D2Q=/(DFc)
3.7 Surface ionization in the GCS model
Surfaces generally do not have a fixed surface charge, described by Σ in the previous sections, but adsorption and desorption of ions adds to, or modifies, the charge, as was already commented on in §1.4 and in §2.2–2.5. For capacitive electrodes, this chemical charge is an extra source of charge besides electronic and ionic charge, see §2.4 and §2.5.
Adsorption of ions can also be a precursor to an electrode reaction in a (capacitive or Faradaic) electrode process.
In the present section we focus on ion adsorption to interfaces other than electrodes, for instance to oxidic materials such as alumina and silica. The descriptions can also be applied to more ‘soft’ interfaces such as polyelectrolytes (charged polymers), either in the form of a network (gel, membrane, layer of adsorbed (bio-)polymers), or as more discrete entities such as globular protein molecules or other charged nanoparticles. Though the theories below refer to an ideal surface with only one type of composition, large parts of the texts are also valid when the surface has distinct patches of different surface composition and/or geometry (for rough or wavy surfaces, cylindrical or spherical surfaces, surfaces with protuberances, etc.).
When a surface has patches, a very interesting problem deals with the diffuse layers corresponding to each patch, and the question is to what extent they ‘mix’, i.e., the question is, does each patch have ‘its own’ diffuse layer, or is there one ‘average’ diffuse layer, the same for both patches. We know this problem exists for the micropores in carbon electrodes (see Ch. 15), in the interior of ion-exchange membranes 5 and for materials such as clays which on their edges have a different chemistry than on the faces of the clay platelets. Many types of bionanoparticles (viruses) have a very heterogeneous surface structure both in terms of charge and shape.
The extent of ‘diffuse layer mixing’ depends on how large are the patches relative to the Debye length: for large patches relative to the Debye length, a treatment with different diffuse layers for each patch is preferable, while for smaller patches relative to the Debye length (low salt concentration), the two diffuse layers mix.
80 The EDL for a planar surface: The Gouy-Chapman-Stern model
One can also think of setting up a numerical theory with two separate diffuse layers –for instance based on the Poisson-Boltzmann equation for a symmetric salt solution, Eq. (3.7)– and at each𝑥-position from the surface we mix the two concentrations in each layer, and this mixing increases with distance from the surface, so the diffuse layers that were fully distinct at the very surface, become more and more intertwined the further we move away from the surface. In this way we solve two adjacent 1D models, instead of having to set up a full 2D calculation.
Let us discuss now a flat surface with a completely homogeneous composition. Questions are: do we need to include a Stern layer (or further refinements to the Stern layer concept), and where (in which theoretical plane) do ions adsorb. On the topic of the Stern layer –to which we return below– it turns out that we do not need to include a Stern layer at all in many calculations with organic materials (gels, membranes, protein molecules and other polyelectrolytes, either in a ‘brush’ or as an adsorbed layer). Or in any case, it does not seem to be necessary. There is just ‘the’ surface to which protons/hydroxyl ions adsorb (and possibly other ions) and we can assume this plane to be ‘directly next to’ the diffuse layer without a Stern layer in between (which would have a voltage drop across it). For ion adsorption in the intercalation materials discussed in Ch. 1 a Stern layer was also not necessary. On the other hand, in carbon electrode micropores, the Stern layer as a mathematical concept is of key importance to describe data of salt adsorption and charge.vii Also for hard surfaces such as oxides, the Stern layer concept is considered of essential relevance, and often further refined, as discussed below.
We next focus on models that include proton/hydroxyl ion adsorption in the 0-plane, see Fig. 3.1, and a further extension can be the adsorption of ‘indifferent ions’ (in most contexts this refers to all ions except for H+ and OH–) in the Stern plane, i.e., the plane located at the interface of Stern layer and diffuse layer, see Fig. 3.1. More refined models are the
‘triple layer model’ which considers that indifferent ions adsorb to a plane ‘inside’ the Stern layer, and in this way an ‘inner Helmholtz plane’ (iHp) is defined where these ions adsorb, in addition to the ‘outer Helmholtz plane’ (oHp), where the diffuse layer starts. This oHp corresponds to the Stern plane that we described before. In the triple layer model there are two capacitances, one between 0-plane and iHp, and one between iHp and oHp.
By now, one wonders perhaps, how do these various planes and their location matter. They
viiIn theoretical modeling of the EDLs in carbon micropores, this Stern layer is assumed to be located ‘behind’ the chemical charge. This suggests that in carbon micropores the Stern layer may relate to the space charge region within the carbon material itself, related to the distribution of electrons there.
Surface ionization in the GCS model 81 matter because in the adsorption isotherm for that specific plane there is an electrostatic term (related to the adsorbing ion), and because the capacitances lead to an increase in potential towards the surface, and this higher potential suppresses adsorption.viii Even more detailed models describe the adsorption of large ions, for instance phosphate, PO43 –, with part of the ion is assumed to adsorb in one plane, and part in another plane.
In the present section, we consider the simplest situation, of adsorption of ‘potential- determining ions’ (often H+ and OH–) in the 0-plane, and ‘indifferent ions’ in the Stern plane, with a single Stern capacitance in between these planes.
Two remarks are now in place relating to the ions. First of all, in some cases the ions adsorbing in the 0-plane, the ‘potential-determining ions’, are not H+and OH–, but are other ions, often cations. This is the case for certain complex inorganic materials such as clays and mica. Nevertheless, from this point onward we assume the ions adsorbing in the 0-plane are H+and OH–. The second point is discussed in the next box.
The relationship between H+- and OH–-ions. It is important to discuss the nature of the H+- and OH–-ions. One might have the idea that they are (to be considered as) two separate species. However, a repeating theme in this book is that it is generally not very helpful to consider these ions in this way, as separate entities.
Instead, because of water self-dissociation, a reaction where two water molecu- les react to H3O+ and OH–, these two ions are intimately connected: knowing the concentration of the one ion, means knowing that of the other. The following example illustrates the relatedness of these ions. The example is, how can one distinguish the situation of an H+-ion moving from bulk and adsorbing to a surface, from the situation that a water molecule adsorbs which consequently releases a OH–-ion which then moves to bulk. We cannot distinguish these two situations: we can set up a theory that focuses on the adsorption/desorption of H+, or we can do the same for the OH–-ion, and these two analyses will give the same results (if correctly done). Reiterating this point: it does not matter whether we consider H+-adsorption, or OH–-desorption. This relates to the assumption underlying this book, that the water is to a large extent an ‘invisible’
solvent, omnipresent in the background. And the equivalence of H+and OH– relates to the assumption of fast enough water self-dissociation. For an equilibrium EDL theory this assumption is valid, and the relation𝐾
w = [H+] [OH−] will be valid. There are many advantages to this approach, and that is why we also use it in Chs. 10, 17, and 18.
viiiIn a more complicated scheme, charge overcompensation leads to the potential not monotonically going up or down which complicates this analysis.
82 The EDL for a planar surface: The Gouy-Chapman-Stern model
One example is: if we have a Faradaic reaction at an electrode with for instance carbonate ions reducing to formate ions, in the ‘H+/OH– are equivalent’-formalism that we advocate, we can use the same model for the Faradaic reaction at all pH values, and we do not need to consider a reaction based on H+at low pH and another formalism using OH– at high pH (and some combination of both at intermediate pH). Instead one and the same model works at all pH-values, see Chs. 17 and 18. Interestingly, in such a model electrode reactions of H+or OH– do not have to be explicitly formulated, but nevertheless after the calculation has completed, one can back-calculate all fluxes of all ions at all position, also of H+and OH–.
Important as well is that in this book we often write H+ as as shorthand for the hydronium ion, H3O+. It is not necessary to explicitly consider that the molecule is actually H3O+ which is formed jointly with an OH–-ion from two water molecules.
Instead, just like how the water self-dissociation reaction is written, as a reaction of a water molecule to a OH–-ion and H+-ion, we can use the H+-ion as shorthand for the hydronium ion. Thus there is freedom in switching between the notation/concept of a H+- and H3O+-ion, and this is only possible because the water is modelled as a continuum fluid around ions. In a more statistical-mechanical framework, this freedom may not exist in this way.
So for adsorption to a surface, we can choose between focusing on H+ or on OH–. Throughout this book, and this holds for most literature in general, the choice is to focus on the H+-ion, and we do so for acidic and basic materials alike. For acidic materials (e.g., a carboxylic acid group−COO– adsorbing/desorbing a proton (protonation/deprotonation) this is rather straightforward. For basic groups this may seem less obvious because they are neutralized because of OH– adsorption, as for the case where a basic group such as an amine,−NH2+, is neutralized because of OH– adsorption. However, as long as we can assume that water self-dissociation is a fast enough reversible reaction, mathematically these two approaches are completely equivalent.
With these elements addressed, what we will arrive at is a situation where we come to one general ‘Langmuir’ isotherm valid for acidic, amphoteric, and basic surface groups, and they are only distinguished based on a constant reference charge number, which is -1,½, and 0, for these three cases.
Surface ionization in the GCS model 83
Amphoteric materials. It is very interesting to discuss the word ‘amphoteric’, which is broadly used and has various meanings. For a discussion on the word ‘amphoteric’
in the context of ionic solutions, see p. 511 For surfaces, it means that the material can charge both positively and negatively. At least three levels of detail can be distinguished.
On the highest level, the term amphoteric material can be applied to a material with distinct regions (each consisting of 10s-1000s of atoms) differing in surface chemistry and geometric structure (as for a coronavirus or a clay particle).
One level down is a material where each surface group can be different, such as a protein molecule. The surface of a globular protein molecule generally consist of several types of acidic and basic groups, each with their own pK-value. An acidic group can be deprotonated (negatively charged) or neutral. A basic group can be neutral or positively charged. So none of these groups is amphoteric, but the material as a whole is, because the total, effective, average, surface charge can go from negative to positive as function of pH, and this average charge follows from a summation over all the groups at its surface.
The smallest scale at which we can define the term amphoteric, is when one and the same surface group (atom) can go from negative to positive. This is the case for amphoteric oxidic materials such as titania and alumina. There is one type of surface group with one pK-value and this group can go from negative at high pH to positive at lower pH. The fractional charge (charge per group) goes from a minimum of -½to a maximum of +½. (Please seeherefor more discussion on the amphoteric behaviour of titania.)
In the present section when discussing an amphoteric material, we refer to this last case, which applies to oxidic material such as alumina and titania. The ionization of protein molecules is discussed in the next box.
We continue with describing these ionizable materials that adsorb and desorb protonic charge, i.e., a material that is charged by (de-)protonation. As discussed, these ions adsorb at the 0-plane, where potential is𝜙
0(relative to the potential outside the EDL). This potential depends on the diffuse layer potential, 𝜙
D, the Stern capacitance, and the charge in the 0-plane,Σ,ixwith𝜙
0 =𝜙
D+𝜙
S. As before,𝜙
Dis the potential across the diffuse layer, and 𝜙Sthe potential across the Stern layer (i.e., both are potential differences).
Next we derive the general Langmuir ionization isotherm. The surface has a density of
ixWe do not include electronic charge for now, or ions adsorbing in the Stern plane. Instead, we consider solely surface charge in the 0-plane and diffuse charge extending beyond the Stern plane, see Fig. 3.1.
84 The EDL for a planar surface: The Gouy-Chapman-Stern model sites to which a proton can adsorb,𝑐
s,max, and the Langmuir isotherm considers that a proton either occupies one of these sites, or not. Thus we take the same approach as in §1.3 in that we describe an equilibrium between adsorbed species, and species in solution.x Thus the equilibrium of a proton exchanging between solution (bulk) and surface is
𝜇aff,∞+ln 𝑐∞,
H+/𝑐
ref
=𝜇
aff,ads+ln 𝑐
0,H+/𝑐
s,max
−ln 1−𝑣 𝑐
0,H+
+𝜙
0 (3.67) where𝑐0,H+is the surface concentration of protons, adsorbed in the 0-plane (unit mol/m2).
A potential𝜙∞is omitted in Eq. (3.67) because𝜙
0is already defined as the surface potential relative to that outside the EDL. Note that in Eq. (3.67) we assume equilibrium between the protons in the 0-plane and the same protons outside the EDL, in bulk. However, when the diffuse layer is not at equilibrium with a solution outside the EDL, then Eq. (3.67) must be written as a balance between the proton at the 0-plane and that at the D-plane. Then on the left side ‘∞’ must be replaced by D, and on the right side𝜙
0is replaced by the voltage difference across the Stern layer,𝜙
S. Note the equivalence of Eq. (3.67) with Eq. (1.15) in Ch. 1.
Next we define a fractional charge, which is the fraction of how many of the surface sites have an adsorbed H+-ion, for which we can use𝜗=𝑐
0,H+/𝑐
s,maxfrom Ch. 1, and note that 𝑣·𝑐
max=1. This converts Eq. (3.67) into
𝜗= 1
1+𝐾·𝑐−1
∞,H+·𝑒𝜙0
= 1
1+103−pK·10pH−3·𝑒𝜙0
= 1
1+10pH−pK·𝑒𝜙0 (3.68) where like in §1.3, 𝐾 = 𝑐
ref · exp 𝜇
aff,ads−𝜇
aff,∞
, and where we implement pH=3−10log 𝑐∞,
H+
and pK=3−10log(𝐾).xi,xii
Eq. (3.68) introduces𝐾 and pK of a certain ionization equilibrium. These are what are calledintrinsic(p)K-values, i.e., they are constants, dependent only on the chemistry of the ion and the surface. Some studies reportapparentpK-values, which are very different. These apparent pK-values are obtained in an experiment to measure the pH at which a material is charged for 50%.
Eq. (3.68) shows how with increasing pH the coverage of the surface with H+-ions will decrease. The same when the surface potential 𝜙
0 goes up. This makes sense, because increasing pH (which refers to bulk, outside the EDL) means less protons in solution, thus
xWe do not derive the Langmuir isotherm as if it is based on a reaction between ‘empty sites’ reacting with a proton to become an ‘occupied site’.
xiThroughout this book, pH without sub- or superscripts refers to bulk, not to pH at the D- or 0-plane.
xiiIn our book, concentrations and 𝐾-values are in mM (mol/m3) and the conversion here to pH and pK, implementing a factor 3– allows for the traditional definition of pH and pK on a mol/L-basis.
Surface ionization in the GCS model 85 also less adsorption, while a higher surface potential would mean that positive ions, such as H+-ions, are pushed away from the surface.xiii
Eq. (3.68) shows how a higher pK-value leads to more H+-adsorption. This corresponds to how we understand basic materials, e.g., going from pK7 to pK9 leads to a higher charge, in line with Eq. (3.68). However, for acidic materials, Eq. (3.68) does not seem right, because we know that a material with∼pK4 (carboxylic acid-type groups) has a lower charge than acid materials with∼pK2 (sulphate-like) let alone∼pK1 (phosphate). At a given pH, these latter materials are charged more than, say, carboxylic acid. However, as we will see below, there is no problem because for acidic material, ‘higher charge’ actually refers to the charge being more negative.
Let us repeat that Eq. (3.68) refers to pH in bulk, thus outside the EDL, with 𝜙
0 the potential at the surface, defined relative to the potential in bulk.xiv The fractional coverage 𝜗times𝑐
maxequals the surface concentration of adsorbed protons, with dimension mol/m2, and multiplying with Faraday’s number,𝐹, we obtain a charge with unit C/m2.
Now, the surface, the material, has of itself a certain number of surface groups, each of which has a ‘natural’ or ‘native’ charge. The surface concentration of these groups is𝑁. The charge density of these groups by themselves, i.e., their native charge, is𝐹 𝛾 𝑁 (in C/m2), where we introduce the charge sign of the native groups,𝛾. For an acidic material𝛾=−1 (each surface group ‘by itself’ has a𝛾=−1 charge). For a basic surface, these𝑁 groups are all neutral, i.e., in the native state, they have no charge, and thus𝛾=0. Amphoteric materials cover the range between these two extremes. We will in this section only consider amphoteric materials which have a native fractional charge of𝛾=−1/2. For these materials the native charge density of the surface groups is 𝐹 𝛾 𝑁. Typical numbers for𝑁 are for instance𝑁=3 nm−2for titania to𝑁=8 nm−2 for silica (to be divided by𝑁av to go to the unit of mol/m2).
Now, how to arrive at the charge density of the material? It may be clear this will be a summation of the native charge and the charge due to proton adsorption. This is indeed the case, and this applies to all three material types (acidic, basic, amphoteric). The interesting step is to recognize that𝑐max is equal to 𝑁, i.e., the site density identified in Eq. (3.67), to which the protons can adsorb,𝑐
max, is the same as the density of surface groups of the ionizable material,𝑁.
Thus, when we add these two terms together, set𝑐
max=𝑁, and multiply by𝐹, we obtain
Σ = 𝐹· (𝛾 +𝜗) ·𝑁 . (3.69)
xiiiSeeherefor a discussion on the involvement of OH–-ions.
xivSeeherefor two remarks extending the validity and use of Eq. (3.68). These statements also apply to the ionization equilibria given below.