The dust a plasma freezes into a crystal

The wall a plasma builds against itself put a flat surface into a plasma and found that it charges negatively, because the light electrons reach it faster than the heavy ions, until the electric field of its own charge slows the electrons enough to balance the currents. The surface ends up sitting a few electron temperatures below the plasma’s potential, and a thin sheath of positive space charge forms in front of it. That essay named a later problem in its closing list: the charging of dust grains, which are surfaces too, but small, spherical and free to move.

The problem turns out to contain a surprise much larger than a correction to the flat-wall answer. A grain a few micrometres across picks up tens of thousands of electrons. A cloud of such grains, held up against gravity by the sheath’s field, repels itself so strongly that the grains arrange themselves into a crystal — a lattice with spacings of a third of a millimetre, sitting inside a glowing discharge, visible to a video camera with a laser sheet shining through it. Three groups reported it within a few months of one another in 1994, in Germany, Taiwan and Japan, and it made the plasma laboratory one of the few places where the melting of a crystal can be watched one particle at a time.

Why a grain charges, and how far #

A grain sitting in a plasma is hit by electrons and by ions. The electrons are some seventy thousand times lighter than argon ions and, in a typical laboratory discharge, a hundred times hotter, so their thermal speed is enormous by comparison, and the first figure finds that an uncharged grain collects electrons about 2,700 times faster than ions. It charges negative at once.

As it charges, the balance shifts. Electrons have to climb the grain’s potential to reach it, and the number able to do so falls exponentially, by the Boltzmann factor, as the wall a plasma builds against itself found for a flat surface. Ions are attracted, and here the sphere differs from the wall. A wall collects only the ions that happen to be heading for it; a small sphere bends the paths of ions passing nearby into itself, so its collection grows with its charge. The standard description, called orbit-motion-limited theory, gives an ion current growing linearly with the potential. The two currents cross where the grain floats: 2.41 electron temperatures below the plasma for argon, 7.2 volts for electrons at 3 electronvolts.

The potential does not depend on the size of the grain, because both currents scale with its area. The charge does. A sphere of radius $a$ holding a potential $\phi$ carries a charge $4\pi\varepsilon_0 a\phi$, so the charge grows in proportion to the radius.

The second figure draws the result. A grain of a tenth of a micrometre, the size of a particle of smoke, carries about five hundred electrons; one of a micrometre carries five thousand; the five-micrometre melamine spheres used in most laboratory experiments carry twenty-five thousand. An ion in the same plasma carries one charge. That ratio is where everything else in this essay comes from: the forces between grains, and the order they impose, grow as the square of the charge.

The proportionality to radius is the capacitance of a sphere at work, and it is worth seeing why it is so generous. How much charge a shape will hold, before anything is charged found that a conductor’s capacitance is set by its size alone, $4\pi\varepsilon_0 a$ for a sphere, about one picofarad per centimetre of radius. Held at a few volts, a body a few micrometres across carries a few femtocoulombs, which is tens of thousands of electrons, simply because an electron’s charge is so small compared with what a volt on a micrometre amounts to. The plasma does not have to push hard. It only has to hold the grain a few volts negative, and the geometry does the rest.

Held up by the sheath #

The grains are heavy as well as highly charged — a five-micrometre plastic sphere weighs as much as ten trillion argon atoms — and in a laboratory on the ground they fall unless something holds them. The sheath does. Above a horizontal electrode in a radio-frequency discharge, the sheath’s electric field points down towards the electrode and grows in strength towards it, and it pushes a negative grain upward.

The third figure compares the push with the weight. Because the charge grows only as the radius while the weight grows as its cube, small grains are held easily and large ones not at all; in a field of ten thousand volts per metre, typical of the lower sheath, the balance comes at about eleven micrometres. A grain settles at the height where the field is just strong enough for its size. Grains of one size therefore settle at one height, a few millimetres above the electrode, and a cloud of identical spheres forms a single thin horizontal layer — which is exactly what is wanted for watching how they arrange themselves within it.

That arrangement is held in from the sides by a shallow dip in the potential, often made by a ring laid on the electrode, so the layer is a disc of grains floating in a bowl. Everything about the system can be seen: a thin sheet of laser light shone sideways through the layer lights up every grain, and a camera above records their positions many times a second.

Coupled more strongly than anything that is not cold #

Whether a collection of particles is a gas, a liquid or a solid depends on one ratio: how strongly they interact with their neighbours compared with how vigorously they jostle. For charged particles the ratio is called the coupling parameter, the electrostatic energy of two neighbours at their typical spacing divided by their thermal energy $kT$. In an ordinary plasma it is tiny — the particles are hot and far apart — and the long-range force that does not reach found that the plasma screens any charge over a Debye length, so that the Coulomb force, nominally long-ranged, dies away exponentially. Dust changes both numbers. The charge enters squared, and is tens of thousands; and the grains’ temperature is set not by the hot electrons but by collisions with the neutral gas, which is at room temperature.

The fourth figure computes the result, with the screening included: the bare coupling is multiplied by a factor that falls with the ratio of spacing to screening length, the form that simulations of screened charged particles have found to collapse their freezing points onto a single value of about a hundred. Five-micrometre grains a third of a millimetre apart sit at forty thousand, hundreds of times beyond the threshold, at room temperature. For comparison, ions reach couplings like that only in the interiors of white dwarfs, where they crystallise as the stars cool, and in traps where a few thousand ions are cooled by lasers to within a thousandth of a degree of absolute zero. The dust in a laboratory discharge does it in a glass chamber at room temperature, and the reason is entirely its charge.

The crystal, and its melting #

The fifth figure simulates the layer. A hundred and sixty grains repel each other with a screened Coulomb force, are held together by a parabolic confinement, and are kicked about by random collisions and slowed by gas friction, with the strength of the kicks set by a temperature. With the grains cold compared with their interaction, they settle into the arrangement that two-dimensional repelling particles always prefer, a triangular lattice in which each grain has six neighbours at the corners of a hexagon; near the edge the confining bowl bends the rows into rings. Heated seventy-five times, the same grains lose the pattern: they still keep their distance from one another, but the sixfold order is gone and they wander.

In the laboratory the melting can be driven by lowering the gas pressure, which reduces the friction and lets energy from the plasma’s fluctuations heat the grains, or directly by pushing them with laser light. Because every grain is visible, the transition can be followed in a way no atomic crystal allows. In two dimensions in particular, theory had predicted since the 1970s that melting might go through an intermediate phase, called hexatic, in which positional order is lost but the orientation of the hexagons survives over long distances, marked by the appearance first of pairs and then of single defects in the lattice. Dusty plasma layers, together with colloidal suspensions and electrons on liquid helium, are among the systems in which the defects can be counted directly as the crystal melts.

Measuring a temperature by watching #

The coupling depends on the grains’ temperature, and in a dusty plasma that temperature is not assumed but measured, by the most direct method there is. The camera records each grain’s position frame by frame; differencing the positions gives velocities; and the spread of the velocities, by half a kT for every way of moving, is the temperature. For a crystal in equilibrium with the gas the measured distribution is close to Maxwell’s at room temperature, which for a grain of $10^{-12}$ kilograms means a thermal speed of well under a tenth of a millimetre per second: the grains tremble about their lattice sites by a few micrometres, as the jiggle that proved atoms found pollen grains doing in water.

The measurement often finds more. In many experiments the grains turn out to be hotter than the gas, sometimes by factors of ten or a hundred, and hotter in the vertical direction than in the horizontal; the excess comes from fluctuations of their charge and from the wake forces described below, which feed energy into their motion. A system whose temperature can be read off a video is unusual, and it is one whose temperature can be found to be several different numbers at once.

The same crystal in water #

Dust in a plasma is not the only way to get large charges onto small particles. Polystyrene or silica spheres a fraction of a micrometre across, suspended in water, dissociate ions from their surfaces and carry hundreds to thousands of charges, screened by the counterions that the water holds around them, which the counterions that never leave the chain found behaving much as they do around a charged polymer. At low enough salt concentration the screening length grows, the coupling climbs, and the suspension crystallises into a lattice that diffracts visible light — the iridescence of opal, which is a natural colloidal crystal of silica spheres. The force between them is the same screened Coulomb repulsion as in the dust layer, with water’s molecules and ions playing the part of the plasma.

The two systems are complementary. Colloids are overdamped, their motion smothered by the water’s viscosity, so they relax slowly and never oscillate. Dust grains move in a gas thin enough that their inertia matters, so a dusty crystal carries waves and its grains can be followed through individual oscillations, and the dynamics of melting, rather than only its end states, can be watched. The like charges that pull together found that colloids can also attract one another when the screening ions are multiply charged; the dusty plasma’s own route to attraction is the ion wake, below. And granular matter, the heap that becomes a solid, shows the contrasting way particles become rigid: by jamming into a disordered pile under load, with no repulsion at a distance and no lattice at all.

A crystal with the dynamics of a plasma #

The grains in the crystal still interact through the plasma, and the plasma gives them properties that ordinary crystals do not have. Their charge depends on where they are and changes when they move, and the forces between them need not be equal and opposite. A grain levitated in the sheath sits in a stream of ions flowing down towards the electrode, and each grain focuses the ions passing it into a positive cloud just below it, a wake. A second grain below the first is attracted to that wake while the first feels no corresponding pull from the second. The forces between grains at different heights are therefore not reciprocal, which a closed system of particles cannot have but a system driven by a flow can, and it lets layers of grains line up vertically into chains and heat themselves.

The grains also carry waves. A disturbance in a cloud of charged grains propagates as a dust-acoustic wave, the same physics as the sound a plasma carries on its electrons’ heat but with the grains as the inertia and the ions and electrons as the spring, and it is slow enough — a few centimetres a second — to be seen by eye. Its natural frequency is the dust plasma frequency, the same quantity the frequency below which nothing gets in defined for electrons, with the grains’ enormous mass in place of the electron’s: a few hundred radians per second instead of billions. In the crystal, the waves become lattice vibrations whose dispersion relations can be measured by tracking the grains, the analogue of the phonons of a solid with each atom on camera.

The same charging, on a larger stage #

The charging the first figure computes is not confined to laboratories. Grains in interstellar clouds, in the tails of comets, in planetary rings and in the dust that lies on the Moon’s surface all sit in plasmas or in sunlight and float at potentials set by a balance of currents. In space the balance usually includes another current, electrons knocked out by ultraviolet light, which the wall a plasma builds against itself noted can make a sunlit surface float positive. On the Moon the dayside soil charges positive and the nightside negative, and the boundary between them, the terminator, is where astronauts and landers reported faint horizon glows, attributed by some to lofted dust. In Saturn’s rings, dark radial “spokes” seen by Voyager and Cassini have been explained by fine grains charged and lifted off the larger ring particles.

The same physics is also a nuisance. In the plasma reactors used to etch silicon chips, grains grow from the reactive gases, charge negative and levitate at the sheath edge above the wafer — exactly where they are least wanted — and fall onto it when the plasma is switched off. Much of the early understanding of dusty plasmas came from engineers trying to keep particles off wafers.

Where the description is approximate #

The charging model treats the grain as an isolated sphere in an unbounded, collisionless plasma, with ions arriving on orbits that conserve energy and angular momentum. Each assumption fails somewhere. Collisions between ions and neutral gas atoms near the grain trap ions in orbits around it and increase the ion current, reducing the grain’s charge, sometimes by half at the pressures of laboratory experiments. In a dense cloud the grains together hold a significant fraction of the negative charge, so the electrons are depleted and each grain charges less. And in the sheath, where the ions are streaming past at supersonic speed rather than drifting randomly, the orbit-limited ion current is the wrong one altogether.

The coupling is equally an idealisation. A single threshold of about a hundred for freezing is a summary of simulations of particles with an isotropic screened interaction in thermal equilibrium. Real grain layers are confined, flowing and slightly heated by the plasma, and their interactions include the non-reciprocal wake forces; the threshold is a guide to where crystals are expected, not a sharp line.

One layer, without the plasma that holds it #

The figures compute charges, forces and couplings, and simulate one layer of grains in two dimensions. They do not show the plasma itself — the electrons and ions whose currents set the charge, the sheath’s field that holds the layer up, the ion flow that makes the wakes — except as the parameters that enter. The simulated grains interact only through a fixed screened potential, with no wakes and no charge that varies with position, so the simulation is the crystal of an idealised system, not of a laboratory one. And the coloured grains in the last figure mark six neighbours within a fixed distance, a crude stand-in for the bond-orientational measures used to identify the hexatic phase.

Still open: how a crystal melts when its forces are not reciprocal #

The non-reciprocal wake forces make laboratory dusty plasmas a system that is not in thermal equilibrium even when it looks steady: energy flows in from the ion stream and is dissipated by gas friction, and the grains’ effective temperature can differ from one direction to another. Whether such a driven crystal melts by the same route as an equilibrium one, whether the hexatic phase survives in it, and how the transition depends on the strength of the wake forces, are being studied in experiments on the ground and in microgravity on the International Space Station, where grains need no sheath to hold them up and can fill three-dimensional clouds.

The habit worth carrying away is to ask which temperature sets the order. A grain in a plasma floats 2.4 electron temperatures below it in argon and so carries about 5,000 charges per micrometre of radius; held in the sheath and jostled only by room-temperature gas, grains of 5 μm a third of a millimetre apart are coupled some four hundred times beyond freezing, and they crystallise into a hexagonal lattice that melts, when heated, in full view of a camera. The plasma is hot; the dust is cold, and its charge decides which of the two governs its order.