Simplified Donnan model

The Donnan model: the EDL in small pores

2.2 Simplified Donnan model

The same as for the extended Frumkin isotherm for intercalation materials, the Donnan model is based on the idea that within a certain small volume, which in this section we call a pore, that is a few nm across at most, with charges on the pore walls, that within this volume ions have a concentration that is different from what it is outside the pore, but within the pore volume, their concentration is independent of position. Related, in this model it is assumed that at all positions in the pore the electrical potential is the same, and likewise other forces and energies are independent of exactly where we are in the pore.

To describe ion concentrations in the pore, relative to outside the pore, we use a general Boltzmann equation, which describes the partitioning of an ion between two phases due to effects such as affinity and volume exclusion, and also includes an electrostatic term which

Simplified Donnan model 37 leads to an attraction or repulsion of the ion into the pore dependent on the charge of the ion and of pore walls (neutral species are not affected)

𝑐_𝑖 =𝑐_∞,𝑖Φ𝑖𝑒^{−𝑧𝑖𝜙}D (2.1)

where𝑧_𝑖 is the valency of the ion (e.g., 𝑧_𝑖=+1 for a monovalent cation), and𝜙

D is the jump in electrostatic potential upon entering the pore (electrical potential in the pore, minus that outside). These dimensionless potentials 𝜙 can always be multiplied by the thermal voltage,𝑉

T=𝑅𝑇/𝐹∼25.6 mV at room temperature, to arrive at a dimensional potential or voltage,𝑉, with unit V. We use the index ‘∞’ to describe (concentrations at) a position outside the pore where we have neutral electrolyte bulk solution.ⁱ We include in Eq. (2.1) a partition coefficient,Φ𝑖, due to contributions to the chemical potential of an ion in a pore other than the Boltzmann, electrostatic, effect. Examples of forces that are included inΦ𝑖

are an affinity (to be in the pore, rather than outside), and another example is the effect of ion volume (excess term). Affinity is a general term encompassing many effects that can lead to a solute-medium interaction energy. The partition coefficient will be discussed in detail in

§2.8, Ch. 4 and Ch. 11. In a multi-ion problem, differences inΦ𝑖between different ions can be the reason why out of several ionic species with the same charge, one adsorbs more than another. We assumeΦ𝑖 =1 until §2.8, and with that assumption arrive at the most common version of the Boltzmann equation, given by

𝑐_𝑖 =𝑐_∞,𝑖𝑒^{−𝑧𝑖𝜙D}. (2.2)

Within the pore we have overall electroneutrality, thus the charge on the walls of the pore, and the charge of ions in the pore volume, add up to zero (the possibility of additional chemical charge will be discussed further on). This can be expressed as

𝐹𝑉pore

∑︁

𝑖

𝑧_𝑖𝑐_∞,𝑖𝑒^{−𝑧𝑖𝜙}D+𝐴

wallΣ_wall=0 (2.3)

where𝐹 is Faraday’s constant (𝐹 = 96485 C/mol),𝑉

pore is pore volume, 𝐴

wallpore wall area, andΣ_wallpore wall charge in C/m². The summation over𝑖 includes all ions in the system.

Next we introduce𝜎_w, which is the charge on the pore walls, defined as a charge density per unit pore volume (thus it has unit mol/m³)

𝜎w= Σ_wall𝐴

wall/𝑉

pore𝐹

. (2.4)

iThis can also be the electroneutral ‘macropore’ electrolyte that fills the larger (transport) pores in a porous electrode, see Ch. 15.

38 The Donnan model: the EDL in small pores The ratio of pore volume over pore area,𝑉

pore/𝐴

wall, is the characteristic pore dimension, ℎp, which depends on pore size and pore geometry: for a planar slit,ℎ

p =½·𝑤, where𝑤 is the width of the slit, while for a cylindrical pore, ℎ

p = 𝑑/4 with 𝑑 the pore diameter.

Combination of Eqs. (2.3) and (2.4) leads to

∑︁

𝑖

𝑧_𝑖𝑐_∞,𝑖𝑒^{−𝑧𝑖𝜙D}+𝜎

w=0. (2.5)

Eq. (2.5) can be simplified when we only have monovalent ions (a 1:1 salt solution, see p. 495). In that case we have a salt solution with cations and anions that all have a charge of 𝑧_𝑖=+1 or𝑧_𝑖=−1, and we arrive at

−2𝑐_∞sinh𝜙

D+𝜎

w=0 (2.6)

where𝑐_∞is the salt concentration outside the pore, which for a 1:1 solution is equal to the total cation concentration (the sum of the concentrations of all monovalent cations), which is also the total anion concentration. Eq. (2.6) can be rewritten to express the Donnan potential as function of𝜎

wand𝑐_∞ⁱⁱ

𝜙_D=sinh⁻¹ 𝜎_w 2𝑐_∞

. (2.7)

At very high wall charge density (relative to the external salt concentration), Eq. (2.7) simplifies to

𝜙D=ln 𝜎w

𝑐_∞

. (2.8)

The wall charge density can have various origins, for instance it can be due to electronic charge in a porous electrode, or it can be the chemical fixed charge of the polymer material of which an ion-exchange membrane is made. In the latter case, the charge of the polymer material is generally denoted by the symbol 𝑋, defined as membrane charge per volume of pores in the material, which for membranes used in reverse osmosis is predicted to be a few mM, while many commercial ion-exchange membranes have charge densities as high as 𝑋∼ ±5 M. The sign of the membrane charge can be denoted by a symbol𝜔, which is 𝜔=+1 or𝜔=−1 for a membrane with positive or negative fixed charges, respectively. A membrane with positive fixed charges provides easy access to anions, and is therefore called an anion-exchange membrane (AEM), while a membrane with𝜔=−1 is a cation-exchange membrane (CEM). Rewriting Eq. (2.8) for such a membrane, thus making the replacement

iiOn the notation of hyperbolic functions sinh, cosh and tanh: index⁻¹implies that the inverse function is used, thus sinh⁻¹(𝑥) = a(rc)sinh(𝑥). However, index²implies the function as a whole is squared. See p. 512 for details on these functions.

Simplified Donnan model 39

𝜎w=𝑋=𝜔|𝑋|, where| |refers to taking a positive quantity, we obtain 𝜙_D=sinh⁻¹

𝜔|𝑋| 2𝑐_∞

. (2.9)

While Eqs. (2.7)–(2.9) describe the relation between Donnan potential and wall charge, another element of an EDL model, i.e., a property that an EDL model must predict, is the concentration of ions in the pores. For each individual ion, this is given by the Boltzmann relation, Eq. (2.2). The total concentration of all anions plus cations in the pore,𝑐

T, assuming a 1:1 solution, is given by

𝑐_T=2𝑐_∞cosh𝜙_D. (2.10)

Making use of cosh

sinh⁻¹(𝑥)

=√

𝑥2+1, Eqs. (2.9) and (2.10) can be combined to arrive at a direct relation between𝑐_∞,𝑋, and𝑐_T, given by

𝑐²

T=4𝑐²_∞+𝑋² (2.11)

independent of the sign of the membrane charge. For the same analysis as in this section but generalized toΦ𝑖≠1, see §2.8.

For applications in ion-exchange membranes, the above equations often suffice.

Modifications are required when the membrane charge is pH-dependent, which we will discuss in §2.4. Other modifications account for electronic charge and are discussed in the next section. Other contributions to the partitioning of ions (for instance due to volume exclusion, because of an energy penalty for ions to fit in the narrow pores) are discussed in

§2.8, Ch. 4, and Ch. 11.

Effect of temperature on electric double layer structure. An interesting topic is the effect of temperature on the EDL structure, for instance according to the just-discussed Donnan model. Often chemical potentials are described as ‘𝑅𝑇ln𝑐_𝑖’ which suggests for this entropic part a significant temperature effect, while the electrostatic part,𝐹·𝑉, is not temperature dependent. However, when converting to a dimensionless electrical potential, 𝜙, we notice the sameRT-term in both contributions and thus temperature turns out to not matter. Indeed, an equation such as Eq. (2.11) correctly does not include temperature. Volumetric effects can be included too, and still temperature does not play a role. Instead, ‘chemical’ affinity effects, such as described by the constantsaandgin the last chapter, have a strong dependence on temperature.

Interestingly, EDL models based on the Poisson equation, evaluated for ions in water, see the next chapter, are also quite temperature-insensitive, for the coincidental reason

40 The Donnan model: the EDL in small pores

that the factor𝜀·𝑇 is quite independent of temperature. This is because the dielectric constant of water decreases with temperature in such a way that after multiplication with temperature𝑇(in K), the result is an almostT-independent factor. When we have transport, the effect of temperature is interesting too, which we discuss in a box on p. 197.

For capacitive processes where small pores are used to store ions, the concept ofcharge efficiency, Λ, plays a key role. Charge efficiency on the one hand is an experimental parameter, obtained from experiments with an electrochemical cell pair, and on the other hand it is a theoretical property of an EDL model. Here we focus on the second aspect, and define charge efficiency as the additional number of ions incorporated in a pore when the charge goes from zero to a particular value, and then divided by that final charge. In this case charge efficiency is given by

Λ = 𝐴⁻¹

√︁

𝐴2+1−1

=tanh(|𝜙_D|/2) (2.12) where 𝐴 = |𝜎

w|/(2𝑐_∞) is a dimensionless pore wall charge. The charge efficiencyΛis a number between 0 and 1, and quantifies how much salt is adsorbed when an electrode is charged. Clearly, a number above unity is not possible, and numbers close to this ideal maximum require that the electrode charge, or Donnan potential, is high.

The simple Donnan model discussed above, is completed by consideration of one final property, the difference in pressure between inside and outside the pore. When the assumptions of the Boltzmann equation are valid, as in the above equations, then the osmotic pressure in the pore is given byⁱⁱⁱ

Π =𝑐

T (2.13)

while the osmotic pressure of a 1:1 solution outside the pore is given byΠ = 2𝑐_∞. Now, because of mechanical equilibrium that we can assume in an EDL, it is the case that the total pressure,𝑃^tot, is invariant across the EDL. Because this total pressure has two contributions,

𝑃^tot=𝑃^h−Π (2.14)

where𝑃^his the hydrostatic (hydraulic) pressure, combination of Eqs. (2.13) and (2.14) leads to the conclusion that the increase of the osmotic pressure across the EDL equals the increase in hydrostatic pressure

Δ𝑃^h=𝑃^h

pore−𝑃^h_∞= Π_pore−Π∞=𝑐

T−2𝑐_∞. (2.15)

iiiThroughout most of this book, ‘reduced’ pressuresΠand𝑃h

are used, as well as a reduced (chemical) potential 𝜇_𝑖. These parameters can be multiplied by𝑅𝑇to obtain pressures in J/m³=Pa and potentials in J/mol.

Donnan model including Stern layer 41 This result is of importance in describing the flow of fluid through a porous medium, such as a membrane, see Chs. 11 and 12, or to calculate the forces that an EDL exerts on particle (pore) walls, which can lead to swelling or fracture of porous charged materials (Biesheuvel, 2017). We will discuss this relation in more detail in Ch. 8.

Mixtures of different ions. Electrolytes generally contain mixtures of ions (e.g., different types of cations), and this situation is described at several points in this book:

in §1.5 we provided the theory to describe differences in absorption between two cations in an intercalation material, and in §2.7 we discuss mixtures of ions of different valencies. The Donnan model is particularly useful because it leads to equations that are tractable and describe many relevant situations.

Here we only address an almost trivial, but very relevant point. Namely, in the Donnan model, and neglecting partition effects due to affinity and ion volume, thus only considering ion charge, then the ratio of concentration of a certain ion𝑖in the pore over that in solution, relates in a very simple way to the same ratio for all other ions 𝑗 (the other ions can have the same valency,𝑧_𝑖=𝑧_𝑗, or have a different valency) by

𝑐_𝑖 𝑐_∞,𝑖

^𝑧𝑖

= 𝑐_𝑗

𝑐_{∞, 𝑗} ^𝑧𝑗

. (2.16)

This result is used in §2.7 to describe the absorption of two cations of a different valency (and one anion) in a negatively charged porous material.