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Mathematical physics

Mathematical physics

Cracking the Brazil nut problem

11 Apr 2001

How do Brazil nuts find their way to the top of a bowl of nuts while the smaller nuts filter to the bottom? The phenomenon of segregation in mixtures of particles takes its name from this effect, but nobody has managed to predict or control the process. Now Daniel Hong of Lehigh University in the US and colleagues have identified a new kind of sifting effect - 'condensation' - and developed a theory that predicts the behaviour of the 'Brazil nut problem' (D C Hong et al 2001 Phys. Rev. Lett. 86 3423).

A system of particles sorts itself through a variety of mechanisms, including percolation – in which small grains migrate downwards through the channels between bigger grains – and ‘convection’, which drives large particles upwards. Now Hong and co-workers have identified a new mechanism – ‘condensation’ – through a series of molecular dynamics simulations. “Both segregation and mixing are extremely important in industry”, Hong told PhysicsWeb. “Mixing is crucial in drug and concrete production, and vibrations can be used to separate foodstuffs like rice”.

The researchers initially considered a system of same-size particles, and equated the kinetic energy of each particle, which is directly related to the ambient temperature in kinetic theory, to its equivalent potential energy. They found that a critical temperature exists below which a layer of particles ‘condenses out’ at the bottom of the vessel. This critical temperature corresponds to a particular ratio of mass and diameter. The condensed particles vibrate in a confined space but cannot swap places with their neighbours or re-enter the ‘fluid’ portion of the system.

Hong’s team observed the influence of condensation during a second simulation in which they mixed two sets of spheres, each with their own ‘critical temperatures’ determined by their masses and diameters. If the temperature is set between these threshold temperatures, the set of particles with the higher critical temperature will ‘condense’ while the other set remains ‘fluid’. Hong and colleagues found that it is the ratio of the masses and diameters of the sets of particles that is crucial: if a sphere A is twice the mass and diameter of a sphere B, the larger particles float. But if they are six times heavier and twice the diameter, they sink – an effect known as the ‘reverse Brazil nut problem’.

“It is extremely important to know the control parameters because then we know when mixing or segregation will occur”, says Hong. From a range of simulations, Hong’s group constructed a phase diagram of different combinations of mass and diameter ratios. The diagram maps the onset of segregation and the flip from the ‘Brazil nut problem’ to the ‘reverse Brazil nut problem’ – and so predicts how certain combinations of particles will mix.

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