The Donnan potential

An ion-selective electrode contains a semipermeable membrane in contact with a reference solution on one side and a sample solution on the other side. The membrane will be permeable to either cations or anions and the transport of counter ions will be restricted by the membrane, and thus a separation of charge occurs at the interface. This is the Donnan potential (Fig. 5 a) and contains the analytically useful information. A concentration gradient will promote diffusion of ions within the membrane. If the ionic mobilities vary greatly, a charge separation occurs (Fig. 5 b) giving rise to what is called a diffusion potential. 

A Donnan membrane, i.e., a membrane impermeable to certain kinds of ions, which results in the occurrence of the Donnan potential. 

A semi-permeable membrane, which is unequally permeable to different components and thus may show a potential difference across the membrane. In case (1), a diffusion potential occurs only if there is a difference in mobility between cation and anion. In case (2), we have to deal with the biologically important Donnan equilibrium e.g., a cell membrane may be permeable to small inorganic ions but impermeable to ions derived from high-molecular-weight proteins, so that across the membrane an osmotic pressure occurs in addition to a Donnan potential. The values concerned can be approximately calculated from the equations derived by Donnan35. In case (3), an intermediate situation, there is a combined effect of diffusion and the Donnan potential, so that its calculation becomes uncertain. 

It can be readily seen that this procedure yields an equation whose left-hand side is the same as that for Eqs (6.1.6) and (6.1.7) and whose right-hand side is not equal to zero, but rather to v+(p — p2) or V-(pi — p2)-Equations (6.1.6) and (6.1.7) yield the Donnan potential A0D = A0M in the form... 

The Donnan potentials contain the individual ionic activities and cannot be measured by using a purely thermodynamic procedure. In the concentration range where the Debye-Hiickel limiting law is valid, the ionic activities can be replaced by the mean activities. 

This theory will be demonstrated on a membrane with fixed univalent negative charges, with a concentration in the membrane, cx. The pores of the membrane are filled with the same solvent as the solutions with which the membrane is in contact that contain the same uni-univalent electrolyte with concentrations cx and c2. Conditions at the membrane-solution interface are analogous to those described by the Donnan equilibrium theory, where the fixed ion X acts as a non-diffusible ion. The Donnan potentials A0D 4 = 0p — 0(1) and A0D 2 = 0(2) — 0q are established at both surfaces of the membranes (x = p and jc = q). A liquid junction potential, A0l = 0q — 0P, due to ion diffusion is formed within the membrane. Thus... 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

Freed of other restrictions, a mobile ion may be expected to diffuse down any concentration gradient that exists between porous solid and liquid. In the particular case of ion exchange, there is an additional requirement that the resin and liquid phases should remain electrically neutral. Any tendency for molecules to move in such a way as to disturb this neutrality will generate a large electrostatic potential opposing further movement, known as the Donnan potential. 

Due to the presence of interactions, the apparent redox potential of a redox couple inside a polyelectrolyte film can differ from that of the isolated redox couple in solution (i.e. the standard formal redox potential) [121]. In other words, the free energy required to oxidize a mole of redox sites in the film differs from that needed in solution. One particular case is when these interations have an origin in the presence of immobile electrostatically charged groups in the polymer phase. Under such conditions, there is a potential difference between this phase and the solution (reference electrode in the electrolyte), knovm as the Donnan or membrane potential that contributes to the apparent potential of the redox couple. The presence of the Donnan potential in redox polyelectrolyte systems was demonstrated for the first time by Anson [24, 122]. Considering only this contribution to peak position, we can vwite ... 

Manipulation of the Donnan potential in random polymer-modified electrodes can also be achieved. In the case of cast redox polyelectrolyte-modified electrodes one can control ion permselectivity by mixing the redox polymer with an oppositely charged polyelectrolyte in an appropriate ratio before film casting [123]. The same strategy can be followed in electropolymerized films by mixing the electroactive monomer with one of opposite charge [124]. 

A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential. By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation. 

In order to obtain the membrane potential, any potential differences present in adhering liquid films must be taken into account, such in addition to the Donnan potentials. It should be observed that the splitting up of the membrane potential in diffusion potential, phase-boundary potentials and film potentials has met with opposition (49,121). 

Neglecting osmotic flow, it is possible to integrate the Nemst-Planck equations including activity coefficients (55, 142) (ref. 55, p. 338), an expression being obtained for the diffusion potential. Adding the Donnan potentials, the result is ... 

This may work well if the process involves only electrically neutral species. However, when ions are discriminated on the basis of size, the partitioning process is affected by the Donnan potential. This potential, which we discuss more fully in Chapter 6, develops at the membrane/electrolyte interface. Another possibility is to discriminate on the basis of charge, as shown in Fig. 7.10 (see Chapter 7). Again, a porous barrier membrane is used, although here it would contain fixed, electrically charged moieties. When placed in front of the transducer, it rejects the like-charged species by electrostatic repulsion. In other words, it is a form of ion exchange membrane. 

We see that for Cp = 0, the concentration of sodium ion in both compartments is the same (i.e., Cn3 = C J and that there is no Donnan potential across the membrane. In other words, it is the presence of the blocked polyelectrolyte that gives rise to the Donnan potential. For low molar concentration of P and high concentration of NaCl, the squared term (zCp/2C)2 can be neglected and (6.11) then changes to... 

Even for this simplest of all situations we had to make a fairly drastic assumption of high concentration of NaCl, in order to get from (6.11) to (6.13). The situation is considerably more complicated when different multivalent ions are present in the solution, although the basic argument is the same. In biological fluids, such as whole blood, the value of the Donnan potential across the dialysis membrane can be tens of millivolts. (In the above derivation of the Donnan potential, concentrations instead of activities have been used for purely historical reasons.)... 

The equivalent circuit corresponding to this interface is shown in Fig. 6.1b. The charge-transfer resistances for the exchange of sodium and chloride ions are very low, but the charge-transfer resistance for the polyanion is infinitely high. There is no direct sensing application for this type of interface. However, it is relevant for the entire electrochemical cell and to many practical potentiometric measurements. Thus if we want to measure the activity of an ion with the ion-selective electrode it must be placed in the same compartment as the reference electrode. Otherwise, the Donnan potential across the membrane will appear in the cell voltage and will distort the overall result. 

As far as the overall potential drop (E) is concerned, the contribution of solute polarization, namely the electric resistance (R ) and junction potential difference (TTj) across any boundary layer, may be neglected. On the contrary, the Donnan potential difference (Eu) in any cell pair, which behaves as a DC generator with inverted polarities with respect to those of the external DC generator (Figure 9), has to be accounted for as the solute concentration difference at both sides of the anionic and cationic membranes increases ... 

It is therefore clear that the Donnan potential for equilibrium potentials (Section 2.11) cannot be used in (17.3), except in special circumstances, since there are various components in the total transmembrane potential. Donnan himself predicted that the phenomenon would be more complicated for ion transfer processes between living cells or tissue and the liquids that surround them9. This transmembrane potential is not, however, the only one that can occur in the membrane10. 

At the present stage, i//c and if/d are still unknown. These values can be estimated if a membrane is sufficiently thick. Suppose that d 1. Then the electrical potential will reach a constant value deep inside the membrane, the Donnan potential [19]. If we let t Don be the scaled Donnan potential, then we have... 

Close meshed membrane electrode membrane electrode with large-meshed membrane. The small pores of its membrane allow diffusion of ions and molecules up to a certain size. Its potential is equal to the -> Donnan potential. 

Consider a gel that carries a certain concentration c,-(r) of immobile negative charges and is immersed in an aqueous solution. The bulk solution carries monovalent mobile ions of concentration c+(r) and c (r). Away from the gel, the concentration of the salt ions achieves the bulk concentration, denoted c0. What is the difference in electrical potential (known as the Donnan potential) between the bulk solution and the interior of the gel [Hint assume that inside the gel the overall concentration of positive salt ion balances immobile gel charge concentration.]... 

In addition to the surface potential, Yg, present at the outer and Inner surfaces of charged vesicles several additional potentials can be created. Of these, the charge separation potential, the diffusion potential and the Donnan potential will be briefly discussed. 

Donnan Potential. Colon exclusion and counterion condensation on the charged vesicle surface creates the well known Donnan potential. The Donnan potential can be derived either by a kinetic or by a thermodynamic approach (32). Using the kinetic approach, the mass transport equation is written by ... 

When a layer of protein molecules is in thermodynamical equilibrium with a solution with a certain pH and ion concentration, the ions in the solution and in the protein layer will be distributed according to the Donnan ratio (the ratio of the ion activities in the two phases). As a results, there will exist an electrical potential difference between the protein layer and the bulk solution, which is known as the Donnan potential. 

As a result of a stepwise increase in electrolyte concentration, the Donnan potentials of the protein layer will diminish, thereby also diminishing the potential of the membrane itself versus the bulk solution. Furthermore, protons will be released by negatively charged protein molecules or taken up hy positively charged protein molecules because the dissociation is changed with the ion concentration. The underlying ISFET measures the change in the potential of the membrane with respect to the bulk solution as well as the pH effect of the protons released or taken up. The pH effect of the protons is responsible for almost the entire response on an ion step. 

First, assume that the surface charge on the membrane particles does not interact with the mobile protons (no proton release or uptake). An ion step will result in an increase in the double-layer capacitances of the particles and consequently in a decrease of the surface potentials fr, because the charge densities remain constant. The ISFET will measure a transient change in the mean pore potential. As a result of the potential changes, an ion redistribution will take place and the equilibrium situation is re-established. The theoretical maximum ion step response is the change in the mean pore potential. This is comparable with the Donnan model where the theoretical maximum is determined by the change in the Donnan potential at the membrane solution interface. 

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