Magic Sizes Found In Metal Alloys

September 22, 1997

BERKELEY, CA -- Scientists at the Ernest Orlando Lawrence Berkeley National Laboratory (Berkeley Lab) have made the surprising discovery that tiny crystals of lead in an aluminum matrix come in only a few specific sizes -- and a handful of unusual shapes -- and avoid others. The discovery sheds new light on the strange behavior of solids at nanometer scale.

Ulrich Dahmen, one of the discoverers of the "magic size" phenomenon and head of Berkeley Lab's National Center for Electron Microscopy (NCEM), says he and collaborators from the University of Copenhagen wanted to find out why the properties of nanoscale particles -- particles measured in billionths of a meter, a scale approaching that of atoms and molecules -- differ so dramatically from those of the same material in bulk.

"We know from the work of Paul Alivisatos here in the Materials Sciences Division, and others, that with free nanoscale particles the melting point may be as little as half that of the bulk material," says Dahmen. "On the other hand, a crystal embedded in a matrix of a different solid may need to be superheated to melt." Dahmen's Danish collaborators, Erik Johnson and colleagues, observed in 1990 that to melt 150-angstrom lead inclusions, they had to be heated 60 degrees Kelvin above their bulk melt point.

"This behavior is like an ice cube refusing to melt in boiling water," says Dahmen. "We didn't understand it, but we hypothesized that it was a shape effect -- the smaller the inclusions, the more likely they are to attain perfect shapes -- so we set out to measure how size and shape were related at this scale."

Dahmen and his colleagues used NCEM's three-story high, one-million volt Atomic Resolution Microscope for their study. They chose to work with lead inclusions in aluminum partly because lead and aluminum, like oil drops in water, don't readily dissolve in each other; boundaries between them remain sharp.

By means of vapor deposition the researchers laid a film of aluminum only a hundred nanometers thick on a silicon wafer, then implanted the lead with an ion beam. Finally the silicon was shaved away until the alloy sample was thin enough to be transparent to the microscope's high- energy electron beam.

The micrographs showed numerous nanometer-sized islands of lead in a shallow aluminum sea, islands that assumed a few different shapes and came in a range of discrete sizes. "What surprised us was that some sizes were preferred, while others were avoided. At first it was puzzling, but on reflection it made perfect sense," says Dahmen, noting that a lead atom is about 20 percent larger than an aluminum atom.

"Putting lead atoms into aluminum has been likened to shoving grapefruits among oranges. If you try to replace oranges with grapefruits three for three, you expend a lot of energy squashing the grapefruits. But replace five oranges with four grapefruits and they fit reasonably well. With lead and aluminum, we found that the most preferred fit was nine for eleven."

The consistently repeating ratio of lead to aluminum was a clue to the forces at work at the nanoscale. Free crystals take shapes that minimize their surface energy, but embedded crystals have to conform to their neighbors in the solid matrix. In this environment, internal elastic energy is important and must be minimized, together with the interface energy.

When the researchers analyzed their micrographs, two features became apparent. First, the "magic sizes" assumed by the lead inclusions corresponded to regularly repeating values of minimal strain energy. Just as multiples of four grapefruits replacing five oranges -- four or eight or 12 grapefruits replacing five or 10 or 15 oranges -- cause less squashing than one-for-one replacements, so some repeating ratios of lead to aluminum atoms cause less strain than others.

Second, many of the lead inclusions, especially the smaller ones, were asymmetrical. The equilibrium shape of a large inclusion (where interface energy is dominant) is a cuboctahedron, a symmetrical double pyramid with its six vertices lopped off. Small inclusions, however -- where residual strain energy is dominant -- are forced to assume magic sizes even at the cost of their equilibrium shape. In the micrographs they appear as parallelograms which have been truncated at one or more vertices.

By adopting asymmetrical shapes, small magic-size inclusions can maintain perfectly flat interfaces with the host matrix. This in turn suggests why small inclusions have to be superheated to melt: more heat energy is needed to overcome the constraint at the perfect interface of an asymmetrical, magic-size inclusion than at the imperfect interface of a symmetrical inclusion.

"To learn something as simple as how ordinary lead-tin solder melts and resolidifies would be fundamental knowledge with far-reaching implications," Dahmen says. He and his collaborators, Erik Johnson in Denmark and Serge Hagüge in France, are currently working with lead- cadmium alloy inclusions in aluminum -- in what Dahmen calls "nanoscale crucibles" -- to understand the relationship of melting behavior and shape in more complicated miniature systems.

Nanoscale manipulation can affect an alloy's electrical, magnetic, and optical properties as well as its mechanical and thermodynamic behavior. Says Dahmen, "As a practical result of these observations we can begin to talk about understanding, perhaps even engineering, inclusions with desired shapes and sizes -- and the resulting properties -- in all kinds of alloys."

The researchers reported their results in a recent issue of Physical Review Letters (Volume 78, No. 3).

Berkeley Lab is a U.S. Department of Energy national laboratory located in Berkeley, California. It conducts unclassified scientific research and is managed by the University of California.



DOE/Lawrence Berkeley National Laboratory

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