Granular materials

Taking the temperature

The 'temperature' of a granular material depends on its entropy, but is hard to measure in the laboratory. So a theory that ties temperature to grain mobility and diffusion is welcome.

Granular materials are collections of solid, macroscopic particles, characterized by a loss of energy whenever the particles interact1 (for instance, through friction when grains collide). On page 614 of this issue2, Makse and Kurchan offer new insight into the statistical properties of granular materials. In particular, their simulations suggest ways of determining a granular 'temperature', defined by the kinetic motion of the grains in much the same way that the random motion of molecules in a gas defines its temperature3.

Typical granular materials include coal, wheat, sand (Fig. 1) and all sorts of powders. The study of such materials has a long history dating back to at least the time of Coulomb4, whose friction law was actually derived for these materials rather than for blocks moving on planes. It was Reynolds5 who first showed that compacted materials must first expand (dilate) if they are to deform. And Faraday's observation6 of convection in a shaken powder provided an insight into the role played by the surrounding air. More recently, investigations of granular materials have been driven by commercial applications. Vast sums of money are expended in handling these materials for applications that range from energy production or extraction to food supplies and pharmaceuticals.

Figure 1
figure1

BILL BACHMAN/SPL

New work2 clarifies the thermodynamics of granular materials, such as sand.

In the past few years, researchers have taken up the issue of the statistical properties of granular materials. Because these materials have flow characteristics that roughly resemble those of ordinary, newtonian fluids, it is tempting to look in that direction for useful analogies. But these analogies should not be pursued too closely: granular systems dissipate energy quickly, so ordinary techniques of statistical mechanics that depend on energy conservation break down.

Undeterred, several groups have worked towards understanding the true statistical properties of granular materials. Such work includes studies of gas-like granular states7, investigations of fluctuations and stress variability in dense systems8, and, perhaps most importantly here, proposed new versions of statistical mechanics that would apply in a granular system where energy is not conserved9,10,11,12.

This issue of energy conservation was addressed early on by Edwards and co-workers9,10. They introduced a new way of calculating the entropy (the degree of disorder) and temperature for a granular system. Entropy depends on the number of states available to a system. Edwards et al. counted the number of configurations possible for fixed volume and energy, and defined the 'Edwards temperature' in terms of the volume derivative of this entropy.

The problem with the Edwards approach is that it is difficult to test directly in real systems. Some experiments have achieved at least a flavour of the Edwards picture, such as studies of granular compaction12,13 that showed very slow increases in the density of a granular column that is tapped repeatedly. But, as with many granular experiments, their interpretation is complicated because there is a non-uniform rate of energy injection and non-uniform density — the key control parameter.

In Makse and Kurchan's numerical model2 of a granular system subject to a gentle shearing force, the rate of energy input is uniform (over a reasonable time scale), and a uniform density is maintained. The authors use a tried and tested means to extract a temperature for the system: a comparison of diffusivity and mobility, whose ratio yields the temperature in conventional fluids. They then show that, for their model granular system, this ratio is identical for different particle sizes, just as one would expect in a fluid. The temperature extracted from their observations of diffusion and mobility is also consistent with the predicted value of the Edwards temperature.

So this work shows that, at least in a model system, it is possible to relate temperatures obtained from transport measurements, such as diffusion and mobility, to the Edwards picture. If similar transport measurements are performed in real granular systems, we would have, arguably for the first time, a measurement of a key statistical property that could be matched, in principle, to a theoretical prescription.

But there is a caveat: when real granular systems are sheared or shaken, they tend to become inhomogeneous often in both space and time. Although, to a certain extent, inhomogeneities can be avoided or accounted for, the link to the theoretical picture would be broken. Nonetheless, this work2 leads us towards a clearer understanding of the statistical properties of real dense granular materials.

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Correspondence to Bob Behringer.

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Behringer, B. Taking the temperature. Nature 415, 594–595 (2002). https://doi.org/10.1038/415594a

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