Donnan theory

J Rieka, T Tanaka. Swelling of ionic gels Quantitative performance of the Donnan theory. Macromolecules 17 2916-2921, 1984. 

The procedure used for testing the ideal Donnan theory is applicable to any model that decouples ionic effects from network elasticity and polymer/solvent interactions. Thus we require that nnet depend only on EWF and not C. While this assumption may seem natural, several models which include ionic effects do not make this assumption. For example, the state of ionization of a polymer chain in the gel and the ionic environment may affect the chain s persistence length, which in turn alters the network elasticity [26]. Similarly, a multivalent counterion can alter network elasticity by creating transient crosslinks. 

To summarize, we have shown that the ideal Donnan model cannot account quantitatively for the ionic effects on gel swelling. Nevertheless, using the heuristic procedure introduced above, the Donnan theory can be used to predict qualitative trends. 

F.G. Donnan, Theory of Membrane Equilibria and Membrane Potentials in the Presence of Non-dialyzing Electrolytes, Z. Elektrochem. 17, 572 (1911). 

To describe macromolecular solutions, the Donnan theory is used in form of the Donnan equation... 

The concept of internal pH together with a suitably modified Donnan theory to take into account ion affinity can be useful therefore in predicting catalytic effects of ions in solution. It does not necessarily follow that the internal solution is a clearly defined phase. Alexander and Kitchener... 

Because of its simplicity, the concept of Donnan equilibrium is an attractive approach for the modelling of fibre-ion interactions. However, despite its general success in many cases, the Donnan theory alone is not sufficient to describe the experimentally observed distributions. In those cases, it is necessary to use ion-specific complexation equilibria with the acidic groups as a part of the model to describe the observed phenomena satisfactorily. Such behaviour has been shown to exist, for example, with chemically modified highly charged fibres. An apparent systematic deviation in the ionic distribution from the predictions of the Donnan theory has also been noted with common oxygen-delignified kraft pulps, partly, but... 

Another necessary modification of the fibre phase data is related to the fact that the amount of water inside the fibres cannot be allowed to vary fieely during calculations. The simplest solution is to keep its amount constant. This is not precisely tme as the swelling of fibres is affected both by pH and the ionic strength of the solution. However, the calculated ionic distribution is not particularly sensitive to moderate changes in fibre phase volume and the same assumption of a fixed amount of fibre phase water has been commonly made with other Donnan theory-based models as well. 

Neale J. Text. Inst., 1929, 373 1930, 225 1931, 349) attributes the attraction of cellulose towards sodium ions to the formation of non-diffus-ible anions of the type (CeHioOj)- or (CgH 9O5.H2O)-. This, it has been suggested, causes a distribution of sodium and hydroxyl ions on each side of the fibre boundary in accordance with the Donnan membrane equilibrium theory. The Donnan theory provides thermodynamic proof that if a solution of an electrolyte containing two diffusible ions is separated by a membrane from another solution containing a salt with a non-diffusible ion, then the equilibrium distribution of the former will not be equal on the two sides of the membrane . This state is illustrated diagrammatically in Fig. 3.12 where, on the side marked A of the membrane (which in this case is the... 

Accordingly to Donnan theory, distribution of mobile ions between the gel and the solution is equilibrium for each ion as follows. 

The uptake of anions, as well as cations, by a cationic exchanger, is predicted by the Donnan theory of membrane equilibria, to increase with the electrolyte concentration in the external solution phase. The cations thus absorbed must be included in the effective resin capacity. Data of Bauman for the HC1 uptake in Dowex 50, taken from an experimental plot (Nl), are as follows ... 

Values of K gx for various cation exchange reactions can be shown to be related to the ratio of activity coefficients in the resin phase by means of the Donnan theory developed by describing membrane equilibria. The Donnan theory, based in thermodynamics, states that the activity product of a salt on either side of a membrane to which it is permeable, is identical. We can consider the resin phase as being separated from the aqueous phase by the equivalent of a membrane which permitted the free passage of all ions except those which are chemically bound to the polymer. Applying this to a system consisting of a resin in the hydrogen form in equilibrium... 

Upon verification under such conditions, the Donnan theory, as applied to swelling, held satisfactorily in various instances. It will be clear that this simple theory cannot bear a general character and that its applicability will be confined to those cases where its premisses may be expected to be consistent i. e. in the swelling and contraction of relatively diluted gels upon changes of the electrolyte concentration in dilute equilibrium solutions. ... 

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