Donnan-Type Ion Partitioning

• Donnan-type ion partitioning is the preferential ion distribution across a charged interface driven by fixed immobile charges and electrochemical equilibrium.
• It plays a crucial role in biological, colloidal, and soft matter systems by influencing osmotic pressure, swelling, and selective localization of macromolecules.
• Analytical and computational models, including cell models and jellium approximations, quantify the entropic penalties and electrostatic effects underlying this phenomenon.

Donnan-type ion partitioning refers to the preferential distribution of ionic species across regions of a system separated by a phase or compartment boundary, arising from the presence of fixed (often immobile) charges in one compartment and governed by electrochemical equilibrium. This phenomenon drives an unequal concentration of mobile ions between two phases or regions—typically a charged matrix (such as a protein-rich phase, hydrogel, membrane, or colloidal droplet) and an adjacent solution—resulting in an electrostatic potential difference, the Donnan potential. Donnan-type partitioning has profound implications for phase equilibria, osmotic pressure, swelling, phase selectivity, and transport in a range of soft and biological systems.

1. Fundamental Principles of Donnan Equilibrium and Partitioning

Donnan equilibrium, originally formulated to describe the partitioning of ions across a semipermeable membrane containing immobile charges, generalizes to any system where fixed charges restrict the full mobility of all ionic species. At equilibrium, the electrochemical potentials of each mobile ionic species are equalized across the boundary,

$\mu_i^\mathrm{(in)} = \mu_i^\mathrm{(out)}$

Assuming ideal solutions and neglecting specific binding, the ion concentrations across the interface satisfy

$$c_i^\mathrm{(in)} = c_i^\mathrm{(out)} \exp(-z_i e \Delta \psi / k_B T)$$

where $c_i$ is the ion concentration, $z_i$ its valency, $e$ the elementary charge, $k_B T$ the thermal energy, and $\Delta \psi$ the Donnan potential (the potential difference across the interface). The system-wide electroneutrality constraint, accounting for both mobile and immobile charges, determines $\Delta \psi$ self-consistently. The resulting distribution causes counterions (opposite in sign to the fixed charge) to be enriched and coions to be depleted in the charged compartment, fundamentally altering the ionic environment.

This partitioning leads to an entropic penalty—often computed as a contribution to the free energy by considering the incomplete mixing of ions and the necessity to maintain local electroneutrality. In dense multicomponent systems, additional corrections may account for finite ion size, dielectric exclusion, Coulombic interactions, and chemically specific affinities (e.g., binding, hydration effects).

2. Analytical and Computational Models of Donnan Partitioning

A range of analytical techniques have been developed to incorporate Donnan partitioning into tractable models and free energies:

• Cell Models: Each solute particle (e.g., a protein) is embedded at the center of a symmetric “cell” representing its local environment. The partitioned ion distributions and associated potentials are solved using Poisson–Boltzmann (PB) or its linearized versions, incorporating both local electrostatics and global electroneutrality. The Donnan potential typically appears as a nonzero reference in the PB solution, and the entropy of ion partitioning is evaluated by integrating the local ionic free energy density (Mishra et al., 2013).
• Jellium Approximation: At high solute concentrations, spatial variations in potential and concentration become weak, allowing the replacement of position-dependent fields by a nearly uniform average value (“jellium”). This approximation provides closed-form expressions relating ion partitioning to the total fixed charge and accessible solvent volume, enabling efficient computation of the entropic cost and direct comparison with numerically exact solutions (Mishra et al., 2013).
• Modified Donnan Models: Corrections to the classic Donnan equation integrate finite-size effects, nonideality (activity coefficients), explicit Coulombic interactions, and spatial heterogeneity. For example, extended Donnan models of the first kind employ a non-electrostatic partitioning coefficient (constant for a given system), while second-kind models incorporate salt-dependent or geometry-dependent corrections, as in explicit evaluations of Coulombic interactions in nanopores (Wang et al., 5 Mar 2024).
• Computational Approaches: Advances include hybrid nonequilibrium molecular dynamics/grand-canonical Monte Carlo methods (e.g., H4D) to sample ion and solvent partitioning in explicit-solvent systems, with detailed charge renormalization procedures matching microscopic and mean-field behaviors even at high charge densities (Kim et al., 29 May 2024).

3. Role in Biological, Colloidal, and Soft Matter Systems

Donnan-type partitioning is central to phase equilibria, macromolecular organization, and functional responses in multicomponent systems:

• Protein Solutions and Phase Separation: In concentrated protein phases, fixed protein charges induce a Donnan potential that drives counterion enrichment and coion depletion. This modulates the electrostatic component of the free energy, influencing phase behavior, osmotic pressure, and critical phenomena such as the salt-dependent shift of the liquid–liquid critical temperature in lysozyme solutions (Mishra et al., 2013). The entropic cost of ion rearrangement is a principal ingredient in reproducing experimental thermodynamics.
• Synthetic and Biological Gels: Polyelectrolyte hydrogels, such as mucus or synthetic networks, exhibit swelling/deswelling transitions driven primarily by the Donnan potential and associated osmotic pressure (the “Donnan pressure”). The fixed charge density and its chemical modulation via binding (e.g., Ca$^{2+}$–mucin crosslinks) directly tune the magnitude of swelling, ion partitioning, and transport rates (Sircar et al., 2015). Altered Donnan partitioning resulting from disease or chemical environment thus impacts rheology and hydration states.
• Colloid–Polyelectrolyte Mixtures: In systems containing colloids and nonadsorbing polyelectrolytes, Donnan partitioning gives rise to potential-driven accumulation of polyelectrolyte near surfaces, strongly modifying depletion forces, phase transitions, and colloidal stability (Landman et al., 2021). The interplay between conformational entropy and the Donnan potential determines whether polymers accumulate or are depleted from interfacial regions.
• Compartmentalized and Phase-Separated Environments: In aqueous two-phase systems (ATPS) formed by mixtures such as polyethylene glycol (PEG)–dextran, differences in polymer charge induce a Donnan potential between coexisting phases. This potential drives cation accumulation in the more negatively charged Dex-rich phase, facilitating the selective partitioning of highly charged macromolecules (e.g., DNA) within droplets (Sakuta et al., 29 Sep 2025). The magnitude and direction of partitioning are modulated by DNA length and salt concentration, establishing Donnan partitioning as a design principle for selectivity and compartmentalization in biotechnological and protocell contexts.

4. Quantitative Expressions and Theoretical Advances

The thermodynamic and mathematical description of Donnan-type partitioning is based on extensions of free energy functionals and the explicit calculation of entropic, electrostatic, and chemical contributions:

• Salt Entropy Term: For a system containing a charged solute in a salt solution, the free energy cost of salt partitioning can be written as

$$f_\mathrm{salt} = k_B T \int [c_{+}(\mathbf{r}) \ln(c_{+}(\mathbf{r})/c_s) - c_{+}(\mathbf{r}) + c_s + c_{-}(\mathbf{r}) \ln(c_{-}(\mathbf{r})/c_s) - c_{-}(\mathbf{r}) + c_s] d^3r$$

where the Boltzmann-distributed concentrations $c_{+}(\mathbf{r})$, $c_{-}(\mathbf{r})$ are determined by the local electrostatic potential and global constraints (Mishra et al., 2013).

• Charge Neutrality Constraint: In the high-density (jellium) approximation, the Donnan potential $\phi$ is determined by imposing charge neutrality within the phase:

$q = 2 v_\mathrm{ion} c_s \sinh(e \phi / k_B T)$

with $v_\mathrm{ion}$ the effective solvent-accessible volume, $c_s$ the external salt concentration, and $q$ the local fixed charge (Mishra et al., 2013).

• Donnan Potential Difference: In systems partitioned into coexisting phases (e.g., Dex-rich and PEG-rich compartments), the Donnan potential difference follows

$$\Delta \psi = \frac{k_B T}{e} \ln \left( \frac{c_+^\mathrm{Dex}}{c_+^\mathrm{PEG}} \right)$$

governing the magnitude of ion enrichment and its effect on macromolecular localization (Sakuta et al., 29 Sep 2025).

• Free Energy Coupling with Osmotic Pressure: Recent work demonstrates that the equilibrium potential across a membrane or compartment boundary comprises both the classical Donnan term and a correction due to osmotic pressure:

$$Q = \frac{RT}{F} \ln\Bigg(\frac{C_{+1}[1 - (2c_2 - 2y^*)]}{C_{+2}[1 - (2c_1 + 2y^*)]}\Bigg)$$

explicitly coupling ion concentration and solvent activity (Chen, 2021).

5. Experimental Manifestations and Observational Challenges

Donnan-type partitioning is responsible for many experimentally observable phenomena:

• Osmotic Pressure Shifts: The Donnan effect leads to osmotic pressure differences between phases, influencing swelling in hydrogels, the formation of colloidal crystals, and equilibrium in protein solutions. In protein-rich phases, despite theoretical predictions of significant salt partitioning, measured salt concentrations may appear unchanged due to the large protein volume fraction, causing an effective cancellation in measurements (Mishra et al., 2013).
• Phase Localization and Selectivity: In ATPS and complex coacervates, Donnan partitioning mediates the selective localization of biomolecules (e.g., nucleic acids), which is not accurately predicted by depletion interactions alone. The magnitude of cation enrichment, the ability to screen or neutralize polyelectrolyte charges, and the influence of polymer length and salt concentration are all key determinants (Sakuta et al., 29 Sep 2025).
• Ion-Specific Effects and Activity Corrections: In narrow pores or highly charged domains, deviations from ideal Donnan partitioning arise due to activity coefficients, Born solvation effects, dielectric exclusion, and finite size/volume exclusion interactions—each of which may be incorporated into extended Donnan models or variational approaches (Wang et al., 5 Mar 2024, Hennequin et al., 2021).
• Experimental Challenges: Detecting the theoretical predictions of Donnan partitioning can be challenging, especially in systems with substantial excluded volume (e.g., concentrated protein solutions) or when the experimental probes are insensitive to the local ionic environment. Crowding, spatial inhomogeneity, and limitations in resolving counterion distribution add further complexity (Mishra et al., 2013).

6. Applied and Emerging Contexts

The implications of Donnan-type ion partitioning extend across multiple fields:

• Membrane and Porous Media: The design and performance of ion-exchange membranes, nanofiltration devices, and capacitive deionization electrodes depend critically on Donnan partitioning to establish selectivity, transport bias, and energy efficiency. Models that incorporate ion activity and non-electrostatic effects enable accurate predictions of partition coefficients and facilitate the optimization of selective separation technologies (Biesheuvel et al., 2014, Sin, 2022, Wang et al., 5 Mar 2024).
• Soft and Biological Materials: Swelling hydrogels, mucous layers in biological tissues, and microcapsule-based diagnostic sensors exploit Donnan partitioning to modulate physical properties (e.g., mesh size, pH, mechanical response) in response to environmental ionic changes or chemical stimuli (Sircar et al., 2015, Tang et al., 2015).
• Confined and Nanostructured Systems: In confined geometries (nanopores, slit-pores, microfluidic channels), Donnan partitioning competes with other energy barriers (solvation, dielectric mismatch, excluded volume). Quantitative agreement between mean-field approaches and molecular simulations can be achieved over extended parameter ranges using effective charge renormalization and explicit inclusion of activity corrections (Kim et al., 29 May 2024).

7. Limitations and Theoretical Developments

While Donnan-type partitioning provides a robust framework, key limitations persist:

• standard treatments often neglect solvent effects, coupling with osmotic pressure, or non-ideal activity corrections, leading to discrepancies at high concentrations or in crowded environments (Chen, 2021);
• for highly charged or multivalent systems, or where explicit structural features (e.g., spatial inhomogeneity, binding site saturation) or specific interactions dominate, Donnan models must be extended or embedded within broader electrostatic and statistical mechanical frameworks (e.g., Poisson–Boltzmann–Langmuir models, field-theoretic variational approaches) (Nikam et al., 2020, Podgornik, 2020).

Continued methodological advances—including hybrid simulation methods, refined free energy expressions, and models that unify electrostatics, chemistry, and molecular structure—are improving quantitative predictions of ion partitioning and enabling new control strategies for soft functional materials and biological systems.

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Donnan-type ion partitioning is a central mechanism underlying selective ion distribution, phase behavior, and functional response in a broad range of complex soft matter and biological environments. The interplay between fixed charges, mobile ions, osmotic effects, and nonideal solution thermodynamics determines both the magnitude of the partitioning and its experimental manifestations, guiding both fundamental understanding and practical applications across disciplines (Mishra et al., 2013, Sircar et al., 2015, Wang et al., 5 Mar 2024, Sakuta et al., 29 Sep 2025).