The Gouy-Chapman-Stern model

The EDL for a planar surface: The Gouy-Chapman-Stern model

3.2 The Gouy-Chapman-Stern model

To describe the distribution of ions in the diffuse layer (DL), the starting point is Maxwell’s first law, also called Gauss’s law in differential form, or the Poisson equation

∇ · (𝜀E)=𝜌 (3.1)

where 𝜀 is the dielectric permittivity of the electrolyte, given by 𝜀 = 𝜀

r𝜀

0, where for water the dielectric constant is𝜀

r ∼78, while𝜀

0 is the permittivity of vacuum, given by 𝜀0 = 8.854·10⁻¹² C/(V·m). In Eq. (3.1),E is the field strength, which is minus the gradient of the electric potential, E = −∇𝑉, and 𝜌 is the local charge density in C/m³. Assuming𝜀to be a constant, we arrive atⁱ

𝜕²𝑉

𝜕 𝑥2 =−𝜌

𝜀 (3.2)

where we also assumed a one-dimensional planar (Cartesian) geometry, with a single coordinate axis, 𝑥. Next we replace the dimensional voltage𝑉 with the nondimensional electrical potential 𝜙, according to 𝜙 =𝑉/𝑉

T, where𝑉

T = 𝑅𝑇/𝐹 is the thermal voltage, which at room temperature is𝑉

T∼25.6 mV, and we arrive at

𝜕²𝜙

𝜕 𝑥2 =− 𝐹 𝜀 𝑅𝑇

𝜌 . (3.3)

The ionic charge density 𝜌 has contributions from all free ions in the electrolyte, 𝜌 = 𝐹Í

𝑖(𝑧_𝑖𝑐_𝑖), and thus Eq. (3.3) becomes

𝜕²𝜙

𝜕 𝑥2 =− 𝐹² 𝜀 𝑅𝑇

∑︁

𝑖

𝑧_𝑖𝑐_𝑖. (3.4)

Up to this point the equation we considered is the Poisson equation, valid irrespective of whether (locally) the system is at equilibrium or not. When equilibrium can be assumed for the EDL, and for ions as ideal point charges, the Boltzmann equation, Eq. (2.1), applies, which can be written as

𝑐_𝑖 =𝑐_∞,𝑖𝑒^{−𝑧𝑖𝜙} (3.5)

iIt is a major assumption to set 𝜀as a constant, independent of composition and frequencies of changes in the electric field. Many fundamental physical theories, for instance to describe the Van der Waals energy of the interaction between polarizable materials across another medium, consider the entire spectrum of the 𝜀-dependence on the frequency of changes in the electrical field due to dipole effects. This is beyond the scope of this book.

The Gouy-Chapman-Stern model 59 where𝜙is the potential at some position𝑥in the DL relative to a position just outside the EDL in a charge neutral bulk phase. This outside position is often identified with an index

∞. We describe here the ion distribution within the same phase, water in most cases, and thus a non-electrostatic contribution to the chemical potential of an ion, which would be described by a partition coefficientΦ𝑖, can be left out in Eq. (3.5). Combining Eqs. (3.4) and (3.5), we obtain a general form of the Poisson-Boltzmann (PB) equation

𝜕²𝜙

𝜕 𝑥2 =− 𝐹² 𝜀 𝑅𝑇

∑︁

𝑖

𝑧_𝑖𝑐_∞,𝑖𝑒^{−𝑧𝑖𝜙}. (3.6)

When we have a symmetric salt solution (1:1, 2:2, etc.) with |z| the magnitude of the valency of the ions in the salt pair, i.e.,|𝑧|={1,2,etc.}, then Eq. (3.6) becomes

𝜕²|𝑧|𝜙

𝜕 𝑥2 =𝜅²sinh|𝑧|𝜙 (3.7)

where𝜅, which is the inverse of the Debye length, is given by Eq. (3.19) that is discussed below. For a 1:1 solution (all ions monovalent),|𝑧|=1, and thus Eq. (3.7) becomes

𝜕²𝜙

𝜕 𝑥2 =𝜅²sinh𝜙 . (3.8)

The charge-voltage relationship in the GCS model can be derived from Eq. (3.6), by multiplying both sides with𝜕 𝜙/𝜕 𝑥and integrating, to

1 2

𝜕

𝜕 𝑥 𝜕 𝜙

𝜕 𝑥 2

=+ 𝐹² 𝜀 𝑅𝑇

∑︁

𝑖

𝑐_∞,𝑖

𝜕

𝜕 𝑥

𝑒^{−𝑧𝑖𝜙}. (3.9)

Now, with the potential𝜙set to zero far away, where𝑥=∞, and with the gradient𝜕 𝜙/𝜕 𝑥 zero there as well, we arrive at

𝜕 𝜙

𝜕 𝑥 2

=+2𝐹² 𝜀 𝑅𝑇

∑︁

𝑖

𝑐_∞,𝑖 1−𝑒^{−𝑧𝑖𝜙}

. (3.10)

This equation is valid at each point in a DL. At the Stern plane, which is boundary of the DL, denoted with label ‘D’, we have an additional condition, which is Gauss’s law

Σ =𝜀 𝐸 (3.11)

whereΣ is the surface charge density (unit C/m²), and where we assume we only have to consider the electric field in𝑥-direction. Withxpointing from the surface into solution, then

60 The EDL for a planar surface: The Gouy-Chapman-Stern model 𝐸 =−𝑉

T·𝜕 𝜙/𝜕 𝑥. Gauss’s law, Eq. (3.11), does not depend on the structure of the diffuse layer, such as ion concentration or valencies of ions. Eq. (3.11) can be rewritten to

Σ =−𝜀 𝑅𝑇 𝐹

𝜕 𝜙

𝜕 𝑥 D

. (3.12)

We combine Eqs. (3.10) and (3.12) and obtain Σ =sgn(𝜙

D)

√︄

2𝜀 𝑅𝑇

∑︁

𝑖

𝑐_∞,𝑖(𝑒^{−𝑧𝑖𝜙}D−1) (3.13) where𝜙

Dis the diffuse layer potential, i.e., the potential at the Stern plane, relative to the potential in bulk solution,𝜙_∞=0. The Stern plane is the position nearest to the surface up to which we still consider the PB-equation to hold.ⁱⁱ Eq. (3.13) is a generalized Gouy-Chapman equation, valid for mixtures of salts, within the PB framework. The sign ofΣis the same as that of𝜙

D. When we have a symmetric salt solution (all ions monovalent, or all ions divalent, etc.), then Eq. (3.13) simplifies to

Σ =sgn(𝜙

D)√︁

4𝜀 𝑅𝑇 𝑐_∞√︁

cosh(|𝑧|𝜙

D) −1=√︁

8𝜀 𝑅𝑇 𝑐_∞ sinh(|𝑧|𝜙

D/2) (3.14) and when we only have monovalent ions, i.e., a 1:1 solution, then Eq. (3.14) simplifies to

Σ =√︁

8𝜀 𝑅𝑇 𝑐_∞ sinh(𝜙_D/2) . (3.15) This is the classical Gouy-Chapman (GC) equation for a 1:1 solution, which can be inverted to

𝜙D=2·sinh⁻¹ Σ

√

8𝜀 𝑅𝑇 𝑐_∞

. (3.16)

We can also express Eq. (3.15) as (here presented in way valid only for 1:1 salt solution) Σ =4𝐹 𝑐_∞𝜆

Dsinh(𝜙

D/2)=2𝜀𝑉

T𝜅sinh(𝜙

D/2) (3.17)

where𝜆

Dis the Debye length and𝜅is the inverse of𝜆

D. The general definition of𝜅is 𝜅=𝜆⁻¹

D =

√︂

2𝐹2𝐼

𝜀 𝑅𝑇 (3.18)

whereIis the ionic strength, which is given by𝐼 =½Í

𝑖𝑧²

𝑖𝑐_∞,𝑖, in which we sum over all ions. For a symmetric salt solution this expression simplifies to

𝜅=𝜆⁻¹

D =

√︂

2|𝑧|²𝐹2𝑐_∞ 𝜀 𝑅𝑇

. (3.19)

iiPoisson’s equation still holds beyond the Stern plane, but we assume there are no more (centers of) ions.

The Gouy-Chapman-Stern model 61 For a 1:1 solution, a linearised version of the GC equation for low𝜙

Dis Σ =√︁

2𝜀 𝑅𝑇 𝑐_∞𝜙_D=2𝐹 𝑐_∞𝜆_D𝜙_D=𝜀𝑉_T𝜅 𝜙_D. (3.20)

Gouy-Chapman equation based on 1:1 salt solution. When we assume from the start that we have a 1:1 salt solution, then Eq. (3.8) applies where we can again multiply each side with𝜕 𝜙/𝜕 𝑥, and integrate by parts, resulting in

1 2

𝜕

𝜕 𝑥 𝜕 𝜙

𝜕 𝑥 2

=𝜅²

𝜕

𝜕 𝑥cosh𝜙 . (3.21)

With the same boundary conditions that potential𝜙=0 and𝜕 𝜙/𝜕 𝑥=0 when𝑥→ ∞, the relation at the Stern plane (position ‘D’) between potential and potential gradient becomes

𝜕 𝜙

𝜕 𝑥 D

2

=2𝜅²(cosh𝜙

D−1) (3.22)

in which we implement Eq. (3.12), resulting in Σ =sgn(𝜙

D) 𝜀 𝑅𝑇 𝐹

√︃

2𝜅2(cosh𝜙

D−1)=sgn(𝜙

D)√︁

8𝑐_∞𝜀 𝑅𝑇sinh𝜙

D/2 (3.23) which for|𝑧|=1 is equal to Eq. (3.14).

Gauss’s law. In the above derivation we made use of Gauss’s law, Eq. (3.11). This law is an integral version of the Poisson equation, Eq. (3.1), but in Eq. (3.11) some assumptions are made in the integration about the electric field outside the EDL on either side (namely that they are zero). No such assumptions are involved in Eq. (3.1).

To explain this, we start with Eq. (3.1) and integrate over a planar layer from one 𝑥-position to another (from position 1 which is the Stern plane, to a position 2 which is outside the DL in the electrolyte; thus we assume a Cartesian, i.e., planar, geometry, with only one one coordinate,𝑥; the system is invariant in directions that are at right angles tox), resulting in

(𝜀 𝐸)₂− (𝜀 𝐸)₁=

∫ ₂

1

𝜌d𝑥 . (3.24)

The right side is equal to the diffuse layer charge,Σ_DL, when we identify position ‘1’

with the Stern plane and position ‘2’ with the solution outside the DL (‘∞’). (When

62 The EDL for a planar surface: The Gouy-Chapman-Stern model

two equal diffuse layers overlap, in a problem of colloidal interaction, then we have 𝐸=0 at a midplane located halfway between the two particles.) If outside the DL we can assume that the field strength is zero, the first term on the left is zero, and we obtain Eq. (3.11) when we realize that the surface chargeΣis equal to minus the diffuse layer chargeΣ_DLbecause the EDL as a whole is electroneutral.

We can also integrate Eq. (3.1) from a position 0^∗(which is just left of the surface chargeΣwhich is located in the 0-plane) and then across the Stern layer to position 1, and then we obtain

(𝜀 𝐸)₁− (𝜀 𝐸)₀∗ =

∫ ₁

0^∗

𝜌d𝑥 . (3.25)

Now on the right is the surface chargeΣ, because there is no charge in the Stern layer.

On the left side we can implement that the field strength at position 0^∗is zero. Thus we again arrive at Eq. (3.11).

Under which conditions is the field strength zero at position 0^∗, i.e., just left of the 0- plane? This is the case when for symmetry reasons the field strength inside the charged material (e.g., a colloidal particle or polymer network) is zero (for instance because ‘on its other end’ there is the same EDL structure). A zero field strength at position 0^∗is also the case for a metallic phase, because in a metal the charge will reside at the surface (such as at plane 0) and thus just inside a metal the field strength is zero.

There are also situations that the surface charge is not concentrated in plane 0, but it is volumetrically distributed to the left of plane 0, possibly ‘mixed’ with fixed charges in the network (e.g., polymeric charges, or the p- and n-dopants in a semi-conductor).

This can be modelled with an extra ‘layer of diffuse and fixed charge’ to the left of 𝑥=0. Exactly at𝑥=0 we now have continuity in potential and chemical potential (like before), and we have Gauss’s law applied to this surface, now without charge in this very 0-plane, which becomes (based on an integration of Eq. (3.1) across this plane)

(𝜀 𝐸)just right ofx=0− (𝜀 𝐸)just left ofx=0=0. (3.26) Based on this understanding we can now model various problems in this book where we have ‘twin’ diffuse layers where one layer is inside a charged polymer network (such as inside an ion-exchange membrane or gel), another in a semi-conductor or other ion-conducting solid state material, or the DL is in a liquid electrolyte such as water.

We continue with the Gouy-Chapman equation, and unless otherwise noted, we consider

The Gouy-Chapman-Stern model 63 a 1:1 solution. A very elegant way to write the GC equation is

𝑄 2

=sinh 𝜙_D

2 (3.27)

where𝑄is a dimensionless charge density 𝑄= Σ𝜆

D

𝜀𝑉T

. (3.28)

The diffuse layer potential𝜙_Dis the potential at the Stern plane. The Sternplaneis the plane that separates the diffuse layer from the Sternlayer. The Stern layer we discuss later.

In the standard GCS model the Stern plane is uncharged. It is solely the plane that borders the region in which (the centers of) the ions in the DL can reside. Thus, the Stern plane can best be considered a plane of ‘closest approach’ of (the centers of) ions to the surface. They cannot approach the hard surface (the 0-plane) any closer. Thus, all ions are volumetrically distributed in the DL, all the way up to the Stern plane, but without any specific adsorption in the Stern plane or Stern layer. In a graphical representation of the GCS model, it would be erroneous to draw even a few, let alone many, counterions as if adsorbed in the Stern plane. In the DL the charge is volumetrically distributed (charge density has unit C/m³), but this does not imply that charge resides in the Stern plane. Nevertheless, we can speak of an (integrated) DL charge, where the volumetric charge over the DL is integrated from the Stern plane out to infinity, to obtain a DL charge with unit C/m². The Stern layer is a layer between the Stern plane and the ‘hard’ surface (which is the 0-plane). The Stern layer is a dielectric layer not containing any charges.

Eqs. (3.10) and (3.15) provide relationships between potential 𝜙 and gradient 𝜕 𝜙/𝜕 𝑥, which at the Stern plane leads to a relation between surface charge Σ and potential 𝜙_D, which is the (generalized) GC-equation. However, this is not yet an explicit solution for the profile𝜙(𝑥). This can be obtained by integrating Eq. (3.10) once again, which results in

𝜙(𝑥)=4·tanh⁻¹{𝑒^{−𝜅 𝑥}·tanh(𝜙

D/4)} (3.29)

with the coordinate𝑥starting at zero at the Stern plane and pointing into solution. For values of𝜙

Dbelow unity, Eq. (3.29) can be approximated to 𝜙(𝑥)=𝜙

D·𝑒^{−𝜅 𝑥} (3.30)

which can also be derived when we combine Eqs. (3.12) and (3.20) (and implement𝜙

D→𝜙) which results in

𝜆D

𝜕 𝜙

𝜕 𝑥

=−𝜙 (3.31)

64 The EDL for a planar surface: The Gouy-Chapman-Stern model which can be rewritten to

∫ ^𝜙

𝜙D

1

𝜙d𝜙=−𝜅

∫ ^𝑥

0

d𝑥 (3.32)

which after integration leads to Eq. (3.30). Even though Eq. (3.30) is not exact at higher potentials, it illustrates well the exponential-like decay of electrical potential with increasing distancexfrom the surface. The charge-voltage relationship compatible with Eq. (3.30) is Eq. (3.20).

An alternative version of Eq. (3.29) is exp(½𝜙(𝑥)) +1 exp(½𝜙(𝑥)) −1

· exp(½𝜙

D) −1 exp(½𝜙

D) +1

=exp(𝜅 𝑥) (3.33)

which is easy to solve in commercial spreadsheet software as we explain next. For a certain value of 𝜙

D we make a list of values for𝜙 between 0 and 𝜙

D and we use Eq. (3.33) to calculate the corresponding value of𝑥, and then we plot𝜙(𝑥) vs. 𝑥. We can implement this𝑥-dependent potential,𝜙(𝑥), in the Boltzmann relation, Eq. (3.5), and directly calculate the profiles in concentration of cations and anions versus𝑥-position, which we can again do in spreadsheet software. This calculation will show that the concentration of counterions goes up towards the surface, to reach a maximum value at the surface (i.e., at the Stern plane), while the concentration of coions drops the closer one approaches the surface, to values that at high voltage are almost zero. [Let it be reiterated that the high concentration of counterions near the Stern plane (expressed as a volumetric concentration in mol/m³) does not imply there are ions adsorbedinthe Stern plane (which would have the unit mol/m²). In the GCS theory, there are no ions adsorbedinthe Stern plane.]

In the limit of low𝜙

D, we can combine the Boltzmann equation with Eq. (3.30) and obtain for the profile of ion concentration

𝑐_𝑖(𝑥)=𝑐_∞,𝑖·exp(−𝑧_𝑖𝜙_D·exp(−𝜅𝑥)). (3.34) Just as Eq. (3.30), Eq. (3.34) is not exact, but illustrates counterion and coion concentration profiles. It is also in agreement with the fact that when the Boltzmann equation applies, for a 1:1 salt, we have at each position in the EDL the relationship𝑐₊𝑐₋ =𝑐²_∞.

In Fig. 3.2 we compare concentration profiles according to the exact GC-equation with Eq. (3.34), for the same surface charge ofΣ =20 mC/m²and for𝑐_∞ =10 mM. For the GC model, this surface charge recalculates to a diffuse layer potential,𝜙

D, of 67 mV, about 2.5× the thermal voltage. However, in the linearised model, the same surface charge leads to a potential of𝜙

Dof 88 mV, about 30% more. The concentration at the Stern plane changes dramatically even though the change in𝜙

Dis only 30%: namely it changes from 135 mM to

The Gouy-Chapman-Stern model 65

0 20 40 60 80 100

0 1 2 3 4 5 6 7 8

ci(mM)

x (nm) c

c_+,lin c_+,PB

c_-,lin c_-,PB

Fig. 3.2:Ion concentration profiles next to a planar charged surface according to the Gouy-Chapman equation (denoted by ‘PB’), and according to a linearised solution, Eq. (3.34). (Surface charge density Σ =20 mC/m². Salt concentration𝑐_∞=10 mM.)

305 mM, a 2.3×increase. Thus in the linearised model, even though at the surface𝜕 𝜙/𝜕 𝑥 is the same as for the non-linearised GC-equation, the counterion concentration starts off at a 2.3×larger value. For the coions, the two models give similar results in the presentation of Fig. 3.2, but also here, the relative difference in concentration (between the two models, for a certain position) going up to a factor of 2.3, with the prediction by the linearised model now closer to zero than for the non-linearised model. The total excess ions adsorption integrated over the DL is overestimated in the linearised model by about 65%. These are significant deviations, and thus, when using any linearisation of the PB-equation, be careful that potentials do not exceed a value of∼30 mV to have sufficient accuracy.

—

Now that we know the profile of𝜙(𝑥), we can calculate the excess adsorption of cations and anions in the diffuse layer. Here, ‘excess’ means the ion adsorption beyond the situation that the surface were uncharged and the concentration at each position in the DL would be equal to the bulk concentration𝑐_∞. First we calculate the excess adsorption of cations and anions combined in the DL (note, we only consider a 1:1 salt)

Γ_ions=𝑐_∞

∫ ∞

0

∑︁

𝑖

𝑒^{−𝑧𝑖𝜙(𝑥)}−1

d𝑥=2𝑐_∞

∫ ∞

0

(cosh𝜙(𝑥) −1)d𝑥 (3.35)

66 The EDL for a planar surface: The Gouy-Chapman-Stern model

where the summation is over anions and cations, andΓhas unit mol/m². The solution to Eq. (3.35) is (Bazantet al., Phys. Rev. E, 2004)

Γ_ions =8𝜆

D𝑐_∞sinh²(𝜙

D/4) . (3.36)

Each individual ion’s excess adsorption (negative for the coion), can be calculated from the above equations, Eqs. (3.15) and (3.36), by making use of

Γ_ions=∑︁

𝑖

Γ𝑖 & Σ =−𝐹

∑︁

𝑖

𝑧_𝑖Γ𝑖 (3.37)

where as before the summation runs over the cations and anions. This evaluation results for both ions in the excess adsorption

Γ_i=2𝜆

D𝑐_∞

𝑒^{−𝑧𝑖𝜙D/2}−1

. (3.38)

Just as in Ch. 2, see Eq. (2.12), we can take the ratio of excess salt adsorption over charge, to calculate the charge efficiency,Λ, of an electrode,

Λ = Γ_ions

|Σ|/𝐹

=tanh

|𝜙

D|

4 (3.39)

which, like in Ch. 2 for the Donnan model, increases from zero to unity when the surface charge, thus𝜙

D, increases. This implies that in the GCS-model (just as in the Donnan model of Ch. 2), the surface charge (when expressed in moles/area) is always larger than the excess number of ions in the diffuse layer.

Does an EDL absorb or desorb salt? Now that we have expressions for the excess adsorption of counterions and coions, we can consider a related, and very interesting, question: does formation of EDLs lead to adsorption or desorption of salt? ¹

When we charge an electrode, we know that we adsorb more counterions than we desorb coions and thus one could say an electrode adsorbs salt. However, we cannot consider just one electrode, we also have a second electrode, from where the charge came. So in this analysis we consider a system without transport of charge.

We analyze how a surface charges up spontaneously when it is brought in contact with water. We consider two scenarios. In the first scenario, ions that were already adsorbed to the material in the dry state, are now released when the material is brought in contact with water (‘is wetted’). In the second scenario, the material is already wetted and one of the ions in the water ‘specifically’ adsorbs to the surface.

The Gouy-Chapman-Stern model 67

So for the formation of an EDL when a dry material is wetted, the first scenario is similar to how we will describe the charging of titania in Ch. 5. In this scenario, we start with a neutral surface, neutralized because of counterion adsorption. For instance the material has COO^–-groups on its surface and they are neutralized by Na⁺-ions adsorbed to these groups.ⁱⁱⁱ This is how many materials can be envisioned when dry.

When the material is wetted, the Na⁺-ions release from the surface, and the surface becomes negatively charged, i.e. the EDL is formed. What happens now with the bulk salt concentration? So these Na⁺-ions desorb, and a diffuse layer builds up. Do part of the desorbed Na⁺-ions go there and another part ‘moves on’ to the bulk? This would mean the bulk salt concentration increases. Or, do they go to the DL, to be joined there by extra counterions coming from solution? Then we have desalination of the bulk water. The answer is easily found by evaluating not these Na⁺-ions, but by considering movement of the co-ions: before Na⁺-desorption, all water volume was available to them as a bulk phase, and there was no diffuse layer. Now, the diffuse layer forms, and the coions are expelled from the DL (they have a negative excess adsorption). Thus their bulk concentration goes up. Because of charge neutrality in the bulk, this will also be the case for the counterions. So the answer is that the first hypothesis is correct:

formation of the EDL (by the chemical mechanism that we described above) expels salt from the EDL and the bulk salt concentration goes up.

The second scenario is that a neutral surface is brought in water and then charges up because one type of ion adsorbs ‘specifically’ to the surface, and in this way an EDL forms. Note that the adsorbing ion will become the coion. [In the first scenario, just discussed, the desorbing ion became the counterion.] Because, for instance, if the adsorbing ion is a cation, then the surface becomes positively charged, and anions become counterions. Thus the cation becomes the coion, and in the final equilibrium situation its concentration in the diffuse layer will be lower than in bulk. So does the bulk water become more concentrated, like we concluded would happen in the first scenario? To find the answer, it is best to consider the non-adsorbing ion. This is the counterion (in this example an anion). So this counterion (anion) is adsorbed in the diffuse part of the EDL. And thus in bulk its concentration goes down. And the same will happen for the cation, and thus also for the salt as a whole. Thus in this second scenario, EDL formation leads to desalination of the bulk water.

Thus we have two different outcomes based on two scenarios, whether (scenario 1) charging of a surface is because of ion release, or (scenario 2) because of ion adsorption.