The angle a voltage can set

The angle a liquid makes derives the contact angle from a balance of three surface energies along a line: the solid–vapour energy pulling the edge of the drop outward, the solid–liquid energy pulling it in, and the liquid’s own surface tension leaning at whatever angle closes the balance. The result, Young’s equation, makes the angle a property of three substances. Change the solid and the angle changes; change nothing and it does not.

Every capillary result since has used the angle that way — how high water will climb takes its cosine as an input, and the pore that fills from dry air sets it to zero and notes what would happen otherwise. It is not, however, a property of the substances in the way a density is. It is a property of a balance of energies per unit area, and anything that adds energy per unit area to one side of the balance moves it.

The cleanest such thing is an electric field.

What the voltage adds #

The arrangement is simple. A conducting liquid — water with a little salt in it — sits on a flat electrode covered with a thin insulating layer, a micrometre or so of a fluoropolymer that water does not wet. A wire dips into the drop, and a voltage is applied between the wire and the electrode. The drop and the electrode are the two plates of a capacitor, and the insulator is its dielectric.

A capacitor stores energy $\tfrac12 CV^2$, and for this one the capacitance is proportional to the wetted area: $C = \varepsilon_0\varepsilon_r A/d$ for a coating of thickness $d$ and relative permittivity $\varepsilon_r$. Spreading the drop over more area stores more energy. That sounds as though spreading costs energy and the drop should resist it, and the resolution is the part that makes the whole effect work.

The battery holding the voltage fixed has to supply the charge the extra capacitance takes, $dQ = V\,dC$, and it supplies it at the full voltage, so it does work $V\,dQ = V^2\,dC$ — which is $\varepsilon_0\varepsilon_r V^2/d$ for every unit of area newly wetted. Half of that goes into the capacitor. The other half is available to do mechanical work, and the free energy of the whole system — drop, capacitor and battery together — falls by $\varepsilon_0\varepsilon_rV^2/2d$ for every unit of area wetted. The same factor of two appears whenever a capacitor is charged from a battery at fixed voltage, and where the energy of a field is is the bookkeeping behind it.

A free energy that falls in proportion to the wetted area is indistinguishable, in Young’s balance, from a lower solid–liquid surface energy. Subtract it and the balance becomes

$$\cos\theta = \cos\theta_0 + \frac{\varepsilon_0\varepsilon_r V^2}{2d\gamma},$$

the Lippmann–Young law. The figure evaluates it and checks two of its features on the drawn curve: that the cosine changes by exactly the electrical term, and that the angle is the same at plus and minus any voltage. The second is why electrowetting works with alternating voltage, which avoids charging up the insulator and is how most devices drive it.

The name goes back to Gabriel Lippmann, who found in 1875 that the surface tension of mercury against a salt solution depends on the voltage between them, and built an electrometer that read voltages off the height of a mercury meniscus in a fine tube. That instrument was sensitive and fast enough that, in 1887, Augustus Waller used one to record the first human electrocardiogram — the electrical activity of a heart, read as the twitching of a capillary meniscus. The modern version replaced the electrolyte’s own double layer with a solid insulator, which lets the effect run to hundreds of volts without electrolysis, and that is the form drawn here.

Thinner insulators, lower voltages #

The electrical term in the Lippmann–Young law has the thickness of the coating in its denominator, so the voltage needed for a given change of angle grows as the square root of the thickness. The figure makes that concrete. The one-micrometre coating of the first figure reaches the law’s complete wetting at 110 volts; a coating a fifth as thick reaches it at 49, and every other angle along the curve is reached at the same factor of 2.24 less. The permittivity enters the same way, so a coating whose permittivity is four times higher buys another factor of two.

That arithmetic is why electrowetting devices came down from hundreds of volts to a few tens. Early ones used thicker insulators, partly because thin polymer films have pinholes through which the liquid reaches the electrode and the device fails by electrolysis. Layers of high-permittivity oxide under a very thin water-repellent top coat, each chosen for one job, brought the working voltage into the range a battery-powered circuit can supply. The price is fragility. The field inside a 0.2-micrometre layer at 49 volts is 245 kilovolts per millimetre, close to what good thin-film insulators survive, and a coating thin enough to need little voltage is thin enough to break down under it. The trade between the square root in the law and the breakdown field of the material is the design problem of every electrowetting chip.

The same drop, spread #

Two features of the drop figure are worth reading off. The first is how little happens at first: 40 volts takes the angle from 115° only to 104°, because the cosine starts on the flat part of its curve near 120°, and the change in angle for a given change in cosine is smallest there. The second is how much happens later: the last 25 volts take the angle through thirty degrees. Electrowetting is a quadratic law in voltage acting through an inverse cosine, and the combination makes a device sluggish at low voltage and touchy near 90°.

The drop itself has not changed at all. Its surface tension is the surface tension of water, its volume is what it was, and nothing mechanical touches it. The solid under it has been made to look more wettable by storing energy in the insulator beneath the wetted area. A contact angle is a free energy per unit area written as a cosine, and it can be moved by anything that adds a free energy per unit area.

Where the pull actually is #

The energy argument says the drop spreads, and it is silent about where on the drop the force acts. A physical force has to act somewhere, and the answer is sharper than the global picture suggests.

Inside the drop, well away from its edge, the liquid is a conductor at a single potential and the field in the insulator beneath it is uniform and vertical: it pulls the liquid down onto the surface with an electrostatic pressure, and that pressure has no sideways component. Nothing there pulls the edge outward. The outward force lives at the contact line, where the capacitor ends. There the field lines fringe out through the edge of the drop, and the charge on the liquid surface near the line is pulled sideways by them.

This is exactly the force that pulls a slab of dielectric into a charged capacitor, and the reason is the same. The energy stored depends on how far the slab has entered, the force is the derivative of that energy, and the derivative is non-zero only where the field is not uniform — at the edge. An ideal parallel-plate formula has no edge and predicts the force without being able to say where it acts. Electrowetting is a capacitor with a liquid plate that is free to move, and the liquid moves for the reason the slab does.

The locality has a consequence that is easy to state and hard to see. The fringing field extends only about a coating thickness from the contact line — a micrometre — and outside that region the liquid surface feels no electrical force at all, while inside it the surface is bent. So at the scale of the eye the drop has the Lippmann–Young angle, and at the scale of the coating the liquid meets the solid at the ordinary Young angle it always had, the two joined by a curve too small to see. The apparent angle changes and the microscopic one does not. Measurements that resolve the region confirm it, and it is one reason the simple law is better than it has any right to be: it predicts an apparent angle, and it is compared with apparent angles.

Drops that move without pumps #

A pull that can be switched on and off along a line is a way to move liquid. Pattern the electrode beneath the insulator into an array of separate pads, put a drop over the boundary between two, and energise the one ahead: the half of the contact line over the live pad is pulled outward, the half over the dead one is not, and the drop moves onto the live pad. Sequence the pads and the drop walks.

The force available is the pull per length times the width of the contact line, and for a millimetre drop at a few tens of volts it is a sizeable fraction of the drop’s weight — enough to move it at centimetres per second, split it in two, merge it with another and mix the result, all on a flat chip with no channels and no pumps. The competing force is the one the angle a liquid makes identifies as the thing that decides whether a drop moves at all: contact-angle hysteresis, $\gamma(\cos\theta_r - \cos\theta_a)$ per unit length, which the electrical pull has to exceed before anything happens. Low-hysteresis coatings are therefore as important to these devices as high-permittivity ones.

The same principle, with the drop confined in a pixel and coloured oil displaced by water, is a reflective display; and with the drop replaced by a curved interface between two liquids, it is a lens.

What it takes to make a drop move #

A drop sitting on a uniform coating under a uniform voltage spreads and stays where it is: its contact line is pulled outward equally all round, and a symmetric pull moves nothing. Moving a drop needs the pull on one side to exceed the pull on the other, and it needs the difference to beat the thing that holds every real drop in place.

That thing is contact-angle hysteresis. The angle a liquid makes found that a contact line advances at one angle and recedes at a smaller one, and that the difference holds a drop against a force per unit length of $\gamma(\cos\theta_r - \cos\theta_a)$. A drop is pinned until something pushes harder than that, which is also why a drying drop leaves a ring — its edge stays put while its volume goes.

Energising the pad ahead of a drop and leaving the pad behind dead adds the electrical pull $\varepsilon_0\varepsilon_rV^2/2d$ to one side only, so the drop starts to move when that pull exceeds the hysteresis force. The numbers make the point about coatings. For water on a 1 µm fluoropolymer layer, a surface with twenty degrees of hysteresis about 110° — advancing at 120°, receding at 100° — needs about 52 volts before a drop moves at all. The same coating polished to five degrees of hysteresis needs about 26. Halving the voltage halves the field in the insulator, which is the difference between a coating that lasts and one that breaks down, so the unglamorous property of a low hysteresis matters to these devices as much as a high permittivity does.

The drop being a conductor is what lets the pull act only where it is wanted. A conducting liquid is at one potential throughout, for the reason the inside of a conductor gives, so the whole of the drop’s underside is one plate of the capacitor and the only place the field departs from uniform is at the edge. Put the edge over a live pad and it is pulled; put it over a dead one and it is not. The drop does not have to be pushed from behind, and nothing in the liquid has to flow except what the moving contact line drags with it.

There is also a reason these devices work with drops a millimetre across and not with puddles. The electrical pull acts along a line, the drop’s weight on its area. Below the capillary length — about two and a half millimetres for water, the size at which a small bubble and a large one stop being governed by the same forces — surface forces dominate gravity and a line force of tens of millinewtons per metre is enough to push a drop around. A puddle is flattened by its own weight into a shape the contact angle barely affects.

A lens whose power is a cosine #

Put water and an oil into a small cylinder, with the water wetting the bottom and the oil above, and the interface between them is a curved surface meeting the cylinder wall at a contact angle. Choose the two liquids with the same density and gravity no longer sags the interface, so its shape depends only on the angle at the wall: a spherical cap whose curvature is the cosine of that angle divided by the cylinder’s radius. Two liquids with different refractive indices on either side of a curved interface make a lens, and its power is the index difference times the curvature.

The electrowetting angle at the wall is set by the voltage, so the lens’s power is too, and it passes through zero when the interface is flat at 90°. The index difference between a water solution and a suitable oil is only about a tenth, which is why the interface has to be strongly curved and the cylinder small; with a radius of a millimetre and a half the range is more than a hundred dioptres, from a strongly diverging lens to a strongly converging one, in a few tens of milliseconds and with no moving part.

That is the autofocus mechanism in some barcode scanners and compact cameras, and it rests on every result in this subject at once: a surface tension, a contact angle, the spherical shape a surface takes when gravity is negligible — the skin that is not a skin minimising its area — and the electrical term that makes the angle adjustable.

Where the law stops #

The angle saturates. The law is exact in its model, and the model says a large enough voltage wets any surface completely. Real surfaces stop: the angle falls with voltage as the law says, then levels off, typically somewhere between 60° and 80°, and raising the voltage further changes nothing or starts to damage the coating. Why is still argued. Charge injected into the insulator would screen the field; ionisation of the surrounding medium at the sharp edge of the drop would leak charge away; the contact line itself may become unstable and throw off tiny satellite droplets once the electrical stress at it is large enough. Each explains some experiments and not others, and the saturation angle is not predicted by any accepted theory.

The insulator has a breakdown field. A hundred volts across a micrometre is a hundred megavolts per metre, comparable with the breakdown strength of the best thin polymer films, so the complete-wetting voltages the law predicts for thin coatings are right at the edge of destroying them. Thicker coatings survive and need more voltage, which is the basic trade every device makes.

The liquid must be a conductor. The derivation assumed the drop is at one potential throughout, which needs free charge to reach its surface faster than the voltage changes. For water with salt that is true up to very high frequencies; for a poorly conducting liquid, or at high frequency, the drop behaves as a dielectric itself, the field penetrates it, and the force comes from polarisation rather than from free charge, with a different law.

And hysteresis and roughness have been left out. The drawn angles are the equilibrium ones. A real contact line sticks, advances at a higher angle than it recedes at, and needs a finite electrical pull before it moves at all — which is exactly the threshold a drop-moving device has to overcome and a lens has to tolerate as a slight dependence of its power on which way the voltage last changed.

What the pictures cannot show #

The drop profiles are spherical caps, and near its edge an electrowetted drop is not one. Within a coating thickness of the contact line the surface bends from the macroscopic angle to the microscopic Young angle, and the charge that feels the force sits in that bend. None of the figures resolves it, and it is where every open question about saturation lives.

Nor do the figures have time in them. A drop switched from one angle to another does not glide to its new shape; it overshoots and rings, with a period set by its mass and surface tension, and damps out over milliseconds. A lens that changes focus rings the same way, and the speed at which such devices can be driven is set by that ringing rather than by anything electrical.

Still open: why the angle stops falling #

Every other feature of electrowetting is understood well enough to design with. The saturation is not, and it is the feature that limits what the devices can do: a lens cannot reach powers the saturation forbids, and a drop cannot be spread past it.

The candidate explanations make different predictions about what should change the saturation angle — the coating’s material if trapped charge is responsible, the surrounding gas or oil if ionisation is, the drop’s size and the electrode geometry if the contact line’s stability is. Experiments that vary one at a time have found cases consistent with each, and the likeliest resolution is that several mechanisms set the limit in different regimes. Which one dominates in a given device is decided, for now, by measurement rather than by theory.

The habit worth carrying away is the one Lippmann’s electrometer already embodied. A quantity treated as a material constant is often an energy balance that nobody has yet put anything else into. The contact angle looked like a property of three substances for as long as the only energies per unit area anybody considered were surface energies, and it stopped looking like one the moment a capacitor was slipped underneath.