Donnan exclusion occurs when ions partition into a material that contains high density of charge ^[10][65]. NF membrane is mainly governed by Donnan exclusion and size exclusion ^[11][29]. Emamjomeh et al. ^[12][66] also determined that the separation mechanism in NF membrane mainly involved the size of the molecules and the electrical response between the surface of the membrane and the ions in the feed. Furthermore, Donnan exclusion occurs due to electrostatic interactions between ionic solutes and the membrane matrix’s fixed electric charges ^[13][17]. Based on these claims from various researchers, one can conclude that NF membrane rejection is not solely determined by the size exclusion; it is also determined by Donnan exclusion. Figure 25 illustrates how Donnan exclusion works for both negatively and positively charged membranes. Negatively charged membranes reject negative ions (yellow ball) but allow positive ions (red ball) to pass through the membrane by transport of various solutes and entraps some of the ions in between the channel pore of the membrane. Solutes are transported by diffusion (movement of solute down a concentration gradient), by convection (solute transported by bulk fluid motion), and by electromigration (ion movement due to the membrane potential gradient ^[9][64]. The same law is depicted in reverse in Figure 2b5b.

To investigate the role of Donnan exclusion in the removal of arsenic by nanofiltration, Seidel et al. ^[14][43] evaluated surface charge, pore size, and rejection behaviour for salt solutions such as sodium chloride (NaCl), calcium chloride (CaCl₂), and sodium sulphate (Na₂SO₄). They determined that the order of rejection rate for these salts is Na₂SO₄ > NaCl > CaCl₂. These findings are consistent with Donnan exclusion. According to Donnan exclusion principle, the rejection of ions increases with co-ions that have higher charges and decreases with counter-ions that have higher charges. Na₂SO₄ has the highest rejection rate because it contains co-ions with the greatest negative charge, which is sulphate (SO₄²⁻) ions. Simultaneously, CaCl₂ has the lowest rejection since it has higher charge of counterions which is calcium (Ca²⁺). The rejection trends shown in their study demonstrate that the ion separation is governed by size exclusion and Donnan exclusion, as the observed trends do not correspond to the size of the hydrated ions.

Donnan exclusion is also more pronounced on divalent ions ^[15][16][67,68]. Nicolini and colleagues evaluated the performance of NF membrane by measuring the saline rejection. The saline rejection for the NF membrane decreases in the following sequence: Na₂SO₄ > K₂SO₄ > CaSO₄ > MgSO₄ > NaCl. The saline rejection sequence is a result of anionic electrostatic repulsion and the preferred attraction of divalent cations.

Furthermore, the performance of fabricated polyamide (PA) NF membranes via interfacial polymerisation using m-xylylenediamine (m-XDA) and polyethyleneimine (PEI) as aqueous monomers were evaluated ^[16][68]. Liu and co-researchers controlled the concentration ratio of m-XDA and PEI from 1:0, 4:1, 3:2, 2:3, and 1:4 to 0:1 and named them M1, M2, M3, M4, M5, and M6. The researchers also investigated the selectivity of Na⁺ and Mg²⁺. The results showed that the addition of PEI to aqueous solution increased the rejection of NF membrane to magnesium chloride (MgCl₂) and magnesium sulphate (MgSO₄), but not to NaCl or Na₂SO₄. This is because both MgSO₄ and MgCl₂ have divalent cations (Mg²⁺), while NaCl and Na₂SO₄ only have monovalent cations (Na⁺). Therefore, Mg²⁺ rejection is determined by Donnan exclusion, whereas monovalent Na+ rejection is determined by steric hindrance (since Na⁺ has weak electrostatic repulsion).

However, the Donnan exclusion mechanism is not sufficiently explained only by the rejection of ionic solutes. A model is needed to describe the rejection of ions by NF membranes. Consequently, a model to calculate ion rejection in single and mixed electrolyte solutions in RO and NF membranes has been proposed ^[17][69]. The researchers designed the model based on the extended Nernst–Plank equation and combined it with Teorell–Meyer–Sievers (TMS) model ^[18][70]. Bowen and Mukhtar proposed in 1996 that an ideal quantitative model for membrane processes should consider the physical properties of the membrane to predict separation behaviour under specified conditions ^[19][71]. The physical properties of the membranes that have been utilized are effective membrane thickness and charge density. They used a hybrid model (HM), which was initially developed for glomerular filtration and incorporated the extended Nernst–Planck equation and Donnan partitioning between the solution and membrane ^[20][72]. Next, in 1997, a new model called the Donnan steric pore model (DSPM) based on the hybrid model was developed by Bowen et al. ^[21][56]. This model has been widely used and evaluated because it describes Donnan exclusion sufficiently. For example, Donnan exclusion was used to predict the movement of ions (containing Ca²⁺, K⁺, and Cl⁻) through a negatively charged membrane.

Schaep et al. evaluated the DSPM model using four commercial NF membranes to investigate the influences of molecular size, polarity, and charge on the retention of organic molecules. The researchers also evaluated the selected membrane with single-species salt solutions such as NaCl, Na₂SO₄, MgCl₂, and MgSO₄ ^[22][47]. The results showed that the membrane charge density was inconsistent and depended solely on salt type and concentration. Furthermore, due to divalent cations in the feed solution, the agreement between DSPM fitting and actual results was less than adequate. The model predicted an unreasonably high membrane charge density, and the presence of divalent cations changed the sign of the membrane charge density from negative to positive ^[23][73]. As a result, dielectric exclusion (DE) is used here as a secondary partitioning effect at the membrane-external solution interfaces, which will be discussed in greater detail in the following section.

As mentioned in a previous section, it has been determined that there is an additional separation mechanism due to the unrealistic magnitude of membrane charge density predicted by DSPM. Therefore, Yaroshcuk et al. ^[24][74] introduced dielectric exclusion during the separation process by providing an in-depth analysis of this exclusion mechanism. The researchers stated that dielectric exclusion occurs when an ion interacts with the bound electrical charges generated by the ion at the interface of two materials with different dielectric constants: in this case, the membrane matrix and the solvent. It was determined that the ion polarizes the two media in accordance with their relative dielectric constants, forming a polarisation charge distribution at the discontinuity surface. This phenomenon is called as production of image forces and is illustrated in Figure 3. This figure shows that if the membrane's dielectric constant is less than the solvent's, this image force will always be repulsive for both anions and cations. 

In addition, Bowen and Welfoot widened the scope of the DSPM by incorporating more complex phenomena into their latest model ^[25][76]. Their study compared the effects of the solvent viscosity and the dielectric constant inside the pore. The results showed that variation of the solvent was insignificant, but the variation of the dielectric constant inside the pore was significant. It is believed that when the dielectric constant inside the pores decreases, it becomes smaller than the dielectric exclusion of bulk solution. Therefore, changes of the dielectric constant in the pores induced an excess energy barrier for ion solvation, preventing charged ions from partitioning into the pores. The effect was named dielectric exclusion (DE) and a Born equation was proposed to quantify it (Born effect). The researchers discovered that the dielectric effect reduced the charge density (extracted from experimental data) to a more reasonable number and improved model fitting when divalent salts were used. By incorporating the dielectric effect into the DSPM, this model is referred to as the Donnan steric pore model with dielectric exclusion (DSPM–DE) model. More recently, this model has been used widely by various researchers ^{[26][27][28][29]}[77,78,79,80]. One of the highlighted studies explored the effect of dielectric constant in NF membrane ^[30][81]. The results showed that when the dielectric constant is decreased, the rejection performance is increased. As depicted in Figure 49, the dielectric constant was lowered from 80 to 60 and the rejection performance was increased as dielectric constant decreased. Therefore, it can be concluded that the dielectric constant holds an important role in the rejection performance of NF membrane.

However, in practice, the DSPM–DE model is inconsistent because dielectric effects are based on slit-like pores, while the transport model is based on cylindrical pores. Later, Szymczyk and Fievet (2005) developed a steric, electric, and dielectric exclusion (SEDE) model. The researchers utilized both effects of dielectric exclusion mechanisms, which are the Born effect and image forces contribution ^[31][82]. The researchers stated that the SEDE model can be used to predict transport via cylindrical or slit-shaped pores. Later that year, Bouranene et al. ^[32][83] studied whether dielectric exclusion should be represented by the Born effect, the image force, or both. The modelling findings demonstrate that the experimental rejection rates are not adequately characterised when the two types of dielectric exclusion mechanisms are included. Hence, the researchers stated that the DSPM–DE developed by Bowen et al. (2002) was adequate for practical applications. Furthermore, as mentioned by Wang, researchers have made many modifications for various scenarios. However, the original DSPM–DE framework remains the most often utilized due to its capabilities of fitting experimental data in most cases ^[23][73].

As stated by Yaroshcuk, the hydration mechanism is not well understood ^[13][17]. It has been proposed that a loss of water dissolving capability of the membrane is likely caused by the changes to the dielectric properties of that particular membrane. The researcher also mentioned that multiple-charged ions should be considerably better excluded from the pores than single-charged ions which are sodium (Na⁺) and potassium ion (K⁺) ^[33][84]. Chen et al. concluded that the rejection of Na⁺ is higher even though K⁺ has a smaller hydration radius than Na⁺. This is because the diffusion coefficient of K⁺ calculated using Stokes’ law and Zwanzig’s theory is larger than that of Na⁺. Furthermore, the relationship between water molecules and hydrated cations passing through the graphene oxide (GO) membrane in a temperature-assisted system was studied ^[34][85]. Cha-umpong and others studied temperature-assisted systems as the driving force instead of pressure-assisted systems since theoretically, temperature as the driving force should not compromise salt rejection.

The researchers discussed why the rejection of Ca²⁺ is stronger than Mg²⁺ in terms of hydration radius. The first hydration radius of Ca²⁺ is bigger than that of Mg²⁺. The unique feature of divalent ions is their structure, as two layers of hydration shell surround the ions. The first layer is dense while the second layer (outer layer) is elastic, and this layer weakly connects with nuclear ions. If ions have a radius larger than the pore size, the ions can still pass through the pore because the water molecules adjust its outer hydration shell so that the inner hydration shell can pass through the pore. Figure 5 explains the mechanism of ions passing through the pore despite having a larger radius than the pore. Figure 5 displays several water molecules from the outer hydration shell being replaced and resulting the first hydration shell can easily pass through the pore. After passing through the pore, the second hydration shell will attract some other water molecules in the permeated region to form a new hydration shell.