# A Metal's Many Faces; a New Mathematics Helps Elucidate How Metals Are Put Together

By Peterson, Ivars | Science News, January 31, 1987 | Go to article overview

# A Metal's Many Faces; a New Mathematics Helps Elucidate How Metals Are Put Together

Peterson, Ivars, Science News

A METAL'S MANY FACES

A metal's etched surface typically showsa jumble of grains jammed together. This grain structure is a natural result of crystal growth. Each grain is a single crystal, made up of atoms in orderly arrays. When the metal solidifies, microscopic crystals that form within the liquid grow until they bump into their neighbors. The subtle interplay between physical forces and the geometric requirements of filling space sets the final grain boundaries.

In many cases, a metal's grain structurelooks a lot like a soap froth, and sometimes it behaves like one. Steel, for instance, is a mixture of carbides and iron. When steel is heated up, grain boundaries shift. The grains act much like bubbles clustered together, where larger bubbles grow at the expense of smaller ones to create a coarser pattern.

Such observations have led metallurgiststo use soap froths as a rough model for a metal's grain structure. The model helps them understand the behavior of metals and suggests ways of manipulating grain structure to get metals with the right properties. "The manipulation of microstructure," says materials scientist John W. Cahn of the National Bureau of Standards in Gaithersburg, Md., "is an important, central feature of modern materials science."

But crystals aren't really like soap bubbles. Crystalsurfaces lack the flexibility of soap films. They don't readily bend around corners. Instead, these surfaces tend to be flat and take on definite directions. This rigidity affects grain boundaries in ways that are not accounted for by the soap bubble analogy. "This is a complexity that we as metallurgists aren't taught to handle," says Cahn. To get a better idea of the types of boundaries that can form between adjacent crystals, Cahn turned to mathematician Jean E. Taylor of Rutgers University in New Brunswick, N.J. More than a decade ago, Taylor, along with Frederick J. Almgren Jr. of Princeton (N.J.) University, had developed a simple mathematical model that accounts for why the many possible configurations of soap-bubble clusters are governed by only a few elementary rules (SN:9/20/75,p.186). Since then, Taylor has been extending her model to what, in effect, are cubic or polyhedral bubbles--forms that have well-defined faces. That's just the kind of mathematics that might apply to crystalline grains in metals.

The collaboration between Cahn andTaylor has now led to a catalog of the different interfaces that may occur between a crystal and a surrounding medium, whether solid, liquid or gas. Although their catalog is merely a first step and covers only one type of interface, the findings already suggest that some geometries, which metallurgists believed were caused by crystal defects, are actually forms that arise naturally in crystal growth within a solidifying metal. Their catalog was published last year in ACTA METALLURGICA (Vol. 34, p.1). A soap bubble's shape is governed by surface tension, which is uniform over the whole bubble. An elastic soap film enclosing a parcel of air stretches only as far as it must to balance the air pressure inside.

The bubble is spherical because asphere has the least possible area for the volume it encloses. A larger area would mean stretching the film and proportionately increasing its surface energy. Hence, a soap bubble's spherical shape minimizes the bubble's surface energy.

The principle of minimizing surfaceenergy also determines the boundaries between crystals or between a crystal and a surrounding fluid. In the case of crystals, the surface energy value depends on the nature of chemical bonds left dangling at particular surfaces. The energy required to break apart a crystal may be much lower in some directions than in others. In that case, a crystal's surface energy would be anisotropic, varying from face to face.

Just as a sphere is the equilibriumshape of a single soap bubble, there is an analogous shape for anisotropic crystals.

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