The liquid a field turns solid

The force whose sign a frequency chooses followed one polarisable particle through a field that varies from place to place. The particle’s induced dipole sits in a gradient and is pulled towards strong field or pushed away from it, according to whether it polarises more or less than the liquid it has displaced. That force needs a non-uniform field, and a single particle. The essay ended by pointing at what happens when neither holds: a uniform field, and so many particles that they sit a few diameters apart. A single particle in a uniform field feels no force at all. A crowd of them does, because each one’s induced dipole makes a non-uniform field at its neighbours.

The result is one of the more startling things a voltage can do to a liquid. Fine particles of a material whose permittivity differs from the oil’s — starch, zeolite, glass, a polymer — are stirred into a silicone oil at a quarter or a third by volume. It pours. A field of a few kilovolts per millimetre is applied across a narrow gap, and the suspension stops pouring and starts behaving like a soft solid that yields only above a threshold stress. Turn off the voltage and it pours again. The change takes milliseconds in each direction and can be repeated indefinitely. Such suspensions are called electrorheological fluids, and every figure here is the same one interaction, between two induced dipoles.

Two neutral spheres that attract #

In a uniform field $E$ a sphere of radius $a$ in a liquid of permittivity $\varepsilon_f$ acquires a dipole moment $p = 4\pi\varepsilon_0\varepsilon_f a^3\beta E$, where the contrast factor $\beta = (\varepsilon_p - \varepsilon_f)/(\varepsilon_p + 2\varepsilon_f)$ is the one the sphere’s surface charges set up and lies between −½ and 1. Two such dipoles, both pointing along the field and separated by a distance $r$ at an angle $\theta$ from the field direction, have an interaction energy

$$U = \frac{p^2}{4\pi\varepsilon_0\varepsilon_f\, r^3}\left(1 - 3\cos^2\theta\right).$$

The factor in brackets is the whole geometry of the problem. It is negative, and the pair attract, when the line joining them lies within $\arccos(1/\sqrt3) = 54.7°$ of the field; it is positive, and they repel, when the line is closer to the perpendicular. End to end the attraction is $-2$ in units of the prefactor; side by side the repulsion is $+1$. The drawing shows the force at points round a ring: arrows pointing in inside two double cones along the field, pointing out across the equator.

There is also a sideways part of the force, from the angle dependence of $U$, and it always acts to turn the pair towards alignment with the field. A neighbour placed anywhere is either drawn in end-on or swung round until it can be. The sign of $\beta$ does not matter, since the energy depends on $p^2$: particles that polarise less than the oil, with negative $\beta$, chain just as well as those that polarise more. This is the same arrangement that holds water molecules together without any net charge, with the difference that there the dipoles are permanent and point every way, and here the field manufactures them already aligned.

Chains, one field line at a time #

The drawing lets forty-eight spheres move under nothing but those pair forces, summed over every neighbour, with the viscous oil turning force directly into velocity and a stiff repulsion at contact. They start scattered with no two touching. They end in chains lying along the field, spaced apart sideways because side-by-side dipoles repel, and every sphere belongs to one.

The order of events is instructive. Pairs that happen to lie roughly along the field close up first, because the end-to-end attraction is strongest and falls as the fourth power of the distance. Short chains then attract other particles and other chains lying along their axes, and repel those alongside them. A chain’s dipole moment is the sum of its members’, so the field at its end is stronger than a single sphere’s, and chains lengthen faster than they are nucleated. Between parallel electrodes the process ends when chains span the gap; in a thicker layer and at high concentrations, chains aggregate sideways into thick columns and sheets, because two chains offset by half a sphere diameter attract, and the long-time structure is a body-centred lattice of columns rather than single strands.

The picture of chains is also where the fluid’s solidity comes from. A chain that reaches from one electrode to the other is a strut anchored at both ends. Shearing the liquid tilts it, and tilting a chain of dipoles stretches its bonds and turns them away from the field, both of which cost energy. The fluid resists shear for as long as the chains hold together.

A chain that hides its own field #

A single chain turns out to be a surprisingly private object, and that is part of why the structure is stable. Along the chain the attraction is not only between neighbours. Every sphere attracts every other sphere in the line, and the energy per sphere of an infinite chain is the nearest-neighbour bond multiplied by $2\sum 1/n^3$, which is $2\zeta(3) = 2.404$ — the more distant members add about a fifth to what the neighbours alone would give. The sum converges quickly because the interaction falls as the cube of the distance.

Sideways the chain’s field does something stranger. Seen from a distance large compared with the spacing of its spheres, a line of identical dipoles has a field that falls off exponentially rather than as a power, with a decay length of about a sixth of the sphere spacing, because the contributions of successive dipoles very nearly cancel. A second chain lying parallel a few diameters away barely feels the first at all, and what little it does feel depends on whether its spheres sit level with the first chain’s or halfway between them: level, they repel; offset by half a diameter, they attract. That weak, registry-dependent attraction is what slowly gathers single chains into the thicker columns seen at long times, and the exponential screening is why the gathering is slow.

Which particles, in which oil #

The effect was found by an engineer, Willis Winslow, who reported in 1949 that suspensions of starch, silica gel and similar powders in oil stiffened between electrodes. His particles all held a little adsorbed water, and for decades it was thought the water was essential, since it made the particles polarise strongly by letting ions move within them. Anhydrous fluids, using particles whose polarisation comes from their structure or from electrons rather than from mobile water, were developed in the 1980s and 1990s because water limits the temperature range and raises the leakage current.

The choice of oil is constrained from the other side. It must be a good insulator, so that the leakage current and the power it dissipates stay small; its permittivity should be low, to make the contrast factor large; it must not break down at the fields used; and its viscosity sets both the fluid’s thickness with the field off and the switching time. Silicone oils meet all four reasonably well, which is why almost every demonstration uses one.

The window in particle size #

A chain is only a chain if thermal motion does not shake it apart, and the drawing compares the end-to-end binding energy of two touching spheres with the thermal energy $kT$. The ratio goes as $a^3 E^2$: the dipole moment has an $a^3$, it appears squared, and it is divided by the cube of the contact distance $2a$. For micrometre particles at a kilovolt per millimetre it is several thousand, and the chains are rigid. For particles of ten nanometres it reaches one only at 13.6 kilovolts per millimetre, near the breakdown strength of the oils used, and a working fluid needs it far above one.

The other end of the size range is set by gravity rather than by the field. A glass sphere of ten micrometres in silicone oil sinks at micrometres per second and settles out of a device in minutes to hours. Working fluids therefore use particles between about a tenth of a micrometre and a few tens of micrometres, and the design problem of keeping them suspended — matching their density to the oil’s, or making them porous or hollow — is as much of the engineering as their electrical properties are.

The stress a tilted chain holds #

Take a lattice of chains between two plates and shear it, so that each chain tilts from the field by an angle $\theta$ whose tangent is the strain, and assume the chains deform with the liquid — each bond stretched from $2a$ to $2a/\cos\theta$. The force each bond exerts along the shear direction follows from the pair force above, and multiplying by the number of chains crossing a unit area, $3\phi/2\pi a^2$ for a volume fraction $\phi$ of spheres, gives the stress:

$$\tau = \tfrac{9}{8}\,\phi\,\varepsilon_0\varepsilon_f\,\beta^2E^2\,\sin\theta\cos^4\theta\,(5\cos^2\theta - 1).$$

At small strain the stress grows in proportion to the strain, which is the elastic response of a solid. It peaks at a strain of 0.389, and past that the stretched bonds weaken faster than the tilt increases the leverage, so a steadily increasing strain takes less stress to maintain: the chain has yielded. The peak is the yield stress, and it is the number that makes these fluids useful — the stress below which the fluid does not flow at all.

Two things in the formula are worth noticing. The stress goes as the square of the field, so doubling the voltage quadruples the stress a clutch or valve can hold. And the particle radius has cancelled. A larger sphere has a bigger dipole, $a^3$, whose square is $a^6$; the force per bond, divided by the contact distance to the fourth power, goes as $a^2$; and the number of chains per unit area, at a fixed volume fraction, goes as $1/a^2$. What remains is independent of size, which is why the size window above matters and the size within it does not.

The yield behaviour is the same kind of thing a paste that holds up its own hill shows — a threshold below which there is no flow — with the important difference that the paste’s structure is always there and the chains exist only while the field does. Like a heap of grains that locks into a solid when pressed, the fluid’s strength is a property of a network of contacts rather than of its material, and the field is what presses it.

How fast the change happens #

The chains form as fast as the particles can move through the oil, and the drawing works out how fast that is for the simplest case: two spheres four radii apart along the field, pulled together by the end-to-end force and resisted by the viscous drag on each. At three kilovolts per millimetre in a silicone oil of fifty times water’s viscosity, they touch in about ten milliseconds. Doubling the field quarters the time.

Once again the particle size drops out. A larger sphere is pulled harder, as $a^2$ at a fixed separation measured in radii, and dragged harder, as $a$, while the distance it has to travel grows as $a$: the three cancel. The time scale is the oil’s viscosity divided by the field energy density $\varepsilon_0\varepsilon_f\beta^2E^2$, a ratio with the units of time, and the same field energy density is what sets the scale of the yield stress. It is why electrorheological devices switch in milliseconds, far faster than a solenoid valve can move its plunger, and why they were proposed for clutches, brakes, shock absorbers that change their stiffness during a single bump, and tactile displays that make a surface feel hard or soft under a fingertip.

Where point dipoles fall short #

The point-dipole model predicts yield stresses of a few tens of pascals at practical fields. Working fluids reach hundreds of pascals to several kilopascals at the same fields — ten to a hundred times more. The model is not wrong about the mechanism but about the field where it matters most.

Two spheres nearly in contact are not two point dipoles. The field in the narrow gap between them is concentrated far beyond the applied field, because the gap is short and the permittivity mismatch drives charge onto the facing surfaces; including the higher multipoles — the quadrupole and beyond, which a distant observer can ignore and a touching neighbour cannot — raises the binding force at contact by an order of magnitude or more when the permittivity contrast is large. And most practical particles are slightly conducting, so over milliseconds their polarisation is set by conductivity rather than permittivity — a dielectric constant depends on how fast it is asked, and a steady field asks very slowly — the same crossover a frequency can switch for a single particle. For conducting particles the gap field is limited only by the conduction of the oil in the gap, and models of that give stresses growing more slowly than $E^2$ at high field, closer to $E^{3/2}$, as measured.

A different class of material, discovered in 2003, goes further. Nanoparticles coated with a polar molecule such as urea give yield stresses above a hundred kilopascals, growing linearly with field rather than quadratically. The mechanism there is thought to be the alignment of the coating’s own molecular dipoles in the enormous field between touching particles — a contact effect with no counterpart in a model of spheres, and one reason these so-called giant electrorheological fluids are argued about.

The same forces with a magnet #

Everything in the drawings has an exact magnetic counterpart, and the magnetic version is the one that reached the market. Iron particles in oil, magnetised by a field, chain up for the same reason and with the same angular dependence, since the interaction of two magnetic dipoles has the same $1 - 3\cos^2\theta$. Iron’s magnetisation is so large that magnetorheological fluids reach yield stresses of tens of kilopascals, enough to use in the shock absorbers of cars and in the dampers that hold some bridges and buildings steady in wind and earthquakes. They need a coil rather than a high voltage, which is easier to arrange safely in a vehicle.

The electrical version keeps one advantage, which is why it is still studied: a voltage can be switched faster than the current in a coil, which fights its own change, and an electrorheological device needs almost no power to hold its state, only a leakage current through the oil.

What the chains do not show #

The drawings follow particles in two dimensions, with no thermal motion, no hydrodynamic interaction between moving spheres — each one feels the oil as if it were alone — and no electrodes except as the ends of a periodic cell. The real structure forms in three dimensions, the viscous flow set up by one sphere drags its neighbours, and chains meeting an electrode are held there by the image of their own dipole in the metal. None of these changes the fact of chaining or its angular rule, and all of them change how long it takes and how the final lattice looks.

Nor do the drawings show the fluid under continuous shear. Above the yield stress chains are broken and re-formed continuously, and the fluid flows with a viscosity that falls as the shear rate rises, because fast flow gives chains less time to re-form. The balance between the rate at which the field rebuilds chains and the rate at which shear breaks them is measured by a single ratio, the Mason number, and much of the practical design of these fluids is getting that ratio right. A dense suspension with no field at all can build load-bearing chains of its own when pushed hard enough for its grains to grip, and there friction rather than polarisation decides which contacts carry the load.

Still open: what holds the giant fluids together #

The coated-nanoparticle fluids break the quadratic scaling and exceed the point-dipole estimate by a factor of thousands. Explanations centre on the molecules at the contact between particles — polar coatings forming aligned bridges, or a layer of adsorbed liquid whose own dipoles line up — but the structure of that contact has not been observed directly while the field is on, and the fluids’ strength depends on their preparation in ways that are not fully reproducible between laboratories. Whether the effect is a new mechanism or an extreme version of the gap-field concentration is not settled, and settling it would say how much further these fluids can be pushed.

The habit worth carrying away is to ask what a field does to the neighbours of the thing it acts on. A uniform field exerts no force on a neutral particle, but it gives every particle a dipole, and dipoles act on each other with a sign that depends on direction — so a field that pushes on nothing can still arrange everything, and a liquid can be given a structure that exists only as long as the voltage does.