A novel composite is stiffer than diamond

The genius of hot and sour soup is that two very different culinary ingredients—chili oil and vinegar—unite to give a distinct flavor, piquant and tangy. Engineers who devise recipes for composite materials apply a similar principle. They might combine a strong material resistant to breakage with a stiff material resistant to deformation and fabricate a composite that is both strong and stiff. But just as hot and sour soup prepared with an ounce of chili–vinegar mixture is not as spicy or sour as it would be were it prepared with an ounce of either undiluted ingredient, one might expect that the engineer’s composite is not as strong as its strongest constituent nor as stiff as its stiffest. A rigorous theorem confirms that intuition. ¹

But in 2001, Roderic Lakes of the University of Wisconsin–Madison and colleagues created a material with a stiffness greater than theoretically allowed. They did it by violating the assumptions of the stiffness-bounding theorem. Specifically, they fabricated composites with negative-stiffness inclusions. Now, six years later, Lakes’s group has reported on a material whose stiffness, at least over a small temperature range, is greater than that of steel, tungsten carbide, and even diamond. ²

Freedom from movement

Three-dimensional materials can be deformed in various ways, and they are characterized by more than one kind of stiffness. The phenomenal stiffness observed by Lakes and company was an elastic stiffness, a resistance to compression or extension. Very stiff materials, says Lakes, are useful in fabricating devices such as computer disks, whose dimensional tolerances have to be very tight.

Elastic stiffness is often quantified by Young’s modulus. Consider a bar, and apply a compressing pressure to its ends. The bar will get slightly shorter until it exerts a force balancing the externally applied pressure. Young’s modulus is the pressure divided by the fractional change in the length of the bar.

Springs offer 1D models for thinking about negative stiffness and the physics behind Lakes and colleagues’ observation. An ideal spring’s stiffness is quantified by the Hooke’s law constant k; for a negative-stiffness spring, k is negative. Such a spring would be unstable. If you attach a mass and extend the spring just a tad beyond its equilibrium length, you’ll see it continue to expand. The spring is likewise unstable to minute compressions.

Although the negative-stiffness spring is unstable in isolation, it can be part of a stable system if conventional springs with sufficiently large positive k are attached to each of its ends and if the displacement at the boundary of the three-spring system is specified. (The 1D system is subtly different from real 3D materials, for which one can fix either the displacement or the force acting on the boundary and still have stability.) Figure 1 illustrates the idea and also shows how a negative-stiffness inclusion can contribute to a system with very high stiffness.

In Lakes and company’s composite, a tin matrix plays the role of the conventional springs. The negative stiffness is realized by a particular crystal form of barium titanate.

At high temperature, BaTiO₃ has a cubic unit cell and resists compression and extension as does a conventional spring. The material undergoes a phase transition at about 120 °C to a form with a tetragonal unit cell. In principle, the phase transition could leave the BaTiO₃ in an unstable equilibrium configuration analogous to the negativestiffness spring. But tetragonal BaTiO₃, more complicated than a 1D Hooke’s law spring, also has available a stable equilibrium form. A real-world cubic-to-tetragonal phase transition carried out on isolated BaTiO₃ produces the stable form.

In the Wisconsin group’s protocol, small bits of BaTiO₃ are added to molten Sn, which then cools. In time, the BaTiO₃ undergoes the cubic-to-tetragonal phase transformation, but now confined in a Sn matrix. Lakes and company reasoned that, like the surrounding springs in figure 1, the Sn matrix might be able to stabilize the BaTiO₃. That is, confined in the matrix, BaTiO₃ might exist in the tetragonal form that would be unstable in isolation. The result is a composite with negative-stiffness inclusions.

Figure 2 shows a micrograph of an extreme-stiffness specimen. The BaTiO₃ accounts for a relatively small 10% of the volume. To determine Young’s moduli of their samples, Lakes and colleagues subjected them to a bending force applied at 100 Hz. The periodic application allowed the group to explore the lag, expressed as a phase offset, between force and sample response.

Bending may seem to be quite different from the linear compression or extension that defines Young’s modulus, but when an object bends, some of it compresses while other parts expand. Thus, the degree to which a specimen bends encodes Young’s modulus. To determine that bending, Lakes and colleagues measured the angular shift of laser light reflecting off the sample. Recognizing that the phase-transition temperatures of BaTiO₃ confined by Sn and BaTiO₃ in isolation could be quite different, the researchers also monitored specimen temperature through several heating and cooling cycles.

Most of the Wisconsin team’s composites exhibited an anomalous stiffness incompatible with having only positive-stiffness components. Figure 3 presents data for one of the minority of samples whose Young’s moduli exceeded that of diamond. Several features are noteworthy. First, the specimen’s behavior changes significantly from one temperature cycle to the next. Lakes suggests that the detailed microstructure of the inclusion–tin interfaces is key to determining a composite’s stiffness. That so-far uncontrollable quality varies from sample to sample and changes with thermal cycling.

Second is that the diamond-surpassing stiffness exists only over a small temperature range. Third, the phase offset is negative in that region of extraordinary stiffness. According to Lakes, the negative phase shift indicates an internal release of potential energy, perhaps a consequence of inclusions rolling down their potential-energy hills away from the unstable equilibrium. “We think the inclusions expand,” he says. “But that’s an inference; we haven’t actually measured the expansion.”

Almost simultaneously with the Lakes and company paper, Walter Drugan, a University of Wisconsin colleague and collaborator of Lakes’s, published a theoretical demonstration that static 3D composites with negative-stiffness elements can be stable. ³ Drugan’s consideration of static systems is important: The extreme stiffness evidenced in figure 3 is a transient effect that occurs during a dynamic temperature scan. Sometimes, notes Drugan, systems can be stabilized by dynamic processes. But if super-stiff composites are to see practical applications, they will need to be stable under static conditions.

In future work, Lakes hopes to expand the temperature range over which his composites exhibit extreme stiffness. One idea that he and Drugan have considered proceeds from the ability of a little bit of negative-stiffness inclusion to have a whopping effect on composite stiffness. Thus, it might be possible to fabricate a composite with small quantities of distinct materials, each of which undergoes a phase transition at a different temperature to a stabilizable negativestiffness form. Over a wide range of temperatures, the composite could exhibit remarkable stiffness, with the several inclusions taking their turn in being responsible for the surprising effect.