A model fluid comprising rotating magnetic particles behaves according to the equations of hydrodynamics, but for a few key differences due to broken mirror symmetry. The resulting active chiral fluid is characterized by parity-odd Hall viscosity.
Humans are fascinated by rotation. As children, we delight in merry-go-rounds, marvel at the rotation of the planets, and then later puzzle over the physics of gyroscopes. As physicists, we are adept at making sense of the things that intrigue us by building models, be they thought experiments or actual physical systems that can be measured. A fascination with rotation might prompt one to imagine what would happen if millions of interacting particles were made to spin synchronously in the same direction. If these particles were to form a fluid, how would it behave? Writing in Nature Physics, Vishal Soni and co-authors have done just that, by rotating a magnet around a colloidal suspension of haematite particles with magnetic moments — effectively creating a two-dimensional chiral fluid1 (Fig. 1).
The word chiral originates from the Greek word for hand, and implies an object distinguishable from its mirror image. The system created by Soni and colleagues can be referred to as chiral because particles rotating clockwise differ from those rotating counterclockwise. In fact, the system of synchronously rotating particles breaks both parity (or mirror) symmetry and time-reversal symmetry, while preserving the combined parity–time-reversal symmetry.
Broken symmetries typically open up possibilities for new effects and processes. In the case of a chiral fluid, it turns out that such effects can be described by old-fashioned hydrodynamic equations with a couple of additional terms usually absent in parity-invariant systems. One of these terms is known as rotational viscosity, which tends to force the fluid as a whole to rotate with the angular velocity of the spinning particles. However, the friction between the fluid and the substrate suppresses the motion of the fluid in the bulk. As a result, the fluid moves mostly at the boundary of the fluid droplet.
This motion shows up in the videos of Soni and colleagues as edge currents1. The penetration depth of the vorticity of the fluid from the boundary into the bulk is controlled by the shear viscosity and substrate friction. If the droplet is not circular, its shape changes in time. Remarkably, the long-wavelength perturbations of the boundary shape are chiral and propagate along the boundary only in one direction with a velocity much smaller than that of the edge of the droplet. Soni and co-authors refer to the propagation mechanism behind these surface waves as edge pumping.
The other term applicable to the hydrodynamics of rotating particles is the so-called Hall viscosity (also known as the odd viscosity). The word ‘odd’ reflects the fact that this term is odd under mirror symmetry. In the presence of shear flow, the Hall viscosity results in stress forces orthogonal to the velocity of the fluid. There is no work done on the fluid by those forces and, therefore, the Hall viscosity is non-dissipative.
It is quite counterintuitive then that the Hall viscosity has been measured through the increased damping of surface waves1. In fact, in the presence of edge velocity, the effect of the Hall viscosity turns out to be similar to that of the surface tension. Whereas both surface tension and the Hall viscosity cannot result in any energy dissipation on their own, together they tend to flatten the surface of the droplet. These flattening forces cause dissipation through the substrate friction and shear viscosity of the fluid.
The mechanism is similar to the more familiar dissipation of energy that occurs when a magnet moves near a conductor. The magnetic field cannot do work and cause dissipation on its own. However, the changing magnetic field results in eddy currents in the conductor, which dissipate energy. The damping of surface waves caused by the Hall viscosity is similar to that resulting from surface tension, but they have a different dependence on the wavelength. This disparity allowed Soni and colleagues to extract both parameters from their damping measurements. The value they obtained for the Hall viscosity was around 30% of the shear viscosity.
Although the rotational viscosity (known as spin viscosity in ferrofluid literature) is well documented, measurements of the Hall or odd viscosity are rare. The first such measurements go back to 19662 when the Hall viscosity was measured in diatomic gases in a magnetic field. The Hall viscosity was reported more recently in a two-dimensional electron gas of graphene in a magnetic field3. And although the non-vanishing Hall viscosity was predicted theoretically in chiral active fluids4, Soni and colleagues have given us the very first measurement of this peculiar hydrodynamic response in such a system. Their model fluid has a rich phenomenology and offers new opportunities for studying two-dimensional chiral fluids.
Soni and co-authors looked at the properties of a chiral fluid, its surface dynamics and stability of surface waves in a particular regime, and found that the Hall viscosity was smaller than the shear viscosity. In the opposite limit in which the Hall viscosity dominates, the physics is expected to be very different. In this regime, the Hall viscosity will not result in damping, but instead it will change the dispersion of surface waves5.
It remains to be seen whether the model fluid reported by Soni and colleagues can be pushed into the regime of Hall viscosity dominance or whether other model fluids fulfilling this requirement could be created. Fluids in external magnetic fields, rotating fluids, chiral active fluids and other systems with broken parity are all candidates. For example, the Hall viscosity is also non-vanishing in dissipationless quantum Hall fluids6, in which shear viscosity is strictly zero. However, the Hall viscosity of electron fluids in the deep quantum regime is yet to be measured.
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