The energy of an electrical double layer
5.2 Electrowetting: The EDL surface pressure on an electrode
5.2.4 Surface tension effects near a charged vertical wall
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 25ois 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.
Electrowetting: The EDL surface pressure on an electrode 121 Thus the maximum height, reached when𝜃=0, isℎ=√
2𝜆. Water (in contact with air) has a capillary length of𝜆=2.7 mm, and thus the maximum height of the meniscus is∼3.9 mm.
[If the vertical wall is at an angle𝛼, the above remains valid, with𝛼added to𝜃in Eq. (5.25).
When the wall is tilted towards the liquid, the height of the meniscus goes up.]
We make a calculation of the energies involved in charging an electrode that is placed vertically in a salt solution, with the meniscus at a height of 3 mm. A height ofℎ=3 mm implies a contact angle of 23.3o, a surface pressure of the wetted electrode of𝑃
s=67.1 N/m, a surface potential of𝜙
D=10.83, a surface charge ofΣ =1.317 C/m2, and a corresponding electrical energy of 0.30 J/m2per unit wetted electrode area. By multiplying with the height of 3 mm, we obtain the electrical energy per unit length in the direction along the contact line of 898.2𝜇J/m. The gravity energy can also be calculated numerically (in lifting up the fluid from the level of the bulk liquid) and is 45.2 𝜇J/m, while the surface energy in the extra created L/G interface is 65.6𝜇J/m. (This number is the product of𝛾
LGand the extra created line length of 0.899 mm; this is the extra ‘length’ of L/G surface as measured in the direction to the electrode.)
In this example of the shape of the L/G surface near a single vertical wall, we find that gravity and surface energy are about equal in magnitude, while the energy of charging the electrode is about 9×larger than the other two energy terms together. Interestingly, the ratio of these numbers is very different here compared to the case of capillary rise, where surface energy was not more than 1% of 1𝜇J, thus negligible compared to the other two energies, see Fig. 5.3, while electrical energy was larger than gravity by a factor of 6–8 (while in the new example the ratio is∼20).
Now, for the example of the single vertical wall, there seems to be an error in all energy numbers. Because how can it be that the electrical energy of 898 𝜇J/m is not the same as the gravitational plus surface energy? Aren’t we investing electrical energy with the aim to raise the liquid against gravity, and expand the surface? But the latter two terms are much lower together, almost a factor of 9. Where is the factor 9 error?
The answer is, there is no error. Instead, we invest electrical energy and that is stored in three ways: 1. charging the surface; 2. lifting up the fluid against gravity; and 3. creating the extra L/G surface. The electrical energy that is storedin the electrode (EDL), is the number of 898𝜇J/m above.
The electrical energy we invest must be equal to the summation of these three energies, which is 1009𝜇J/m. Let us see if we can calculate that number via an independent route.
In a first method, for each height (from zero to the 3 mm) we determine the contact angle via Eq. (5.25), thus the surface pressure via Eq. (5.11), and integrate, i.e., we push against the surface pressure along the path to push the contact line up to its final position. Thus we
122 The energy of an electrical double layer obtain for the invested energy
𝐸inv=
∫ ℎ
0
𝑃sdℎ=𝛾LG
∫ ℎ
0
cos𝜃dℎ=𝛾LG·𝜆−2·
∫ ℎ
0
√︁
𝜆2ℎ2−1/4ℎ4dℎ (5.26) which results in a number of𝐸inv=110.8𝜇J/m, but this is not the number of 1009𝜇J/m that we are after! Actually it equals the sum of the gravitational and L/G surface contributions.
Thus, integration of the surface pressure over height, only gives us the energy in the L/G interface, plus the gravitational energy. In hindsight it makes sense that this last calculation gave an energy that was too low, because when we push charge in an electrode, we are not pushing against the contact line, but we push against the potential of the EDL already formed. Thus we have to make a calculation where we charge an electrode and calculate the energy that is invested by integrating from one charging state to another. In a standard calculation, the electrode has a fixed surface area and thus we integrate from one charge density to another, see Eq. (5.9). However, in the present case, while we charge the wetted part of the electrode expands, and this increase in area must be part of the integration process, because we integrate over charge (C), not charge density (C/m2). This becomes
𝐸inv=𝑉
T
∫ 𝑆
0
𝜙Dd𝑆=𝑉
T
∫ 𝑆
0
𝜙D(ℎdΣ+Σdℎ) (5.27) where𝑆 is a charge with unit C/m, andΣas before the charge density,Σ =𝑆/ℎ. For both terms in the integration, we must know howΣand𝜙
Dchange with height, which we obtain in a calculation where we combine Eqs. (3.15), (5.11), (5.13), and (5.25). In a numerical calculation, this entire procedure leads to an energy that is invested of ... 1009𝜇J/m, exactly the same as the number for total energy arrived at earlier!
It is interesting that this last integration results in the total energy that is invested in the process, although this calculation seems to be about electrostatics only (EDL formation), and nowhere are the gravity and L/G surface energy involved. So how can this calculation also provide us the gravity energy and the energy of the L/G interface? The answer is that
‘information’ of these two energies is included in the dependency of charge on meniscus height, i.e., on the ‘shape’ of theΣ(ℎ)-function. This relationship is part of the integration to obtain𝐸inv. For another surface tension or surface shape, this function is different, and thus the outcome of the integration would be different. Thus, the electric work as calculated by Eq. (5.27) was in effect also pushing against the contact line.
In conclusion, we have consistency between all equations that we used in this problem of single electrode electrowetting, and consistency with the same theory applied earlier on for the electrowetting of a capillary.
Electrowetting: The EDL surface pressure on an electrode 123 5.2.5 Influence of Stern layer on contact angle and electrowetting
The above two sections considered several elements of an EDL model, such as the diffuse part of the EDL, described by the Gouy-Chapman equation, the ionization of titania, an amphoteric material, and the related chemical energy of surface (de-)protonation. The next extension of these models would be to include a Stern layer (dielectric layer). As explained before, the Stern layer is a layer of constant capacitance, that can be considered to be located between the diffuse layer and the underlying 0-plane or electrode. This 0-plane is where in the case of titania we assume the surface groups (titania charged groups) to reside. The Stern layer changes the effective pH at the 0-plane, thereby reducing𝛼(pushing it closer to zero), and it adds a term to the surface energy. Also in the case of electrowetting the Stern layer plays a role in EDL surface energy, surface pressure and the resulting capillary rise. Actually, the Lippmann equation of capillary rise assumes only a constant (Stern, Helmholtz) capacitance, and a diffuse layer is not considered. Despite their relevance, we must nevertheless postpone discussion of these Stern layer effects to a later time.
Electrocapillary curves. A problem related to the topics in this section, is the phenomenon of electrocapillarity, where the fluid inside a capillary is a liquid metal, mercury, which can be externally electrified. The mercury column at its lower end is in contact with electrolyte. The EDL structure at this electrolyte/mercury interface can now be studied.
Compared to the electrowetting case discussed above, in this device electronic charge is not injected into the conductive wall of the capillary, but it is injected into the fluid itself, which is a quite different experiment. This electronic charge will move to the mercury/electrolyte interface. The required gas phase pressure applied to the top of the column to keep it in place is a measure of the surface tension of the mercury/electrolyte interface, and data of this surface tension versus voltage are plotted as electrocapillary curves, which have a maximum at a certain voltage (when the interface is uncharged), and they drop off, roughly symmetrically, at lower and higher voltages. [The asymmetry is because when mercury is positively charged it attracts anions as counterions, but when negatively charged cations are the counterions, and these ions may have different properties such as hydrated ion size and shape.] From these curves (which depend moderately on salt concentration) by analysing the slope (first derivative) we obtain the charge for each voltage, and taking a second derivative results in the capacitance of the EDL as function of charge.
Detailed discussion of this methodology in this book must wait for a later moment.
124 The energy of an electrical double layer