The dance of the diamonds

You probably haven’t heard of the Chladni effect but you’ve likely seen it in action. Sprinkle some grains of sand on a thin metal plate and play a violin bow across it, and you’ll notice that the grains bounce around for a bit before settling down into a pattern, and refuse to budge after that.

This happens because of a phenomenon called a standing wave. When you drop a rock into a pond, it creates ripples on the surface. These are moving waves taking the rock’s kinetic energy away in concentric circles. A standing wave on the other hand (and like its name implies) is a wave that rises and falls in one place instead of moving around.

Such waves are formed when two waves moving in opposite directions bump into each other. For example, in the case of the metal plate, the violin bow sets off a sound wave that travels to the opposite edge of the plate, gets reflected and encounters a newer wave on the way back. When these two waves collide, they create nodes – points where their combined amplitude is lowest – and antinodes – pointes where their combined amplitude is highest.

In 1866, a German physicist named August Kundt designed an instrument, now called a Kundt’s tube, to demonstrate standing waves. A short demonstration below from user @starwalkingphoenix:

The tube is made of a transparent material and partially filled with a soft, grainy substance like talc. One end of the tube opens up to a source of sound of a single frequency while the other end is stewarded by a piston. As the piston moves, it can increase or decrease the total length of the tube. When the sound is switched on, the talc moves and settles down into the nodes. The piston is used to identify the resonant frequency: it is used to increase or decrease the tube’s length until the volume suddenly increases. That’s the sweet spot.

In the Chladni effect, the sand grains settle down into the nodes of the standing wave formed by the vibrations induced by the violin bow. These nodes are effectively the parts of the plate that are not moving, or are moving the least, even as the plate as a whole hosts vibrations. Here is a nice video showing different Chladni patterns; notice how they get more intricate at the higher frequencies:

The patterns and the effect are named for a German physicist and musician named Ernst Chladni, who experimented with them in 1787 and used what he learned to design violins that produced and emitted sound better. The English polymath Robert Hooke had performed the first such experiments with flour in the late 17th century. However, the patterns weren’t attributed to standing waves until the early 18th century by Sophie Germain, followed by Horace Lamb, Michael Faraday and John Strutt, a.k.a. Lord Raleigh. (The term ‘standing wave’ was itself coined only in 1860 by [yet] another German physicist named Franz Melde.)

Now, both Chladni and Faraday had separately noticed that while the patterns were formed most of the time, they did not when finer grains were used.

A group of scientists from a Finnish university recently rediscovered this bit of strangeness and piled some more weirdness on top of it. They immersed a square silicon plate 5 cm to a side in a tank of water and scattered small diamond beads (each 0.75 mm wide) on top. When they applied vibrations at a frequency of 9,575 Hz, the beads moved towards the parts of the plate that were vibrating the most instead of the least – i.e. towards the antinodes instead of the nodes.

A schematic illustration of the experimental setup. Source: PRL 122, 184301 (2019)

This doesn’t make sense – at least not at first, and until you stop to consider what you might be taking for granted. In the case of the metal plate, the sand grains are bounced around by the vibrations, and those that are thrown up do come back down due to gravity – unless they’re too light or the breeze is too strong, and they’re swept away.

Water is over 800-times denser than air and would exert a stronger drag force on the diamond beads, preventing them from being able to move around easily. Then there’s also the force due to the vibrations and gravity. But here’s the weird part. When the scientists combined the three forces into a common force, they found that it always pushed a bead towards the nearest antinode.

And this was just at the resonant frequency: the frequency at which an object is most amenable to vibrate given its physical properties. In other words, the resonant frequency is the frequency of the vibration that consumes the least amount of energy to cause in the body. For example, the silicon plate resonated at 9,575 Hz and 11,175 Hz.

But when the scientists applied vibrations at a non-resonant frequency of 10,675 Hz, the diamond beads moved around in swirling patterns that the scientists call “vortex-like”.

In 2016, another group of scientists – this one from France – had reported this swirling behaviour with polystyrene microbeads on a polysilicon membrane, both suspended in ultra-pure water. On that occasion, they had compared the beads’ paths to those of dancers performing a farandole, a community dance popular in Provence, France (see video below).

Polystyrene beads each 70 μm wide in a cavity rotating in a farandole-like manner at an applied frequency of 61,000 Hz. The time frame between each picture is 0.5 s. Source: PRL 116, 184501 (2016)

The scientists from the Finnish university were able to record over 96,000 data points and used them to try and figure if they could obtain an equation that would fit the data. The exercise was successful: they obtained one that could locate the “nodal, antinodal and vortical regions” on the silicon plate using two parameters (relatively) commonly used to model magnetic fields, called divergence and curl. Specifically, the divergence of the “displacement field” – the expected displacement of all beads from their initial position when a note is played for 500 milliseconds – denoted the nodal and antinodal regions and the curl denoted the parts where the diamonds would do the farandole.

However, to rephrase what they wrote in their paper, published in the journal Physical Review Letters on May 10, the scientists can’t explain the theory behind the patterns formed. Their equations are based only on experimental data.

The French group was able to advance some explanation rooted in theoretical knowledge for what was happening, although their experimental conditions were different from that of the Finnish group. Following their test, Gaël Vuillermet, Pierre-Yves Gires, Fabrice Casset and Cédric Poulain reasoned in their paper that an effect called acoustic streaming was at play.

It banks on the Navier-Stokes equations, a set of four equations that physicists use to model the flow of fluids. As Ronak Gupta recently explained in The Wire Science, these equations are linear in some contexts and nonlinear in others. When the membrane vibrates slowly, the linear form of these equations can be used to model the beads’ behaviour. This means a certain amount of change in the cause leads to a proportionate change in the effects. But when the membrane vibrates at a frequency like 61,000 Hz, only the nonlinear forms of the equation are applicable: a certain amount of change in the cause precipitates a disproportionate level of change in the effects.

The nonlinear Navier-Stokes equations are very difficult to solve or model. But in the case of acoustic streaming, scientists know that the result is for the particles to flow from the antinode to the node along the plate’s surface, then rise up and flow from the node to the antinode – in a particulate cycle, if you will.

Derek Stein, a physicist at Brown University in Rhode Island, wrote in an article accompanying the paper:

… this migration towards antinodes is a hallmark of particles being carried in acoustically generated fluid streams, and the authors were able to rule out alternative explanations. … [The] streaming effect in a liquid is only observable within a restricted window of experimental parameters. First, the buoyancy of the beads has to closely balance their weight. Second, the plate has to be sufficiently wide and thin that its resonant vibrations have large amplitudes and produce high vertical accelerations. The authors also noticed that tuning the driving frequency away from a resonance coaxed the particles to move in regular formations. This motion begged to be anthropomorphised, and the authors duly likened it to the farandole…

After this point, both research papers break off into discussing potential applications but that’s not why I am here. My favour part at this point is something the Finnish university group did: they built a small maze and guided a 750-μm-wide glass bead through it simply by vibrating its floor at different frequencies. They just had ensure that at some frequencies, the node/antinode would be to the left and at others, to the right.
Credit: K. Latifi et al., Phys. Rev. Lett. (2019)

And because they also possessed the techniques by which they could induce a particle to travel in straight lines or in curves, they could the move the beads around to trace letters of the alphabet!

Source: PRL 122, 184301 (2019)