Donnan exclusion

OH

C at h o d e

H²O

A n o de

Figure 1.8 Schematics of bipolar membrane. Cations are rejected by the anion-exchange layer (AEL) and anions are rejected by the cation-exchange layer (CEL). The electrical potential drop across the transition layer catalyses dissociation of water that diffuses into the membrane.

The applications possible by this watersplitting process are many. By producing hydrogen and hydroxide ions while transferring electrical current and serving as an impermeable wall to both cations and anions, it is possible to regenerate acids and bases from mixed solutions.

1.2.4 Donnan exclusion

To explain the mechanisms of the ion-exchange membrane's selective behavior towards positive and negative ions, consider a cation-exchange membrane in contact with a dilute ionic solution as sketched on Figure 1.9. In this case the cations are the counter-ions and the anions are the co-ions.

For the exclusion to be effective, the membrane’s concentration of fixed charges (the membrane’s ion-exchange capacity) must be higher than the concentration of the electrolyte. Under normal operating conditions, the concentration of mobile cations inside the membrane is much higher than in the dilute solution to preserve electroneutrality of the fixed negative groups. The concentration of mobile anions, however, is much lower (if existing) inside the membrane than in the solution.

Figure 1.9 The distribution of cations and anions between a cation exchange membrane and a dilute electrolyte solution. Inside the membrane fixed negative charges are balanced by mobile counter-ions (cations). The axis gives a rough estimation of the concentration levels of the ions inside the membrane (indexed with ^) and in the solution at equilibrium.

For neutral molecules, ordinary diffusion would take place to level out the concentration differences. The cations and anions all carry charges and attempts to balance the concentration differences are disturbing electroneutrality both inside the membrane and in the external solution.

Cations diffusing out of the membrane and into the external solution create a positive surplus of charge (space charge) in the solution just outside the membrane and anions from the external solution diffusing into the membrane create a negative space charge just inside the membrane surface. This creates an electrical field across the membrane-solution interface that opposes further diffusional flow as shown in Figure 1.10.

Figure 1.10 Cations diffusing out of the membrane while anions diffuse into the membrane resulting in the establishment of an electrical field ∆Ψ = Ψmembrane - Ψsolution that balance out the driving diffusion forces.

At electrochemical equilibrium this electrical field balances the driving diffusion forces of the ions.

This potential allows the counter-ion concentration to be higher, and the co-ion concentration to be lower inside the membrane than in the external solution. Since the co-ions are repelled from the membrane, the electrolyte itself is repelled, because of the electroneutrality requirement. This exclusion of electrolyte is named Donnan exclusion and the electrochemical equilibrium is named Donnan equilibrium in honor of his pioneer work in this area (Donnan 1911).

The electrical potential ∆ψ that arises can be calculated from the electrochemical potentials µ of the mobile ions. Neglecting pressure differences between the membrane’s liquid phase and the external solution, the electrochemical potential for every ionic species can be expressed:

Equation 1.1 µ_j =µ^Ο_j +RTlna_j +z_jFψ

µ° is the chemical standard potential, R is the gas constant (8.31 J/mol·K), and T the temperature (K). a and z are the activity and ionic charge, respectively. F is the Faraday number (96485 C/eq.) and ψ the electrical potential.

At steady state the chemical potential for each component is equal on both sides of the electrolyte/membrane interface. Using index m for membrane conditions and s for solution conditions, this can be expressed as:

Equation 1.2 µ_j^Ο,m +RTlna^m_j +z_jFψ^m =µ^Ο_j^,s+RTlna^s_j +z_jFψ^s

At equilibrium the chemical standard potential µj° is equal in both phases. Hence the potential Equation 1.3 can be changed to:

Equation 1.4 _m

From Equation 1.5, some general considerations concerning Donnan equilibrium is reflected on in the following examples.

Example A

For a single monovalent salt (z+ = 1, z- = -1) like sodium chloride in a dilute (0.01 M) solution, assuming activity contants γ± ≈ 1, Equation 1.5 rearranges into:

Equation 1.6 _m

In this example, C+s = C-s = 0.01 M. For a cation-exchange membrane the membrane's ion-exchange capacity (concentration of fixed ionic groups) is assumed to be around 1 M, which means that the concentration of mobile cations in the membrane, C+m is around 1 M to balance the fixed charges.

From Equation 1.6 it then follows that C_-^m must be in the region of 0.0001 M. This example of a typical electrodialysis setup, demonstrates that the membrane's concentration of mobile cations is about 10.000 times larger than the concentration of mobile anions, meaning the selectivity of cations over anions in this membrane in this solution is 99.99%.

From Equation 1.6 it is clear that the higher the counter-ions' concentration ratio between membrane and solution (C+m/C+s), the higher the ratio of the solution/membrane concentration of the co-ions (C_-^s/C_-^m). This results in higher selectivity. The opposite is also applies. The lower the counter-ion ratio becomes, the result is a lower co-ion ratio, which results in lower selectivity. Ion

exchange membranes generally perform poorly in high electrolyte concentrations. When the electrolyte concentration matches the membrane capacity, there is no effective selectivity.

For low electrolyte concentrations (100 times lower than the membrane’s ion-exchange capacity) it is safe to assume that co-ion concentration is insignificant inside the membrane.

Example B

In a dilute electrolyte solution with both monovalent and divalent cations in contact with a cation-exchange membrane, Equation 1.3 still applies. Consider a mixed electrolyte solution with sodium chloride and calcium chloride and for simplicity, the charge equivalent concentrations of both cations are the same. The Donnan potential that arises at equilibrium affects both ions. Following the same routine from Equation 1.3 as before for the two types of cations and still neglecting activity coefficients, the ensuing expression is reached:

Equation 1.7 _s

The higher the ionic charge of a counter-ion, the more readily it will enter and fill the membrane's capacity spots.

Example C

To calculate the intermembrane concentration of mobile counter- and co-ions, consider the need for electroneutrality of both phases in the system at equilibrium. For a salt in the initial solution:

Equation 1.8 =0

The same equation can be set up for the membrane’s mobile phase, including the membrane's ion exchange capacity CR:

Equation 1.9 + _{R R} =0

Assuming a cation-exchange membrane with monovalent fixed charges (zR = -1) in contact with a monovalent salt (z+ = +1, z- = -1), Equation 1.8 and Equation 1.9 rearranges into:

Combining these two equations with the Donnan exclusion condition in Equation 1.6, the membrane’s co-ion concentration C_-^m can be obtained from the following expression:

Equation 1.11 1

For dilute solution, the ratio CR/C-m >> 1 and Equation 1.11 can be simplified to:

Equation 1.12

( )

Equation 1.12 confirms the theoretical estimation of the co-ion concentration of 0.0001 M inside a cation-exchange membrane with a capacity of 1 M in contact with a 0.01 M monovalent electrolyte as treated in example A.

For more complex electrolyte solutions, the results must be evaluated separately from case to case, and the assumption of activity coefficients at unity must be questioned. But for dilute, monovalent electrolytes Equation 1.12 yields very good estimates.