-density regions, the analog of vacancies in crystals.

In 1983, physicists Michael Shlesinger and Elliott Montroll 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. (See An Unbounded Experience in Random Walks with Applications by Michael F Shlesinger.)

They suggested that defects, in order to move, have to overcome different energy barriers scattered throughout the material. Whereas small barriers are easy to hurdle, larger ones significantly reduce mobility.

In the early stages of relaxation, defects that experience low barriers don't have any trouble. There's enough thermal energy for them to jump such barriers, and relaxation ensues. Others, faced with moderate barriers, take 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. Relaxation stretches over a long period of time.

Mathematically, the situation is closely related to the problem of determining the length of a fractal. 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 answers, depending on the scale used.

Scale matters. Taking long steps carries you past a lot of tiny indentations (top). Taking shorter steps means that you end up traveling a longer distance along such an indented shoreline (bottom).

On a world globe the size of a basketball, the eastern coast of the United States looks like a fairly smooth curve, which, according to the globe's scale, may be 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 now evident 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 be included in the measurement.

Just as every distance scale occurs in the coastline problem, every time scale occurs for relaxation in amorphous materials. Each shift in time scale—from days to minutes to seconds—adds new features to be included in a relaxation measurement.

Although it isn't as picturesque to think of infinitely many time scales as it is to think of patterns within patterns on different length scales, the analogy is mathematically exact.

This comparison leads to the concept of fractal time. Instead of occurring in a sequence of regular, equally spaced intervals, events that occur in fractal time are clustered.

Instead of occurring at regular intervals, events that happen in fractal time are clustered in a self-similar pattern that features rapid bursts interspersed with long pauses.

Such clusters consist of events that happen rapidly, one after the other, interspersed with long stretches of nothing happening in between.

To support this theoretical picture, researchers have discovered that in polymer relaxation, some phenomena occur within picoseconds whereas other effects aren't apparent for years. Such an astonishing array of time scales shows how tricky it is to do experiments investigating the phenomenon because it's hard to measure physical characteristics over so many orders of magnitude in time.

Time to Relax III