Donnan equilibrium at the membrane/solution interfaceinterface

In this section we discuss the Donnan layer at the interface between a solution phase (electrolyte) and a charged gel or membrane, where we derive and then use the general Boltzmann equation, Eq. (2.1). Thus we now combine the electrostatic Donnan effect with partitioning due to volume effects or due to the affinity of an ion with a certain phase, but other contributions are possible too. Solute partitioning is of importance for instance in Ch. 11 that deals with ion adsorption in charged membranes and transport across them. For some membranes the water-filled pore phase is highly constricted, thus partitioning because of size exclusion plays an important role besides Donnan electrostatics.

We can derive the general Boltzmann equation from an expression for the chemical potential of an ion𝑖, including an affinity term, and an excess (volume exclusion) term, which for an ion in each separate phase is written as

𝜇_{𝑖 , 𝑗}=𝜇

ref,𝑖+ln 𝑐_{𝑖 , 𝑗}/𝑐

ref

+𝜇

aff,𝑖 , 𝑗+𝜇

exc,𝑖 , 𝑗+𝑧_𝑖𝜙_𝑗 (2.28)

where𝜇_aff,𝑖includes all possible effects not related to charge or ion volume, generally called affinity, which can include contributions such as dielectric exclusion, image forces, and ion dehydration energy. In Eq. (2.28) index 𝑗refers to position, which can be in solution, ‘∞’, or in the membrane, ‘m’.^vi,viiIf we equate the chemical potentials in solution with that just

vAt very low𝛽we must include the anions in the charge balance, i.e., Eq. (2.27) is not valid at low𝛽. At very low 𝛽,𝑐c

Ca²⁺

/𝑋increases again. The lines in Fig. 2.2 are for𝛽high enough that anions have a low concentration in the collagen and Eq. (2.27) is valid.

viSee p. 507 for an explanation how each𝜇-term in Eq. (2.29) can be multiplied by a term𝑅𝑇to arrive at a dimensional potential with unit of J/mol.

viiEDL models are solved for conditions ofchemical equilibrium, and thenmechanicalequilibrium is also established. In problems of flow of solutes, these equilibria conditions do not apply, and then Eq. (2.28) has an additional term,𝜐_𝑖𝑃tot

that is discussed in several upcoming chapters. At mechanical (and thus also at chemical) equilibrium, this term is zero.

50 The Donnan model: the EDL in small pores inside the membrane, we arrive at

ln𝑐_∞,𝑖+𝜇

aff,∞,𝑖+𝜇

exc,∞,𝑖+𝑧_𝑖𝜙_∞=ln𝑐^∗

m,𝑖+𝜇

aff,m,𝑖+𝜇

exc,m,𝑖+𝑧_𝑖𝜙

m (2.29)

where constant terms 𝜇

ref,𝑖 and ln𝑐

ref, that show up on both sides, are cancelled. On the right side of Eq. (2.29), we use a term𝑐^∗

m,𝑖to denote a concentration of ions or other solutes defined per unit total membrane volume (i.e., for the totality of the water- and ions-filled pores together with the solid polymer membrane matrix structure). When we are not at mechanical equilibrium, an ion’s chemical potential also includes the insertion pressure (a term which is zero at equilibrium), which is the last term in Eq. (7.63) in Ch. 7.

One contribution to 𝜇_exc,m,𝑖relates to the exclusion of ions from the membrane matrix, where by ‘matrix’ we refer to the polymer network and structure of which the membrane is made. This exclusion effect simply implies: where there is polymer (membrane structure), there can not be an ion (and neither a water molecule). We can quantify this excess term by using the Carnahan-Starling equation of state (CS-EOS), or any of its extensions to multicomponent systems, and to porous structures, such as Eq. (4.11) in Ch. 4. When we assume that the ions do not have any volume, i.e., they are point charges, only being excluded from the volume occupied by the membrane structure, the consequence of the CS equation of state is that𝜇_exc,m,𝑖=−ln𝑝_mwhere𝑝_mis the porosity of the membrane, i.e., the fraction of the total volume that is available to water and ions, i.e., the fraction of the total volume that consists of pores, see Fig. 7.7. If we combine this term with the concentration term, we have

𝜇exc,m,𝑖+ln𝑐^∗

m,𝑖=−ln𝑝

m+ln𝑐^∗

m,𝑖=ln𝑐

m,𝑖 (2.30)

where the ion concentration per unit pore volume,𝑐

m,𝑖, is given by 𝑐

m,𝑖 =𝑐^∗

m,𝑖/𝑝

m. We include this result in Eq. (2.29), and we can do the same for an ion in solution, where𝑝_∞=1 and thus the related term is𝜇

exc,∞,𝑖=0. In general there is still an excess contribution on top of the effect just described, so the conversion just discussed does not make volume effects go away, and thus in the equations below,there still is an excess term. Detailed expressions for these excess, volume, effects will be discussed in §4.2, §7.8, and §8.1. Volume is important to consider because it has a significant effect on ion partitioning and thus on ion selectivity in membrane processes.

To further simplify Eq. (2.29), we implement the definition of the Donnan potential, Δ𝜙

D =𝜙

m−𝜙_∞, and we use the notation ofΔ𝜇_{𝑘 ,𝑖} to describe a difference in each of the 𝜇_{𝑘 ,𝑖}-contributions between inside the membrane, and in solution. We then arrive at

ln𝑐_∞,𝑖=ln𝑐

m,𝑖+Δ𝜇

aff,𝑖+Δ𝜇

exc,𝑖+𝑧_𝑖Δ𝜙

D. (2.31)

Donnan equilibrium at the membrane/solution interface 51 Now, all of these contributions to an ion’s chemical potential (except the ln𝑐-term) can be rewritten to a contribution to the partition coefficient,Φ𝑘 ,𝑖, wherekrefers to the type of energy (excess, affinity, etc.), andito the species, according to

Φ𝑘 ,𝑖=exp −Δ𝜇_{𝑘 ,𝑖}

, Δ𝜇_{𝑘 ,𝑖}=𝜇_{𝑘 ,}

m,𝑖−𝜇_{𝑘 ,∞,𝑖} (2.32)

and if we apply it to the affinity and volume (excess) terms, we then have Φ_aff,𝑖=exp − 𝜇

aff,m,𝑖−𝜇

aff,∞,𝑖 , Φ_exc,𝑖=exp − 𝜇

exc,m,𝑖−𝜇

exc,∞,𝑖 . (2.33) Including these conversions in Eq. (2.31) we arrive at a detailed general Boltzmann equation 𝑐_m,𝑖 =𝑐_∞,𝑖·Φ_aff,𝑖·Φ_exc,𝑖·exp(−𝑧_𝑖𝜙_D) (2.34) and if we combine the contributions to the partitioning of a species across an interface (those that we identified here) into a single term,Φ𝑖, thusΦ𝑖= Π𝑘Φ𝑘 ,𝑖 (except for the term exp(−𝑧_𝑖𝜙

D)), then Eq. (2.34) simplifies to the general Boltzmann equation, Eq. (2.1). In Eq. (2.34), and throughout this book, we described the Donnan (Boltzmann) term separately, and only subsume non-Donnan effects into theΦ𝑖-function. However, one can also define a Donnan (contribution to the) partition function,Φ_D=exp(−𝑧_𝑖𝜙

D)and consider this Donnan partitioning as another contribution toΦ𝑖.

In a real membrane process, there are many types of ions to consider, and for all of them we evaluate Eq. (2.1). All ions have a different value forΦ𝑖, and will also have different valencies,𝑧_𝑖. Also for neutral species such as carbonic acid and ammonia we use Eq. (2.1), with𝑧_𝑖=0. All of these Donnan equations for each ion separately are solved simultaneously with local electroneutrality in the solution phase just outside the membrane, as well as local electroneutrality just inside the membrane (just ‘beyond’ the Donnan layer), where local EN inside the membrane also includes the membrane charge,𝑋, and thus we have

∑︁

𝑖

𝑧_𝑖𝑐

m,𝑖+𝑋 =0. (2.35)

For all ions, Eq (2.1) can be inserted in Eq. (2.35), to arrive at a relation between membrane charge 𝑋, solution concentrations, and Donnan potential, 𝜙_D. Note that the membrane charge, 𝑋, is defined –like ion concentrations– as a concentration per unit pore volume.

The resulting Donnan potential can be used again in Eq. (2.1) to calculate for each ion the concentration just inside the membrane. When membrane charge is a function of local pH (just in the membrane) or dependent on any other ion concentration, this adds an additional relation to the set of equations to be solved simultaneously, such as 𝑋 = 𝑓 𝑐

m,H⁺

, but otherwise this addition does not change the Donnan model equations.

52 The Donnan model: the EDL in small pores

Thus we can insert the general Boltzmann equation, Eq. (2.1), in Eq. (2.35) and that leads to

∑︁

𝑖

𝑧_𝑖𝑐_∞,𝑖Φ𝑖exp(−𝑧_𝑖𝜙

D) +𝑋 =0 (2.36)

which can always be solved if for all ions we knowΦ𝑖. [Note that in the derivation which followsΦ𝑖is assumed to be independent of ion concentrations. Equations until the present point did not require that assumption.] If we have a symmetric salt (with𝑧=1 for a 1:1 salt, 𝑧=2 for a 2:2 salt, etc.), and all cations have the valueΦ+, and all anionsΦ−, this simplifies to

𝑧𝑐_∞(Φ+exp(−𝑧 𝜙

D) −Φ−exp(𝑧 𝜙

D)) +𝑋 =0 (2.37)

which can be rewritten to 𝑧𝑐_∞

√︁Φ+Φ−(exp(− (𝑧 𝜙

D+½ln(Φ−/Φ+))) −Φ−exp(𝑧 𝜙

D+½ln(Φ−/Φ+))) +𝑋=0 (2.38) and thus

−2𝑧𝑐_∞

√︁Φ+Φ−sinh(𝑧 𝜙

D+½ln(Φ−/Φ+)) +𝑋=0 (2.39) and thus

𝑧 𝜙D+½ln(Φ−/Φ+)=sinh⁻¹

𝑋 2

√Φ+Φ−𝑐_∞

. (2.40)

We can write an equation for the total ions concentration in the membrane, 𝑐

T,m = 𝑐m,++𝑐

m,−, similar to Eq. (2.40), which leads to 𝑧 𝜙D+½ln(Φ−/Φ+)=cosh⁻¹

𝑐

T,m

2

√Φ+Φ−𝑐_∞

(2.41) and thus combination of Eqs. (2.40) and (2.41) leads to

𝑐²

T,m=(𝑋/𝑧)²+Φ+Φ−(2𝑐_∞)² (2.42) and this relation depends on the geometric mean partition coefficient,

√Φ+Φ−, but not on the individual factorsΦ− andΦ+. It turns out that the same holds for the cation and anion concentration in the membrane, as long asX is fixed and as long as

√Φ+Φ− is the same.

Thus, even when for one ion the factorΦ𝑖is increased, if we lower that for the other ion such that the geometric mean stays the same, the cation and anion concentrations stay the same.

This result implies that for a symmetric binary solution, with one value ofΦ𝑖for all cations, one for all anions, the only property of relevance is the geometric mean partition coefficient,

√Φ+Φ−. This implies that we cannot experimentally distinguish between the two values,

Donnan equilibrium at the membrane/solution interface 53 and theory only requires consideration of the geometric mean.^viii So from this point onward, we useΦ𝑖 as if both ions have the same partition coefficient, or as the geometric mean in case they are different.

In that latter case, differences betweenΦ𝑖 for cat- and anions (as long as the geometric mean is the same) will influence the Donnan potential 𝜙

D, and this will influence proton adsorption thus pH, and then individual values forΦ𝑖do matter, but except for such indirect effects, the use of oneΦ𝑖-value for both ions does not influence the predicted EDL structure, thus does not influence𝑐_T,mand neither does it influence co- and counterion concentrations just in the membrane. So we continue now usingΦ𝑖as the geometric mean of the individual partition coefficients, or, in a simplified situation, we just assume it is the same for all ions.

When all ions are monovalent (or some are zero-valent, i.e., neutral), and we now use the sameΦ𝑖for all ions, then the Donnan potential follows from

𝑋=2𝑐_∞Φ𝑖sinh𝜙_D (2.43)

and the total ions concentration (anionspluscations) in the membrane becomes 𝑐T,m=

√︃

𝑋2+ (2Φ𝑖𝑐_∞)² (2.44)

which is always larger than|𝑋|. With𝛼= 𝑋/(2Φ𝑖𝑐_∞), the counterion (‘ct’) concentration in the membrane is given by

𝑐m,ct

Φ𝑖𝑐_∞

=√︁

1+𝛼2+ |𝛼| (2.45)

which is always larger than|𝑋|, just like𝑐_T,m, while the coion (‘co’) concentration is 𝑐m,co

Φ𝑖𝑐_∞

= Φ𝑖𝑐_∞ 𝑐_m,ct

=

|𝛼| +√︁

1+𝛼2

−1

= 𝑐

T,m−𝑐

m,ct

Φ𝑖𝑐_∞

=√︁

1+𝛼2− |𝛼|. (2.46) which is smaller than|𝑋|as long as𝑐_∞is smaller than

√

2|𝑋|/Φ𝑖.

This concludes the description of the Donnan layers formed at the interface of membrane and solution. These results are used again in Ch. 11 to describe desalination of water.

references

1. M. Higa, A. Kira, A. Tanioka, and K. Miyasaka, “Ionic partition equilibrium in a charged membrane immersed in a mixed ionic solution,”J. Chem. Soc. Faraday Trans.89, 3433–3435 (1993).

viiiTo find the individual values we need very precise experiments with at least three different ions.

3

The EDL for a planar surface: The