Incorporation of ion volume in the GCS model

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

3.3 Incorporation of ion volume in the GCS model

Ion volume effects are discussed in many chapters of this book, either using a Langmuir/lattice gas approach or based on the Carnahan-Starling equation-of-state. The Stern layer (thickness) is also an approach to incorporate ion volume effects in an EDL model, as it can be interpreted as a layer next to the surface that is inaccessible to ions because ions have a size. To be more precise, in this approach it is a layer which is

Incorporation of ion volume in the GCS model 71 inaccessible to thecentersof the (hydrated) ions, where we envision the charge to reside, and its thickness is then equal to the radius of the (hydrated) ions. With a dielectric constant in the Stern layer set to𝜀

r∼7 (instead of the value in bulk water of∼78), and the thickness of the Stern layer equal to the radius of a typical hydrated ion, say 0.3 nm, then a Stern layer capacitance follows of𝐶

S∼0.2 F/m², a value in line with data for𝐶

Sin an EDL model of oxidic materials in water.

In this section we describe ion volume effects in the diffuse layer (DL) of the GCS EDL model using the Langmuir isotherm, a model with a long history.^iv,vIn this approach, using the Langmuir-description for the excess volume term, the Langmuir-Boltzmann chemical potential of an ion is

𝜇_𝑖 =𝜇

ref,𝑖+ln𝑐_𝑖−ln(1−𝑣 𝑐

tot) +𝑧_𝑖𝜙 (3.44)

where we assume that all ions in the system have the same molecular volume,𝑣, and where 𝑐_totis the total ion concentration (summation over all ions, not including water molecules).^vi In this approach all ions can pack to fill all space up to 100%, when𝑐

tot→1/𝑣.

If we set𝜇_𝑖 =𝜇_{𝑖 ,∞}, and just as in §1.3, we use𝜗for the volume fraction occupied by all solutes (in the context of the Langmuir model), thus𝜗=𝑣 𝑐

totand𝜗_∞ =𝑣 𝑐

tot,∞, Eq. (3.44) can be developed into (𝜙_∞ =0)

𝑐_𝑖 =𝑐_{𝑖 ,∞} 1−𝜗 1−𝜗_∞

𝑒^{−𝑧𝑖𝜙}. (3.45)

If we now have a 1:1 solution we have 𝜗 = 𝑣 (𝑐₊+𝑐₋), while in bulk solution we have 𝑐_{𝑖 ,∞}=𝑐_∞and𝜗_∞=2𝑣 𝑐_∞. We can now solve Eq. (3.45) for both ions, and arrive at

𝜗= 𝜗_∞ cosh𝜙

1−𝜗_∞(1−cosh𝜙) = 𝜗_∞ cosh𝜙

1+2𝜗_∞sinh²(𝜙/2) (3.46) which we can once again combine with Eq. (3.45) and implement in the modified PB equation to obtain

𝜕²𝜙

𝜕 𝑥2 =𝜅² sinh𝜙

1+2𝜗_∞sinh²(𝜙/2) (3.47) which results in the Langmuir-Gouy-Chapman (LGC) equation for the chargeΣas function of diffuse layer potential,𝜙_D, given by

𝑄= Σ𝜆

D

𝜀𝑉_T

=sgn(𝜙

D)

√︂

2 𝜗_∞ ln

1+2𝜗_∞sinh²(𝜙

D/2)

(3.48)

ivFor a full historical overview, see §3.1.2 in ref.²

vFor a brief discussion of other possible contributions to the EDL structure, seehere.

viTo include that ions have different sizes, an extension of the Carnahan-Starling equation-of-state can be used, see for instance §4.2.

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

where we use the same dimensionless surface charge𝑄that we introduced in Eq. (3.27). For 𝜗_∞→0, Eq. (3.48) simplifies to the GC equation, Eq. (3.15). Eq. (3.48) can be rewritten to

𝜙D=sgn(𝑄)cosh⁻¹

𝜗_∞⁻¹(𝛼+𝜗_∞−1)

(3.49) where𝛼=exp 𝜗_∞/2·𝑄²

.

The Bikerman-Freise (BF) capacitance, based on Eq. (3.48), is 𝐶D= 𝜀

𝜆D

sinh𝜙

D

𝛼 𝑄

= 𝜀 𝜆D

(𝛼|𝑄|)⁻¹

√︃

𝜗_∞⁻¹(𝛼+𝜗_∞−1)2−1 (3.50) which for𝜗_∞ →0 simplifies to the GC capacitance, Eq. (3.41). The expansion of the BF capacitance around zero charge is

𝐶D·𝜆

D

𝜀

=1+ 1 8

(1−3𝜗_∞)𝑄²+ O 𝑄⁴

(3.51) where the additional negative term−3𝜗_∞ (which is not in Eq. (3.42)) shows that volume effects reduce the capacitance compared to situation that ions have no volume. Indeed, with ions having a volume, with increasing charge the capacitance will level off and come down again (in the GC model it only increases without limit), see Fig. 3.4.

For high charge, Eq. (3.50) simplifies to 𝐶D= 𝜀

𝜆D𝜗_∞

𝑄⁻¹= 𝜀 𝐹

𝑣Σ (3.52)

which shows how at high charge, and with ions having some volume, the capacitance no longer increases as the GC capacitance predicts, but decreases with charge following a -1 power dependence. Bazantet al.show an inverse square root-dependence of capacitance on diffuse layer potential, which as Eq. (3.55) shows (to be discussed further on; take the last term on the right side) also leads to a -1 dependence on charge. In agreement with Bazant et al., Eq. (3.52) shows no dependence on salt concentration. In Eq. (3.52), parameter𝑣is as before the molar volume of the ions.

We next derive an approximation for the location and maximum of the capacitance curve.

First we discuss the charge𝑄at the maximum. As Fig. 3.4 shows, if we take the intersection of the curves for the GC capacitance, Eq. (3.41), and the high-charge expression, Eq. (3.52), we obtain an excellent prediction for charge𝑄at the maximum. Assuming (correctly) that 𝜗_∞≪1, we then obtain

𝑄|maximum capacitance∼

√︂

2

𝜗_∞ (3.53)

Incorporation of ion volume in the GCS model 73

0 50 100 Q 150 200

80

60

40

20 CD

HIGH CHARGE LIMIT

BIKERMAN-FREISE

Fig. 3.4:The capacitance of a diffuse layer according to Gouy-Chapman, Bikerman-Freise, and the high-charge limit (ion volume𝑣=0.4³nm³,𝑐_∞=10 mM, other conditions apply for water at𝑇

room).

On they-axis,𝐶

Dis scaled to𝜀/𝜆

D|_{1 mM}=72 mF/m².

which indicates that the maximum shifts to lower charge when the ion volume𝑣increases and when salt concentration𝑐_∞ increases (both influence𝜗_∞ = 2𝑣 𝑐_∞), and this is in line with Fig. 6 of Bazantet al. (2009)² Eq. (3.53) very accurately describes the charge𝑄at which𝐶

Dis at a maximum.

To find the value of the capacitance at the maximum, we can again take the intersection of the two branches just discussed and implement Eq. (3.53) to obtain

𝐶D|maximum capacitance∼𝑐

1

√︂

𝜀 𝐹2

2𝑅𝑇

·

√︂

1

𝑣 (3.54)

showing that a 10×smaller ion volume,𝑣, should lead to a∼3×larger capacitance maximum.

This prediction matches exactly Fig. 5b in Bazantet al. (2009)². Furthermore, in their Fig. 6, Bazantet al. show how the maximum in𝐶_Ddoes not depend on salt concentration, and Eq. (3.54) also agrees with that.

Because Eq. (3.54) is based on the intersection point of two limiting curves, see Fig. 3.4, compared to the correct BF capacitance it predicts a too high capacitance. The deviation is a constant factor (as far as we can tell independent of𝑐_∞or𝑣) for which we find a value of 𝑐1∼0.64. This correction factor can be implemented in Eq. (3.54) to provide an excellent prediction of the maximum in the diffuse layer capacitance𝐶

D.

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

In a numerical scheme, it is known that the above equations can be hard to work with at high voltages³ and thus the following approximation to the LGC equation is proposed, which is very close to Eq. (3.49),

|𝜙

D| ∼2 sinh⁻¹(|𝑄|/2) +𝜗_∞/2·𝑄². (3.55) Eq. (3.55) heuristically puts two terms together, first the GC equation valid in case ions have no volume, and second the high-charge branch as discussed above. That this term must be quadratic in charge is based on the following observation in calculations with the FGC equation (also reported by Bazantet al., Fig. 5a) that beyond a certain charge the counterions form a close-packed layer of which the thickness is proportional to the charge, and proportional to the ion volume (the layer must be thicker the larger is the charge, and the larger are the counterions). The gradient in voltage across this layer is also proportional to charge (Gauss’s law), and this together leads to a dependence on 𝑄², and on𝑣, and an inverse dependence on𝜀.

Based on Eq. (3.55) we can also derive for capacitance 𝐶D

−1∼𝜆

D

𝜀

1+1/₄𝑄² −1/2

+𝜗_∞𝑄

(3.56) which has the correct limits at low and high𝑄and also has the maximum located at the correct𝑄-value. However, around this maximum, it underpredicts the capacitance by around 20%. [Note that Taylor expansion of Eq. (3.56) around𝑄=0 does not give the correct behaviour.]

In the next chapter we discuss in more detail the Carnahan-Starling equation for hard sphere mixtures, and extensions thereof for mixtures of spheres of different sizes, even allowing for molecules that are better described as two or more connected spheres. These more detailed models can also be used in EDL models instead of the lattice-based models discussed above. A typical result is that when volumevin the lattice models is equated to𝜐 in the CS-models, that in the latter models volume exclusion effects are much stronger, thus ions are rejected much more from regions of high ion density (near charged walls).

Incorporation of polarization in EDL models 75