# Electrowetting: from basics to applications

## Summary (4 min read)

### Historical record

- The study of electromotive forces of contact has up to now provided a complete chapter distinct from the study of capillary phenomena with variable results.
- They are linked to motion of mercury electrodes.

### A.1.1 Definitions

- The name capillary constant or interfacial tension has been given to the coefficient A of the Laplace’s equation (A.1): p = A ( 1 R + 1 R′ ) (A.1) p is the normal pressure in any point of the surface; R and R′ are the principal radii of curvature in this point.
- Electrical equilibrium being established, the electrical potential inside the mercury is known to have a unique value V ; likewise the electrical potential inside the body of water (or acidic water) has a uniform value V0.
- I will call this difference electrical difference at the water-mercury surface.
- This being defined, experiments show that if the apparatus is set such that x takes a fixed value x0, A takes a value A0 perfectly defined, i. e. the perturbations observed with the ordinary settings do not occur.
- If x is set to another value x1, A takes another value A1, value perfectly determined and without perturbations.

### A.1.2 First method

- A vertical glass capillary GG’ communicates at its bottom with a mercury reservoir A (see Fig. A.1).
- This mercury penetrates into the capillary tube where it undergoes a depression.
- The same apparatus has been used to demonstrate that if the electrical difference takes a new value x1, the capillary constant takes a new value A1 which remains constant as long as x1 does.
- The microscope is left in place and the meniscus is moved back to its original position, given by the horizontal mark of the retic- 8NOTE:.

### A.2 Second Law

- When this surface is mechanically deformed, in order to increase its area, the value of the electrical difference at the surface increases.
- One can set the experiment has follow: two glass beakers containing both mercury covered by acidic water are set side by side; they are set in electrical communication together using a coton wick or some filter paper.
- One will notice besides that the First Law and the Second Law govern two reversed series of phenomena.
- Inversely, when the surface is increased, the electrical difference increases; this is the Second Law.
- When a liquid surface is deformed the electrical difference varies in such a way that the interfacial tension developed under the terms of the First Law opposes to the continuation of the motion.

### A.3.1 Notations

- One can graphically represent the state of the surface of contact for each time as follow.
- When the state of the surface changes, P moves along a certain curve.
- The external work dT supplied during a deformation infinitely small of the surface has for expression dT = −A× dS This work is thus graphically represented by the area generated by the displacement of the ordinate A. dq being the quantity of electricity flowing through the surface from water to mercury, one can write: dq = XdS +.
- Y Sdx X represents the quantity of electricity flowing through the surface when the surface S varies of 1 mm2, x being kept constant.
- One can obtain this condition by two different ways.

### A.3.2 Conservation of energy

- I consider first the case where the representative curve is closed.
- When the quantity of electricity dq flows through the surface from water to mercury, the charge of the water volume (whose potential is V0) decreases of a quantity dq; the electrical energy of the free-surface of water decreases of V0 × dq.
- After having followed a closed cycle, the state of the surface is the same as in the origin, the external work supplied is thus necessarily equal to the decrease of the electrical energy; one has then for any closed cycle∫ xdq = − ∫ AdS, or ∫ (xdq + AdS) = 0.
- Let φ be the total inner energy of the surface unit water - mercury.
- The hydrogen has not been given off neither.

### A.3.4 Consequences: Expression of the two electrical

- Capacities of a surface Eq. A.4 and Eq. A.2 determines X and Y : X = −dA dx (A.5) and Y = −d 2A dx2 (A.6) These equations can be translated in everyday words: 1. The electrical capacity of the surface unit at constant electromotive force is equal to the opposite ot the first derivative of the interfacial tension.
- The electrical capacity of the surface unit at constant surface is equal to the opposite of the second derivative of the interfacial tension.
- One will notice that the signs are in agreement with the experiments.
- The first of these consequences has already been demonstrated using different arguments (see Chapter A.2.3)11.

### A.4 Electrocapillary motor

- Description and func- tioning of the machine A mercury surface running on a closed cycle is a device able to transform an indefinite quantity of electricity into mechanical work, or inversely, without undergoing any changes, some electrical work is supplied or absorbed depending on the direction of the motion of the point P on the cycle (clockwise or anticlockwise).
- The apparatus having been built up symmetrically, the vertical thrust undergone by the bundles BB and transmitted to the arcs UU are in equilibrium at then end of HH, like the weights of the plates of a balance are in equilibrium at the end of the beam.
- The electrical motors have the same efficiency as the electromagnetic motors [Verdet13].
- When the mercury surface is allowed to contract it executes a mechanical work; in the same time its electrical difference decreases, as well as its capillary constant which depends on it, unless a current keeps these quantities constant.
- There is then absorption of electrical work.

### A.5 Measurement of the capillary constant of

- Necessity to measure the capillary constant keep- ing the electrical difference constant The apparatus of the Fig. A.1 allows to keep the electrical difference constant and to measure in the same time the capillary constant.
- One sees that the numbers of the column V do not differ from their average more than 1/10 mm.
- These variations are the source of the perturbations.
- Let us assume for instance that at the bottom of the beaker containing the water a large drop of mercury is introduced using a pipette in order to measure its height and to deduce from that the value of the capillary constant, method used by Mr. Quincke.

### A.6.1 Description of the apparatus

- Experiments have shown that the laws stated in the Chapter A.1 can be applied to the measurement of electromotive forces.
- In the tube A, a mercury column is poured, sufficiently high so that the mercury penetrates by its own pressure in the capillary tip (750 mm for example).
- The microscope is then moved to put the zero of the micrometer tangent to the meniscus; the microscope will then always keep this position.
- Measurement of the electromotive force of a Daniell cell loaded with acid.
- The capillary electrometer has been used for a large number of measurements;.

### A.6.4 Properties of the electrometer. Invariability. Ac-

- The invariability of the indications of the capillary electrometer has always been perfect.
- When the apparatus is charged the mercury level goes out of the origin; when it is discharged by connecting α and β with a metal wire the mercury comes back immediately and perfectly at zero.
- The divisions were sufficiently separated to estimate a fraction of this interval, for example 1/6.
- In both experiments one can check directly using an auxilliary electrometer that the interuption of the motion is due to the electrization that the motion produces following the Second Law.

### A.7.1 Historical background

- Henry from Manchester has observed in 180019 that the mercury he was using as an electrode changed its shape while tarnishing.
- Draper24 in 1845 observed that the depression of the mercury column in a capillary tube filled with water decreases with the flow of an electrical current from water to mercury; this physicist admitted as an explanation that water acquires the property to wet the mercury during the flow of the current.
- Assuming a stationary state, if one considers lines of equal electrical differences that could be drawn on the surface, these lines would be in the same time lines of equal capillary tension (First Law, Chapter A.1); thus each point of the surface will then have a tangential velocity perpendicular to these lines of constant tension.

### Summary

- Two distincts laws have been demonstrated by the experiments.
- The First Law (Chapter A.1) links the capillary constant to the electrical difference.
- These two laws, established separately from the experiments are linked together by a crucial theoretical argument.
- The electrometer described in the Chapter A.6 is the most accurate of all the electrometers known up to know.
- The Chapter A.7 gives the explanations of Gerboin’s whirls.

### A. – The junction angle

- The junction angle between glass and mercury under pure or acidic water is equal to zero at the condition that the water wets the glass.
- This condition is fulfiled with water only for a short time; with acidic water for several hours or even days.
- When the motor described above is left a long time without working it is wise to raise the bundles above the mercury to wet them again with the acidic water.
- The same condition holds for the electrometer after any deviation in the instrument.
- 28The hypothesis which assigns the variation of the electrical difference at the electrodes to the chemical action of the current (which leads to the terms polarization by hydrogen, by oxygen) has neither been called nor discussed in the present work; I will try to show that the chemical action and the electrical phenomenon can be produced separately.

### B. – Precaution when checking the First Law

- In the Chapter A.1 and A.6 it has been assumed all the time that the wire α is connected to the negative pole of the electromotive force included between α and β.
- And when the electromotive force x0 stops being constant the value of x cannot be determined anymore (Chapter A.1); the zero of the apparatus has been changed.

### C. – Measurement of the coefficient X

- Measurements performed to determine the numerical value of X (Chapter A.3) have shown that X is independent on s; this value has been determined with only a low accuracy and by default.
- The reason is that the values of x are getting closer to a constant value x0 with time and this faster when they are much different than x0; this kind of loss of charge has been exactly compensated during the verification of the First Law.

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### "Electrowetting: from basics to appl..." refers background or methods in this paper

...The Thermodynamic and Electrochemical Approach Lippmann’s classical derivation of the electrowetting or electrocapillarity equation is based on general Gibbsian interfacial thermodynamics [32]....

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...) The local ion concentration and the electrostatic potential φ are coupled via the PoissonBoltzmann equation [32]...

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### Additional excerpts

...[26, 88–90])....

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### "Electrowetting: from basics to appl..." refers methods in this paper

...A popular numerical tool is the public domain software package SURFACE EVOLVER [87]....

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