Donnan EquilibriumEdit
Donnan equilibrium describes how ions distribute themselves across a semipermeable membrane when some charged species cannot cross the membrane. Named for the early 20th‑century work of Frederick Gibs Donnan, the effect arises from the interplay of chemical potential and electric potential across the barrier. When a membrane is permeable to small ions but impervious to certain fixed or impermeant charges, the system reaches a steady state in which the chemical potentials of the permeant ions are balanced on both sides, producing a characteristic electric potential difference known as the Donnan potential. This framework is foundational in physical chemistry and biophysics and helps explain phenomena observed in cells, filtration, and various industrial processes. Donnan equilibrium and semipermeable membrane concepts are closely linked, and the effect sits at the crossroads of electrostatics, diffusion, and chemistry.
In biological contexts, the Donnan effect contributes to the resting properties of membranes and to osmoregulation in tissues. The presence of fixed charged groups inside cells and extracellular matrices creates an imbalance that permeant ions partially compensate for, influencing ion distributions and the electrochemical gradient that drives many cellular processes. The Donnan framework thus helps illuminate why cells maintain particular ionic compositions and how membranes establish potential differences across their interfaces. It is often introduced alongside broader topics such as cell physiology and biological membranes to show how simple equilibrium ideas scale to living systems. However, real biological systems involve active transport and ion channels that modify or partially override the Donnan picture, which is an important caveat for practitioners. Membrane potential and electrolyte concepts frequently enter discussions of Donan-type systems.
This article explains the core ideas, traces the basic mathematics at a conceptual level, and highlights both applications and limitations in practice. It also situates the Donnan framework within the broader landscape of membrane science, including its relevance to engineering applications such as dialysis and desalination, where ion distributions and electrical potentials across membranes govern performance. Dialysis and desalination technologies rely on principles related to Donnan equilibrium to some extent, while more advanced models account for non-ideal behavior in concentrated solutions. Osmosis is another related phenomenon, as it reflects how differences in solute concentrations and membrane permeability drive water movement in Donnan‑like settings. Gibbs–Donnan equilibrium is a broader descriptor sometimes used to emphasize the thermodynamic underpinnings of the same phenomenon.
Principles
Impermeant charges on one side of a membrane create an electrostatic constraint: the side with fixed charges must balance its positive and negative charges, while the opposite side has no fixed charge. This electroneutrality condition drives the distribution of permeant ions across the membrane. Impermeant ions are central to the effect, and their presence prevents a straightforward equalization of concentrations by diffusion alone.
Permeant ions redistribute to balance chemical potential across the membrane, leading to a Donnan potential that aligns the electrochemical potentials on both sides. For a simple case with monovalent ions, the concentrations of cations and anions on each side adjust according to the membrane potential difference Δφ, following relationships such as c_+^a / c_+^b = exp(-FΔφ/RT) and c_-^a / c_-^b = exp(+FΔφ/RT). These relations arise from setting the chemical potentials equal for permeant species: μ_i^a + z_i F Δφ = μ_i^b, where z_i is the ionic charge. Nernst equation concepts are often brought in to connect these ideas to potential differences.
Electroneutrality must hold on both sides of the membrane: the sum of positive charges equals the sum of negative charges, including both permeant ions and the fixed, impermeant charges. This condition constrains the possible ion distributions and is a key reason why the Donnan potential arises.
The classic Donnan picture is most transparent for relatively simple, dilute solutions where activity coefficients can be neglected or treated approximately. In real systems, especially at higher ionic strength or with multivalent ions, activity coefficients and finite ion size become important, and the simple exponential relations are replaced by more complex formulations. Electrochemistry and activity coefficient concepts are helpful here.
Mathematical formulation (conceptual)
In a two‑compartment system separated by a semipermeable membrane that is permeable to the main salts but impermeable to fixed charges, the balance of chemical potential for each permeant ion i yields:
μ_i^a + z_i F Δφ = μ_i^b,
which, under ideal conditions, reduces to RT ln a_i^a + z_i F Δφ = RT ln a_i^b. If activities are approximated by concentrations, the ratios of concentrations across the membrane depend on the membrane potential Δφ. The impermeant charge X on side a enters the problem through electroneutrality constraints:
c_+^a = c_-^a + X.
Solving these relations together with the corresponding equation on side b (which typically has no fixed charge, so c_+^b = c_-^b) yields the Donnan potential and the distribution of permeant ions. For more rigorous treatment, see discussions of the Gibbs–Donnan framework and related membrane models. Gibbs–Donnan equilibrium.
Biological and technological relevance
In cells and tissues, the Donnan effect contributes to ion partitioning between compartments and can influence the magnitude of the resting membrane potential, especially in regions with high fixed charge density such as extracellular matrices. This helps explain why ionic gradients can persist even when membranes are permeable to many ions. Biological membranes and membrane potential are the natural entry points for these ideas.
In dialysis, desalination, and sensor design, Donnan considerations inform how membranes separate ions and how the electrical field develops across a barrier. Designers use these ideas to optimize selectivity and efficiency, while advanced models incorporate non‑ideality and multi‑ion effects to improve accuracy. Dialysis and desalination illustrate practical contexts where ion distributions and potentials matter.
Osmotic effects accompany Donnan behavior: an unequal distribution of ions can create osmotic pressure differences that influence water flow and swelling in tissues or porous media. Understanding these effects helps interpret experiments involving saline solutions and porous membranes. Osmosis.
Controversies and debates
The classic Donnan model rests on several simplifying assumptions (ideal solutions, simple ionic species, and negligible activity corrections). In concentrated or multicomponent systems, activity coefficients depart from unity, and finite ion sizes lead to deviations from the simple exponential relations. Researchers debate the conditions under which the Donnan framework provides a good approximation and when more sophisticated treatments are required. Activity coefficient and ionic strength concepts are often invoked in these discussions.
Real biological systems are not passive barriers. Active transport, ion channels, and complex intracellular/extracellular architectures modulate ion distributions in ways that can surpass or override the pure Donnan equilibrium. As a result, Donnan theory is typically used as a foundational, first‑principles model or a simplifying assumption in a broader, dynamic membrane context. This nuance is a common point of emphasis in modern physiology and biophysics teaching. Ion channel and ion transport topics illustrate where Donnan logic complements, rather than replaces, more complete transport theories.
In the literature, the Gibbs–Donnan perspective is sometimes invoked to emphasize thermodynamic consistency, but it is equally important not to overstate its reach in complex systems. Critics remind practitioners to verify that the conditions for Donnan equilibrium hold in any given application and to account for competing effects that can dominate in real materials. Gibbs–Donnan equilibrium.