The energy of an electrical double layer
5.1 Wetting of titania: The free energy of an EDL with ionizable surface charge
In previous chapters we discussed the EDL structure according to the Donnan and the GCS models, also including the possibility that surface charge responds to local pH, i.e., an ionizable surface or material, see §2.5 and §3.7. In the present section we use the GCS model for a single surface that does not have a fixed surface charge, but the charge responds to local pH (i.e., at the surface). We analyze the surface energy of the EDL, and describe how this energy impacts the contact angle when a droplet of water is in contact with such an ionizable surface.
EDL theories for single surfaces are often used to calculate properties such as the zeta- potential and charge of a surface as function of pH and salt concentration, as we also do in Ch. 3. However, in this and the next sections we extend the range of applications of EDL theory and demonstrate the use of GCS theory for several very different purposes. In the present section we discuss the pH- and salt-dependence of the contact angle of an air bubble in contact with a flat titania surface submersed in water (Virgaet al., 2018). Titania has a surface charge that is strongly pH-dependent, from positive at low pH to negative at high pH, i.e., titania is an amphoteric material, and in the example calculation in this section we show the effect of pH and salt concentration on the EDL structure and how that impacts the contact angle, see Fig. 5.1 for an illustration.
A balance of forces acting on the G/L/S contact line determines the contact angle. In this balance, three forces play a role, one force related to the L/G interface, one to the S/L interface, and one to the S/G interface.
We can define the force by which each surface pulls on the contact line, and this is the surface tension,𝜎. The reverse is the surface pressure,𝑃
s, which is the force that pushes on the contact line. Because in the present section all three surfaces pull on the contact line, the𝜎’s are positive and the𝑃s’s are all negative.
The force balance on the contact line, in the direction along the solid surface, is 𝑃s,LG·cos𝜃+𝑃
s,SL=𝑃
s,SG (5.1)
where𝜃is the contact angle measured in the liquid phase (water).
For each of the three surfaces, the surface pressure is related to the surface energy 𝛾 according to
𝑃s =−𝜕(𝛾 𝐴)
𝜕 𝐴 (5.2)
where𝐴is the area of that surface, and𝛾is an energy density, i.e., defined per unit area.
Wetting of titania: The free energy of an EDL with ionizable surface charge 109
Fig. 5.1:The contact angle of a water droplet on titania. The charge of titania in contact with water changes from positive at low pH, to zero at the point of zero charge (PZC), to negative at high pH. A droplet of water placed on an otherwise dry titania surface would be the most flattened out at very low and very high pH (low contact angle), when the energetic reward of the surface being in contact with water is the highest.
Now, when𝛾is a constant when we change the area (i.e., when upon stretching the surface there is no internal variable such as concentration or charge that changes), then𝑃
s =−𝛾, and we can rewrite Eq. (5.1) to the classical Young’s equation,
𝛾LG·cos𝜃=𝛾
SG−𝛾
SL. (5.3)
In Eq. (5.3) there is the water/air surface energy which is𝛾
LG∼73 mN/m. The other two energy terms however are yet unknown, though we only need to know the difference between the two, to be able to predict the contact angle.i For the dry titania (S/G interface)𝛾
SGis a constant, while for the wetted surface, i.e., titania in contact with water, there is a constant term, 𝛾unch
SL , which is the energy when the surface is uncharged, and when the surface is charged, there is also a term due to the free energy of EDL formation on this surface.
This term due to the EDL is negative and reduces the surface energy of the S/L interface (the titania in contact with water), and the more so the higher is the charge of the titania, thus at very high and very low pH. Reducing𝛾
SLbecause of EDL formation will make the contact angle𝜃smaller, i.e., more of the titania surface will be wetted, see Eq. (5.3).
iOr vice-versa, data on the contact angle will only provide information on the difference between these two 𝛾-terms.
110 The energy of an electrical double layer
For titania in contact with water (the S/L interface), the EDL surface energy can be written as the sum of a chemical and an electrical contribution, where the chemical contribution is negative (this actually drives the spontaneous formation of the EDL) while the electrical energy is always positive. The sum is negative because EDL formation is spontaneous in this case.
Instead of a distinction between chemical and electrical work, it is easier to write the EDL energy as a sum of a contribution from the diffuse layer,𝐹D, and from the surface,𝐹S(Chan and Mitchell, 1976). These two𝐹-contributions are both added to the surface energy of the S/L interface in the absence of charge, which is𝛾unch
SL , and in this way these two terms end up in Young’s equation.
For a single surface, within the GC model for a 1:1 salt, the contribution of the diffuse layer isii
𝐹D=−16𝑐∞𝑅𝑇 𝜆
Dsinh2(1/4𝜙
D) (5.4)
where𝜙
Dis the electrical potential at the start of the diffuse layer (at the Stern plane), i.e., at the titania surface.iii
For an ionizable material with only acidic or basic groups, the surface contribution to the free energy is
𝐹S=𝑅𝑇 𝑁ln(1−𝛼) (5.5)
where𝑁is the concentration of surface groups (in mol/m2), and where the ionization degree, 𝛼, is based on a standard Langmuir model (both for acidic and basic groups,𝛼in Eq. (5.5) is defined as a number between 0 and 1), with the surface charge given byΣ =±𝐹 𝛼 𝑁.
However, materials such as titania and alumina are amphoteric materials, and the charging degree of each surface group is between -½ and +½. For such a material, the isotherm describing the charge as function of surface potential is
𝛼= 1 2
− 1
1+10pK−pH·𝑒−𝜙D (5.6)
where pH is that in bulk solution (sufficiently far away from the surface) and for an amphoteric material pK is the pH at which the material is uncharged, which is the point of zero charge, PZC. For titania, pK=PZC=4.4.
For a material that obeys Eq. (5.6), the expression for𝐹Sis
𝐹S=1/2𝑅𝑇 𝑁 ln( (1+2𝛼) (1−2𝛼)) (5.7)
iiIn this and the next section, pressures and energies are presented with unit N/m or N/m2; in many other parts of the book the term𝑅𝑇is omitted, see p. 507.
iiiWe neglect the Stern layer in this section, and use𝜙
Dinstead of𝜙
0in Eq. (5.6).
Wetting of titania: The free energy of an EDL with ionizable surface charge 111
Fig. 5.2:The contact angle of the air-water-titania contact line, as function of pH for two values of salt concentration. The GCS model including surface ionization very well reproduces the experimental data. (𝑁=3 nm−2, pK=4.4,𝛾
SG−𝛾unch
SL =56 mN/m).
which like𝐹Dfrom Eq. (5.4) is negative. Thus, EDL formation will decrease𝛾
SLand thus reduce𝜃and thus increase the wetted area on the titania.
For each salt concentration and pH, we can solve Eq. (5.6) together with the Gouy- Chapman equation for an isolated surface, Eq. (3.15), to obtain𝛼and𝜙
D. These values enter the expressions for𝐹Dand𝐹Sand these are combined with the S/L surface energy for the uncharged surface,𝛾unch
SL and jointly used as𝛾SLin Eq. (5.3).
Results of this theory are presented in Fig. 5.2 as function of pH and𝑐∞and they show a very good agreement with data. The model predicts symmetry of the contact angle𝜃around the PZC (at pH=pK=4.4), and the data corroborate this prediction. The much broader plateau in contact angle at low salinity (1 mM) compared to the situation at high salinity (100 mM), as experimentally found, is also very well reproduced in the theory.iv
ivA very different approach to Young’s equation is based on minimization of the total energy in the system, which is a summation of the surface energies of the three areas, times the respective area. This energy is minimized with the constraints that the volume of bubble or droplet is constant, with the shape thereof a (truncated) sphere, and that the solid surface is either wet or dry. This entire minimization (also possible including gravity, resulting in the shape no longer being spherical), leads exactly to Young’s equation, without having to explicitly consider forces. This analysis also shows that the L/G interface ideally forms a hemisphere, with𝜃ideally at 90o, because at this point the L/G area is at a minimum. Thus, the L/G interface always pulls to bring𝜃closer to 90o. If the solid material is a partially wetted electrode (see next section), this minimization must be done for a fixed total charge. Then we do not end up with Young’s law based on surface energies,𝛾.
112 The energy of an electrical double layer