The surface pressure in electrowetting according to the GC model

The energy of an electrical double layer

5.2 Electrowetting: The EDL surface pressure on an electrode

5.2.1 The surface pressure in electrowetting according to the GC model

112 The energy of an electrical double layer

5.2 Electrowetting: The EDL surface pressure on an

Electrowetting: The EDL surface pressure on an electrode 113 also leads to a novel representation of Eq. (5.4), which is

𝐹^D=−2𝑉

T

√︁Σ²+8𝜀 𝑅𝑇 𝑐_∞−√︁

8𝜀 𝑅𝑇 𝑐_∞

(5.10) which mathematically is the same as Eq. (5.4).

Thus, in this case the energy of Eqs. (5.4) and (5.8) combined is positive and thus EDL formation increases the surface energy of the S/L interface, and the more so at higher surface charge. Then, how can it be that charging such an electrode leads to electrowetting, i.e., the spreading of liquid over an electrode, when it increases the surface energy, whereas for titania we had more wetting when the energy of the S/L interface decreased?

The reason is that for electrowetting we cannot use Young’s law, Eq. (5.3), but must return to the force balance, Eq. (5.1). For the L/G interface, and for the dry part of the electrode, we can still use as before𝑃_s=−𝛾. For the S/L interface, there is again a contribution from the uncharged surface,𝛾^unch

SL , and in addition we again have a contribution from the EDL.

But for this contribution it is no longer the case that surface energy and surface pressure are simply given by𝑃

s =−𝛾. Instead of being of opposite sign, they are even of the same sign! All of this is because of a difference in the differentiation of the EDL-contribution to the energy of the S/L interface by Eq. (5.2). For electrowetting, the EDL contribution to the surface energy,𝛾^EDL, is not constant when we expand the surface. Instead, in such an experiment one inserts a certain amount of charge in thetotalelectrode,Σ_tot(per unit total, dry+wet, area), but the charge can only be stored in the wetted part of the surface.^v Thus it is now the case that when the S/L interface expands, the charge density in the wetted area decreases, and the surface energy,𝛾, decreases as well. Thus for electrowetting, for a given amount of injected charge, this surface energy depends on how much the surface is stretched, and is not independent of that, which was the key assumption on which Eq. (5.3) is based.

In this case we must use a modification of Eq. (5.3) where we replace the term𝛾_SLby the constant term𝛾^unch

SL (the contribution to the surface energy of S/L interface due to its contact with water, which is independent of the charging process), and add to that a term−𝑃^EDL

s,SL. Thus we obtain for the electrowetting problem

𝛾_LG·cos𝜃=𝛾_SG−𝛾^unch

SL +𝑃^EDL

s,SL. (5.11)

What is the surface pressure of the S/L interface due to the formation of the EDL,𝑃^EDL

s,SL? The surface pressure can be derived from the total EDL free energy density (𝐹EDL

SL =𝐹D+𝐹S) by using Eq. (5.2), which can be modified to

𝑃s = Σ²𝜕(𝐹/Σ)

𝜕Σ = Σ𝜕 𝐹

𝜕Σ −𝐹 (5.12)

vIn the dry part of the electrode, the S/G interface, a very tiny charge already leads to a large voltage, and thus hardly any charge will be stored here.

114 The energy of an electrical double layer

whereΣis the surface charge density. Implementing the two contributions to𝐹^EDL

SL according to Eqs. (5.4) and (5.8), and making the derivation in Eq. (5.12) leads to

𝑃^EDL

s,SL =8𝑐_∞𝑅𝑇 𝜆

D(cosh(1/₂𝜙

D) −1)=16𝑐_∞𝑅𝑇 𝜆

Dsinh²(1/₄𝜙

D) (5.13) which, after inserting the GC-equation, Eq. (3.15), we can write as function of charge as

𝑃^EDL

s,SL =2𝑉

T

√︁Σ²+8𝜀 𝑅𝑇 𝑐_∞−√︁

8𝜀 𝑅𝑇 𝑐_∞

. (5.14)

With this addition, the relevant force balance for electrowetting, Eq. (5.11), will predict that 𝜃goes down when we increase the electrode charge, as expected.

For a sufficiently low surface charge (or low diffuse layer potential), Eq. (5.14) simplifies to

𝑃^EDL

s,SL = 𝜆

D

2𝜀

Σ²+ O Σ⁴

(5.15) which has a dependency on salt concentration by a power -½, and a quadratic dependency on charge.

For a high surface charge (or high diffuse layer potential), instead Eq. (5.14) simplifies to the ‘counterions only’ limit

𝑃^EDL

s,SL =2𝑉

TΣ−8𝑐_∞𝑅𝑇 𝜆

D (5.16)

which is accurate when the the first term on the right side is at least twice as large as the second term, and thus the criterion to apply Eq. (5.16) isΣ>8𝑐_∞𝐹 𝜆

D. 5.2.2 Electrowetting and capillary rise

Using electrowetting we can quickly change the shape of a bubble or droplet on an electrode, which is described by the theory of the previous section. In addition, electrowetting is often used to induce fluid motion, for instance against gravity inside a vertical capillary. The question in a study of capillary rise is, how much does a fluid move against gravity inside an electrode-coated capillary that we charge up?

We can solve this problem of capillary rise by considering the two energy contributions involved, which are gravity and electrical energy. The energy in the L/G surface, which is the L/G area×surface energy𝛾

LG, is actually very low for a thin capillary (around 1% of the other energies in the example we discuss below).^vi

viThis is very different in the problem we discuss further on that is about a single vertical plate in contact with water.

Electrowetting: The EDL surface pressure on an electrode 115

0 1 2 3 4 5

0 20 40 60 80 100

height h (mm)

Ene rgy (  J)

electrical energy

maximum height

eq

Fig. 5.3:The energies involved in electrowetting, which are electrical energy and gravity, as function of capillary rise (heighth). Electrical energy as function of total charge,Σ^∗. The maximum fluid rise isℎ

max =100 mm, andΣ^∗is based onℎ^∗ =ℎ

max(𝜌

L=1 g/mL,g=9.81 m/s²,𝑎=147𝜇m, 𝛾LG=73 mN/m,𝑐_∞=10 mM).

The electric energy of charging is given by𝐹^D+𝐹^S of Eqs. (5.8) and (5.10), times the total wetted area, which is height ℎ times 2𝜋 𝑎, where 𝑎 is the capillary radius.^vii The gravitational energy of the fluid that is lifted against gravity is¹/₂𝜌

Lgℎ² times𝜋 𝑎² which is the cross-sectional area.^viii This total energy, electrical plus gravitational, is minimized by varyingℎfor each value of a total charge,Σ^∗. ThisΣ^∗is a charge density (in C/m²) for a certain reference area (2𝜋 𝑎 ℎ^∗). Thus we make use ofΣ·ℎ = Σ^∗·ℎ^∗.^ix For this given total charge, when the wetted area expands, the charge per unit wetted area decreases, and this reduces the electrical energy. Thus, as function of height ℎ, the gravitational energy increases quadratically, while the electrical energy decreases, see Fig. 5.3. The summation of these two terms has a minimum atℎ_eq, which is the height up to which the capillary will be filled with liquid.

The equilibrium height, ℎ

eq, can also be found from a force balance acting on the L/G

viiHeightℎis defined relative to a starting heightℎ

0when the capillary is uncharged.

viiiWe use the gravitational constantgas a positive number in this section.

ixWe neglect a technical complication that also for zero charge part of the electrode-coated capillary is below the fluid level, i.e., already wetted. Charge will also go to this region, but the charging of the EDL here will not lead to fluid rise.

116 The energy of an electrical double layer

interface. The first term relates to gravity and is minus the derivative of the gravitational energy with height, which gives a downward force equal to𝜌

Lgℎ×𝜋 𝑎²by which the liquid column pulls the L/G interface downward. This gravitational force is balanced by the EDL surface pressure given by Eq. (5.13) times the circumference, thus 𝑃^EDL

s,SL ×2𝜋 𝑎.^x This balance of forces leads to the equilibrium height in the capillary given by

ℎeq=2𝑃^EDL

s,SL

𝑎 𝜌Lg (5.17)

in which we can insert the various expression for the EDL surface pressure,𝑃EDL

s,SL, given by Eqs. (5.13)-(5.16). The general solution is

ℎ_eq=4𝑉_T (𝜌_Lg𝑎)⁻¹√︁

Σ²+8𝜀 𝑅𝑇 𝑐_∞−√︁

8𝜀 𝑅𝑇 𝑐_∞

(5.18) which shows that the capillary rise increases with charge and increases when the diameter of the capillary becomes smaller. Eq. (5.18) exactly matches the minimum, or equilibrium, depicted in Fig. 5.3, which was based on an energy analysis. For low charge, we can do a Taylor expansion of Eq. (5.18) aroundΣ =0, resulting in

ℎ_eq =𝜆_D· (𝜌_Lg𝑎)⁻¹·𝜀⁻¹·Σ²+ O Σ⁴

(5.19) which shows that in this limit height depends quadratically on chargeΣ, and depends with a -½power on salt concentration.

In the high-charge limit, we can insert Eq. (5.16) in Eq. (5.17), and obtain for the capillary rise

ℎ_eq−ℎ_ref=(𝑎 𝜌_Lg)⁻¹·4𝑉_T·Σ (5.20) which predicts that the height depends linearly on surface charge,without any dependence on any EDL property.^xi This is quite remarkable. If correct, then one expects also the same equation for height versus charge when instead of evaluating a GC model for the diffuse layer, we use a more complicated EDL model, or a model only dependent on a constant dielectric, or Stern, capacitance. In that case capillary rise is independent of the Stern capacitance but only depends on the charge injected into the electrode, as described by Eq. (5.20).^xii

xNon-electrostatic contributions to the surface energy are not part of this balance becauseℎis defined relative to the height of the column when the capillary wall is uncharged.

xiOnly the constant termℎ

ref(which is small compared to the right side at high charge) has a½power dependency on salt concentration.

xiiOf course, the EDL voltage that is required to store a certain charge, depends on properties of the EDL and on the properties of the dielectric coating.

Electrowetting: The EDL surface pressure on an electrode 117 5.2.3 Derivation of the Laplace equation from Young’s equation Can we continue charging the inner surface of the capillary, and thereby have the fluid move up more and more in the capillary? Interestingly, no, we cannot. There is a natural limit, and that follows from the next analysis.

This analysis is based on a combination of the modified Young’s equation for the contact angle in case of electrowetting, Eq. (5.11), with the balance of forces given by Eq. (5.17), which results in^xiii

ℎ_eq= 2

𝑎 𝜌Lg ·𝛾_LG·cos𝜃 (5.21)

which shows the dependence of height on the capillary radiusa, and on contact angle,𝜃, with height increasing whenagoes down or𝜃goes down. Now, Eq. (5.21) shows that there is a maximum in height arrived at when𝜃is zero, and for this height we obtain

ℎeq,max= 2 𝑎 𝜌_Lg·𝛾

LG. (5.22)

The maximum height for the calculation settings of Fig. 5.3 was ℎ

max=100 mm, for a capillary radius of𝑎∼150𝜇m. This is already a fairly thin capillary and this analysis shows that more than 100 mm capillary rise is not possible for an electrode-coated capillary of this inner diameter filled with water.^xiv

We can also derive another result from this analysis. We first note that for thin enough capillaries, the shape of the L/G interface is spherical, i.e., the radius of curvature𝑅 is a constant, independent of radial position. In that case we can relate𝑅to the contact angle 𝜃 and to the capillary radius 𝑎, according to 𝑎 = 𝑅 cos𝜃, which we can implement in Eq. (5.21), resulting in

𝜌Lgℎ

eq=𝛾

LG

2 𝑅

. (5.23)

Now, the left side of this equation is also equal to the increase in pressure when we go vertically down the capillary, starting at a position in the liquid near the L/G interface down to the height of the bath in which the capillary is positioned. This latter pressure equals the external gas phase pressure.^xv Thus, the left side of Eq. (5.23) is the gas pressure minus the pressure in the liquid just below the L/G interface (i.e., the latter is lower than the gas phase

xiiiWe assume𝛾

SG−𝛾unch

SL =0 (this term only leads to a constant change in height).

xivOf course microporous materials such as membranes, gels, or electrodes, can have much smaller pores (down to the nm-range), and thus the wetting of such materials occurs throughout the sample against gravity to a much larger height. Thus water transport against gravity will make use of this microporous phase, not through larger pores if they exist. These will be dry.

xvAll of this is correct irrespective of details of the S/L interaction on the capillary wall (such as EDL charging), and of the L/G interface.

118 The energy of an electrical double layer

pressure). For this pressure difference across the curved G/L interface,Δ𝑃, we therefore arrive at

Δ𝑃=𝛾

LG

2

𝑅 (5.24)

and this is the Laplace equation for a thin capillary! Thus, we derived the Laplace equation (for a thin capillary) based on two inputs: 1. A force balance of the G/L/S contact line on the capillary wall; and 2. An overall force balance relating gravity with the surface pressure of the wetted capillary wall. Thus, we derived the Laplace equation –relating curvature and pressure difference across the L/G interface– by combining two previously discussed force balances.

Derivation of Laplace equation. Let us again derive the Laplace equation based on a vertical capillary, but starting with an energy analysis. We bring a vertical capillary in contact with a volume of water. The capillary just touches the upper surface of the water, and this position we assign as heightℎ=0. The water will rise in the capillary until an equilibrium heighthis reached.

The energy has a gravity contribution,½𝜌

Lgℎ2×𝜋 𝑎2, and a contribution from the energy of the wetted surface minus that of the dry surface,(𝛾

SL−𝛾

SG) ×2𝜋 𝑎 ℎ. Adding up and calculating at whichhthis total energy is at a minimum, leads to 𝜌

Lgℎ 𝑎+2(𝛾

SL−𝛾

SG) =0. This last equation is a ‘column-based’ force balance, and does not include the G/L interface. It is not a force balance on the G/L/S contact line such as Eq. (5.1) which does include the G/L interface.

We can leave out the G/L interface in this balance because the capillary is thin and thus this area small, and thus the energy related to the G/L interface is small too. We implement Eq. (5.3) and the geometrical relationship for a (hemi-)sphere,𝑎=𝑅cos𝜃, leading to𝜌

Lgℎ=2𝛾

LG/𝑅. The column heighthtimes𝜌

Lgequals the pressure in the gasphase minus that just below the curved G/L meniscus,Δ𝑃, and thus we again arrive at the Laplace equation.

A very different derivation of the Laplace equation is based on creating a bubble in a liquid.

If we have a bubble and increase the radiusRby dR, we must push against the surface energy.

The energy required to enlarge the bubble, to push in the extra volume, isΔ𝑃·4𝜋 𝑅2d𝑅, and the energy increase of the surface is𝛾

GL·8𝜋 𝑅d𝑅. [The increase in area when the radius of a bubble increases by dRis 4𝜋(𝑅+d𝑅)²−4𝜋 𝑅2which for small dRbecomes 8𝜋 𝑅d𝑅.] Each of these terms is a force, pushing outward and inward on the bubble surface, and the equilibrium condition is when these forces add up to zero. Doing so, we again arrive at the Laplace equation, Eq. (5.24), now derived on the basis of forming a bubble in a liquid. Thus, with𝛾

SLa positive number, the pressureΔ𝑃, measured as that in the gas phase minus in the liquid, is positive for a gas bubble or in any other situation where the G/L surface ‘bends towards the gas phase’, ‘tries to enclose it’. ThenΔ𝑃is higher than in the liquid around it, the more so the smaller the bubble, and the more so the higher the surface energy of the G/L interface.

The same result also follows if we minimize the energy of a bubble that is compressible, placed

Electrowetting: The EDL surface pressure on an electrode 119

inside a bath of liquid, water for instance. We start with a small bubble at high pressure which we let expand to end at a size where the pressure inside (which is larger than in the bath) is such that we reach mechanical equilibrium. So the expansion (increase of volumeV) leads to a decrease of pressure energy. [If we would compress it against an inside pressure higher than outside, then the energy would go up, so expanding a volume that has higher pressure than outside, means the energy in the object goes down.] Thus counted from the initial volume, the pressure energy is𝐸

p=−∫

Δ𝑃d𝑉with pressureΔ𝑃that inside minus outside (outside must be a large system at constant pressure). The surface energy is𝐸

s =∫

𝛾GLd𝐴, withAarea, and this creation of (more) surface means an energy increase. So what must be minimized is𝐸

p+𝐸

s. Making use of d𝑉=4𝜋 𝑅2d𝑅and d𝐴=8𝜋 𝑅d𝑅we indeed end up with the Laplace equationΔ𝑃=2𝛾

GL/𝑅, and thus the pressure inside the bubble is higher than outside. The same result should be arrived at for a water droplet in air, but then the derivation of an expanding droplet is less intuitive.

Charging beyond the maximum. One final question is, what happens in a charged capillary when we charge the surface beyond the maximum value? [In the example of Fig. 5.3, this maximum was at a charge density ofΣ = Σ^∗∼1.43 C/m²(EDL voltage 283 mV) when we reach a height ofℎ

eq,max=100 mm.]

What then happens is that the surface continues to be wetted, but only with a very thin layer forming, just enough for an EDL to form. This layer therefore can be as thin as just a few nm.

More and more of the capillary will be coated with a nanoscopic thin layer of fluid, and charge is stored in this EDL. The system ‘pays’ in electric energy that must be invested in forming this EDL, as well as in forming the new L/G interface, while S/G interface is replaced by S/L interface.

There can also be a ‘stabilizing’ effect of a repulsive Van der Waals force (repulsive between the dissimilar materials air and metal, interacting across the water film).

For the assumptions made above, charge and potential follow from a force balance for the expansion of the surface. In this balance capillary radius𝑎and heightℎdo not enter. This force balance is𝑃EDL

s,SL =𝛾

LG, with𝑃EDL

s,SLcalculated for instance by Eq. (5.14). Interestingly, we know the answer must be the charge and potential that we have when we reach the maximum capillary height,ℎ^∗. And indeed in our previous calculation we again find a value for the EDL voltage of 283 mV as a threshold value beyond which the nanoscopic layer will form. Or in other words, this nanoscopic film starts to form when (the electrostatic contribution to) the surface pressure is about to exceed the air-water surface energy of𝛾

SL=73 mN/m.

Interestingly, if we run the electrowetting experiment by making steps in voltage, and we now arrive at the EDL threshold voltage identified above (the value required for the nanoscopic layer to start being formed, when the capillary has reached the maximum height, where𝜃becomes zero), then a further increase in EDL voltage will immediately lead to the rapid formation of the nanoscopic layer until some system limit is reached (e.g., the nanoscopic layer reaches the top of

120 The energy of an electrical double layer

the capillary). This may look like an electric breakdown or Faradaic loss current, because now suddenly a small increase in voltage leads to a large current spike. But it is the current that goes into charging the nanoscopic surface layer that is rapidly growing beyond the L/G meniscus.

Contact angle saturation. An outstanding problem in the field of electrowetting is contact angle saturation, which is the phenomenon that a droplet deposited on an (often polymer-coated) electrode surface will not expand further when a contact angle of around 25^ois reached (i.e., the contact angle will not go down further). It is as if a further expansion of the G/L interface just doesn’t pay off anymore, and to accommodate the extra charge, formation of a thin layer on the electrode outside the droplet is preferred. It would be interesting to find out if the electrode voltage –which is measured against a (pseudo-reference) electrode inserted in the droplet– at the point that the lowest contact angle is reached, is the same voltage as measured for the same liquid and same electrode in the geometry of a vertical capillary for the condition that the maximum height ℎ^∗is reached. If these voltages coincide, a nanoscopic film can also be expected to form for a droplet on a horizontal electrode when charge continues to be injected after this threshold voltage is reached. In combination with a repulsive Van der Waals force acting across the nanoscopic layer –which stabilizes the layer, i.e., lowers the energy of formation– this may then explain why the macroscopic G/L interface does not further expand, and that may explain the phenomenon of contact angle saturation.