The aperture sieving effect, also known as the steric hindrance effect, refers to the phenomenon where substances larger than the membrane pores are more easily selectively rejected when ions pass through the NF membrane pores, while smaller solutes can penetrate the membrane, thereby achieving solute separation. It provides an important fundamental principle for the pore model and plays a crucial role in size-based sieving and salt rejection, especially for neutral solutes and uncharged membranes, where larger solutes (or smaller membrane pore sizes) generally correspond to higher rejection rates. The aperture sieving effect and the Donnan effect are key separation mechanisms for Li⁺/Mg²⁺ sieving^[32-33], working synergistically during ion sieving. However, while the Donnan effect enhances Mg²⁺ rejection, it simultaneously increases the repulsion of Li⁺ ions, making it difficult to significantly improve Li⁺/Mg²⁺ selectivity. Therefore, the challenge in achieving high-selectivity separation lies in designing NF membrane pores on demand based on the size differences between Li⁺ and Mg²⁺ ions. However, in traditional interfacial polymerization (IP) processes, the instantaneous nature of the condensation reaction ultimately leads to an uneven distribution and chemical instability of the polyamide (PA) layer^[34-35]. Thus, more effective methods are required to regulate the IP process and enhance the separation selectivity of the PA layer.

Since the traditional NF membrane preparation process makes it difficult to precisely control the membrane pore structure, many researchers have attempted various approaches, such as altering the diffusion rate of PIP monomer from the aqueous phase to the organic phase to adjust the structure and pore size of the PA membrane. Wang et al. effectively controlled the IP process between PIP and TMC by using a weakly polar organic solvent, ethyl formate (EF) [37]. The involvement of EF significantly reduces the interfacial tension, thereby promoting the rapid diffusion of amine monomers from the aqueous to the organic phase, which helps construct a PA membrane with higher crosslinking degree. As shown in Figure 3b and Figure 3c, the optimally modified membrane (NF-10) exhibited a significantly reduced pore size (~0.195 nm), demonstrating precise solute separation while maintaining ideal permeability (18.4±0.9 L·m-2·h-1·bar-1). Chen et al. fabricated a positively charged intermediate layer between the polysulfone (PSF) substrate and the PA layer through a co-deposition reaction of polyphenol and PEI [38]. The introduction of this intermediate layer flexibly regulated the diffusion rate of PIP, and by controlling the physicochemical properties (thickness and charge) of the intermediate layer, NF membranes with smaller average pore sizes and narrower pore size distributions were obtained. Under the synergistic effect of size sieving and Donnan exclusion, the rejection of Mg2+ was enhanced. As shown in Figure 3e and Figure 3f, the optimally performing NF membrane (TFC-9) achieved a high Li+/Mg2+ separation factor of up to 88.6 while maintaining high permeability (22.5 L·m-2·h-1·bar-1), outperforming most NF membranes reported in the literature. These findings indicate that by regulating the diffusion of PIP-based monomers, the PA layer structure can be altered, enabling the preparation of ideal NF membranes primarily governed by the size sieving effect. However, precisely controlling the impact of aqueous-phase monomer diffusion on the PA layer (promoting diffusion to form a dense separation layer or slowing diffusion to enhance flux), and overcoming the “trade-off” phenomenon between selectivity and permeability to fabricate high-performance NF membranes remains a significant challenge.

Alkali metal ions typically exist in aqueous solutions in the form of hydrated ions. When passing through channels with sub-nanometer pores, the hydration shells of ions always undergo energy-consuming dehydration or distortion^[39-41]. The difficulty of dehydration is usually determined by the strength of hydration around the ions, which is commonly expressed in terms of hydration energy. Table 1 lists the radii, hydrated ion radii, and hydration energies of common ions. As shown in Table 1, the hydration energy of Mg²⁺ is greater than that of Li⁺, indicating a higher dehydration difficulty. Experimental results from Zhang et al.^[32] revealed that during the dehydration process, Li⁺ experiences a greater reduction in hydrogen bonding among surrounding water molecules compared to Mg²⁺, confirming that Li⁺ more readily undergoes dehydration when entering the membrane. Additionally, multiple researchers have emphasized the important role of dynamic changes in hydration structures in explaining ion selectivity^[42-43]. Liu et al.^[43] prepared a quaternary ammonium (QA)-functionalized Troger's base (TB) microporous polymer membrane with sub-nanometer channels (window sizes ranging from 5.89 to 6.54 Å) by introducing positively charged QA groups into TB. The incorporation of QA groups promoted the formation of a microphase-separated structure and provided channels for the transport of dehydrated ions. This structure prevents hydrated ions from passing directly, thereby promoting ion dehydration. The differences in hydration energy between monovalent and divalent ions lead to distinct dynamic changes in their hydration structures during passage, ultimately affecting the migration rates of ions within the channels.

Due to the significant enthalpy change (changes in ion chemical bonds with surrounding molecules) and entropy change (changes in ion spatial structure) exhibited at the molecular level during ion dehydration, the transition state theory (TST) has been widely used to investigate ion dehydration phenomena in membranes^[44]. Pavluchkov et al.^[45] applied TST to measure the activation enthalpy and entropy of ions to explain the ion selectivity mechanism in polyamide (PA) membranes, indicating that ion selectivity in a simple diffusion system is enthalpy-driven and primarily dependent on the hydration enthalpy of ions. Furthermore, they quantified the intrinsic permeability of a set of cations and anions, as well as their corresponding Eyring enthalpy and activation entropy in PA membranes, aiming to better understand ion dehydration phenomena. Shefer et al.^[44] clarified the contributions of enthalpy changes (e.g., ion dehydration) and entropy changes (e.g., size sieving) to the selectivity between monovalent and divalent ions under varying pH conditions, offering new insights into the role of membrane surface charge and pore size in inducing ion dehydration under constrained conditions. However, it is important to note that most dehydration mechanisms are currently discussed through theoretical simulations, and studies focusing on regulating channels suitable for mono-/divalent ion dehydration via interfacial polymerization (IP) remain limited. Moreover, few studies have specifically emphasized and thoroughly explored the significance of the dehydration effect during the separation process.

The Donnan effect explains the electrostatic interaction between charged solutes and charged membranes. Ions with opposite charges to the membrane are attracted to the membrane through electrostatic attraction, while ions with the same charge are rejected due to electrostatic repulsion. Furthermore, the strength of the Donnan effect is influenced by the ionic environment and ion concentration of the solution. Choi et al.^[48] indicated that ions with the same charge as the membrane and higher valence exhibit stronger electrostatic repulsion compared to those with lower valence, making them more likely to be rejected. The introduction of the Donnan effect provides a crucial theoretical foundation for the charge model, which, based on in-depth analysis and quantitative description of ion behavior on both sides of the membrane caused by the Donnan effect, enables a more accurate explanation and prediction of the membrane's charge mechanisms during the separation process.

Inspired by the Donnan effect, researchers have attempted to achieve selective separation of specific ions by altering membrane charges, such as designing positively charged NF membranes for Li⁺/Mg²⁺ separation^[49]. Currently, common methods for changing membrane surface charges include selecting reactive monomers with high amine group density^[50-51], surface modification^[52], and altering reaction conditions^[53]. Foo et al.^[54] utilized PA membranes treated with acid, enhancing the Li⁺/Mg²⁺ selectivity by 13 times (Fig. 4a). This improved selectivity is attributed to the protonation of carboxyl groups under low pH conditions, which amplifies the Donnan potential, generating a positive potential on the membrane surface. As a result, the repulsion against Mg²⁺ ions becomes significantly stronger than that against Li⁺, thereby enhancing the membrane's selectivity. Moreover, PEI contains abundant amine groups and has been widely used in the fabrication of NF membranes. Liu et al.^[55] first prepared a polyethyleneimine-based PA membrane (PEI-TMC). Subsequently, through surface modification, they reacted a newly synthesized electrolyte monomer containing quadruple imidazolium salts and hydroxyl groups (QTHIM) with residual acyl chloride groups on the membrane surface, producing a high permeate flux NF membrane (PEI-TMC-QTHIM) (Fig. 4b). As shown in Fig. 4c and Fig. 4d, the modified membrane exhibits a loose structure and enhanced positive charges, demonstrating high water permeate flux and good MgCl₂ rejection. Wang et al.^[56] performed surface modification using a BPEI/ethanol solution, reacting it with residual acyl chloride groups on the surface of the original hollow fiber (HF) NF membrane to form a dual-charged layer structure (Fig. 4e). Due to the combined effects of Donnan repulsion and steric hindrance, the prepared HF membrane demonstrates excellent selectivity for divalent cations, while monovalent cations passing through the modified membrane surface experience an attractive force from the negatively charged PA layer, thus promoting greater penetration of monovalent cations (Fig. 4f).

Dielectric repulsion is an effect caused by the interaction between ions and bound charges induced by the ions at the interface between media with different dielectric constants (especially membrane matrices and solvents)^[57-58]. It can be divided into two parts: the ion solvation mechanism of dielectric repulsion and image forces^[59]. The former arises due to an energy barrier for ion solvation into the channel caused by the orientation of water molecules within the membrane matrix. The latter occurs because when an ion resides in a medium with a higher dielectric constant (water or other polar solvent), it induces charges of the same sign at the interface with a medium having a lower dielectric constant (polymer or inorganic membrane matrix), hence the name. Moreover, the polarization charge is proportional to the square of the ion charge, which is why both cations and anions are excluded from the pores^[60].

It is worth noting that dielectric exclusion differs from Donnan exclusion. Donnan exclusion repels co-ions from the pores and attracts counterions inside the pores, while dielectric exclusion is independent of ion charge. Dielectric exclusion always hinders ion transport when the dielectric constant of the solution inside the pore is smaller than that of the external solution and larger than that of the membrane matrix. The dielectric exclusion effect is rarely used alone and usually works synergistically with the Donnan effect^[61] to enhance the sieving capability for mono- and divalent ions. Zheng et al.^[62] incorporated carbon dots with cationic amine groups (PEI-CDs) into the PA selective layer of thin-film nanocomposite (TFN) membranes, generating charged nanogaps between the carbon dots and the surrounding PA network. As shown in Fig. 5a, these nanogaps surrounding the carbon dots effectively reduce the resistance of the dense selective layer to water permeation by creating highly permeable pathways for rapid water molecule transport. Moreover, the ionizable groups within the nanogaps can reduce the dielectric constant of the membrane channels, thereby increasing the energy barrier for ion solvation into the nanopores. Therefore, under the combined effects of the Donnan effect and dielectric exclusion, the prepared TFN membrane exhibits higher rejection of divalent ions and improved fouling resistance (Fig. 5b). This study will aid in the design and development of carbon dot-based TFN membranes with charged nanogaps for efficient mono-/divalent ion separation.

Hydrated ions interact with groups on the pore wall when undergoing dehydration through the channel, specifically manifested as the pore-wall groups entering the ion's hydration shell and participating in the hydration process^[63], thereby reducing the energy barrier for ions to enter the nanopore. These interactions can compensate for the loss of ion hydration (deformation), hence referred to as the compensation effect. Dissolved ions typically have two hydration shells, with the second hydration shell playing a decisive role in ion transmembrane permeation^[32,64]. Research by Zhu et al.^[65] indicates that the determining factor for Li⁺/Mg²⁺ separation depends on whether the modified groups can fully replace the role of water molecules stripped from the second ion hydration shell (see Figure 6a). Xu et al.^[66] grafted four different functional groups with the same chain length but distinct properties onto the pore walls of covalent organic frameworks (COFs)—hydrophobic group —CH₃, hydrophilic group —NH₂, moderately charged group —NH₃⁺-50%, and strongly positively charged group —NH₃⁺-100%—and found that a balanced proportion of hydrophilic and positively charged groups on the pore wall ensures an appropriate compensation effect within the nanopore, allowing for accelerated Li⁺ permeation while simultaneously repelling Mg²⁺ (see Figure 6b). This demonstrates that the pore-wall groups' compensatory effect on ion hydration plays a crucial role in Li⁺/Mg²⁺ separation. However, designing pores with compensatory interactions to selectively stabilize target solutes remains highly challenging; therefore, studies utilizing the compensation effect for selective separation of monovalent and divalent ions are still very limited.

Adsorption on the membrane can be influenced by hydrogen bonds, and hydrophobic solutes tend to adsorb onto the membrane surface or within the membrane structure. Braeken et al.^[67] investigated the effect of solute hydrophobicity on rejection rates in aqueous solutions and pointed out that solutes with high hydrophobicity typically exhibit low rejection rates, whereas hydrophilic solutes show the opposite behavior. Li et al.^[68] demonstrated that perfluorinated compounds (PFCs) can interact with the NF membrane surface, leading to adsorption onto the membrane and affecting solute rejection. Hydrophobic adsorption is currently widely used to explain the rejection of organic micropollutants, and the solute-membrane interactions described in this theory help quantify the real mechanisms by which solute rejection efficiency decreases or increases under different conditions. Some researchers^[69] have indicated that different ions can compete for adsorption with membrane functional groups, thereby influencing the membrane surface sites. This provides a theoretical basis for using NF to remove monovalent and divalent ions.

In conclusion, research on the separation mechanism of NF membranes is crucial for improving the sieving efficiency of Li⁺/Mg²⁺. The interaction of pore size sieving, dehydration effects, Donnan effects, and dielectric repulsion provides theoretical support for the construction of NF membrane models. Future studies could further explore the relationship between the structure and performance of NF membranes, optimize membrane preparation processes and modification methods based on separation mechanisms, and thereby enhance the separation efficiency and stability of NF membranes in lithium-magnesium sieving, offering more effective technical support for the extraction and utilization of lithium resources.

The non-equilibrium thermodynamics model, also known as the irreversible thermodynamics model, assumes that the internal structure of the NF membrane is unknown and treats it as a "black box." This model is commonly used to describe the relationship between driving forces and fluxes. Kedem and Katchalsky^[70] developed the first membrane-based irreversible thermodynamic model for single solute non-electrolyte solutions. Subsequently, several researchers^[71-73] applied this model to charged NF membranes for solute rejection.

The non-equilibrium thermodynamic model indicates that the solute flux is the sum of the diffusive flux caused by the pressure gradient across the membrane and the convective flux caused by the concentration difference across the membrane. Therefore, changing the pressure and solute concentration that drive NF membrane filtration can both affect the membrane's physicochemical properties. Murthy et al.^[72] applied the non-equilibrium thermodynamic model to explain the solute rejection by charged NF membranes, and their experimental results showed that both the permeability and rejection rate increased with increasing applied pressure. Wang et al.^[56] used the same model to explain their observation that the water permeability of both LiCl and MgCl₂ solutions decreased with increasing feed concentration. This is because the presence of non-permeating solutes alters the chemical potential, thereby changing the driving force^[70]. In addition, this model gives rise to the well-known Spiegler-Kedem (SK) equation^[74], which is widely used in other structural models to describe the relationship between rejection rate and flux.

where F = \mathrm{e} \mathrm{x} \mathrm{p} \left[-J_{\mathrm{v}}(1-\sigma )/P\right], P is the solute permeability coefficient, m/s·Pa; C_p is the solute concentration in the permeate, mol/L; C_m is the solute concentration at the feed-membrane interface, mol/L; \sigma is the membrane reflection coefficient. When J_v approaches infinity, σ=Rmax. The membrane parameters such as σ, P, and L_p can be determined from experimental data.

However, it is worth noting that the necessary condition for establishing this model is the neglect of the internal structure of the NF membrane, making it impossible to analyze the solute transport process within the membrane from a physicochemical perspective. Therefore, combining the non-equilibrium thermodynamic model with a structural model to establish the interrelationship between membrane structure and membrane transport parameters is of great significance.

Based on non-equilibrium thermodynamics and the Stokes-Maxwell friction model (which describes the friction and resistance during substance transfer within the membrane), Nakao and Kimura^[75] developed a small pore model (SHP model) for estimating membrane structural parameters. This model assumes that the membrane consists of uniform pores much smaller than the membrane thickness, and considers the influence of friction and spatial hindrance effects on the transport of spherical ions through cylindrical pores^[76], providing calculation formulas for membrane-related coefficients. The membrane pore size distribution can be determined by the relationship between solute rejection and the radius of neutral solute molecules^[77], and is represented by the molecular weight cutoff (MWCO)^[78-79], due to the strong correlation between the Stokes radius of molecules and their molecular weight (MW) (as shown in Figure 8). The small pore model indicates that the solute radius is obtained according to the Stokes-Einstein equation (Equation 2):

where: K is the Boltzmann constant, J/K; T is the temperature, K; μ is the solution viscosity, Pa·s; D_s is the solute diffusion coefficient of neutral molecules or the generalized diffusion coefficient of 1-1 type electrolytes, defined as D_s=2D₁D₂/(D₁+D₂), m²·s^-1, where D₁ and D₂ represent the solute diffusion coefficients of two different solutes.

The aperture sieving effect is crucial, hence it is also very important to clarify the relationship between particle size and membrane pore size. Nair et al.^[76] evaluated the hypothetical pore radius of six membranes through permeation experiments with charged ions by using the pore model. Seman et al.^[80] obtained membrane parameters such as membrane reflection coefficient and solute permeability through the SK equation and SHP model, and conducted experiments using NaCl solutions to determine the membrane's ion retention. The pore model can adequately describe the separation mechanism of NF membranes for non-electrolytes and is suitable for structural evaluation of NF membranes. However, since it only considers the sieving effect and not the charge effect, it is more applicable to neutral solute systems.

The charge models can be divided into the fixed-charge model and the space charge model according to the assumptions about the charge distribution in the membrane.

The fixed charge model was proposed by Teorell, Meyer, and Sievers in 1935-1936, and is also known as the TMS model^[81]. This model assumes the membrane to be in a gel phase with uniformly distributed charges, while neglecting parameters that affect the solute separation process, such as membrane pore size^[82-83]. As shown in Figure 9a, the TMS model considers the membrane potential as the sum of the Donnan potential occurring at the membrane-solution interface and the diffusion potential within the membrane^[84], which can be used to predict the theoretical membrane potential and surface charge density. It plays an important role in studying the practical mechanisms of NF membranes applied in ion sieving and improving membrane filtration efficiency. Romero et al.^[85] used the TMS model to electrochemically characterize charged membranes with different materials and structures, determining the effective concentration of fixed charges in the membrane and the ion transport numbers by analyzing membrane potential values, thereby evaluating the ion selectivity of the membrane. Zehra et al.^[86] found that the experimentally obtained membrane potentials were consistent with the predictions of the TMS model. Furthermore, by plotting a set of theoretical and experimental potential curves against the negative logarithm (-logC₂) of the NaCl electrolyte solution concentration, they determined the actual fixed charge density D of the membrane phase from the coincidence points between the experimental and theoretical curves (Figure 9b), confirming that the TMS model can accurately describe ion transport phenomena within the membrane and providing a reliable theoretical basis for evaluating membrane performance.

The space charge model assumes that the potential and ion concentrations in the membrane pore capillaries are radially distributed. Its fundamental equations mainly consist of the Poisson-Boltzmann equation, which characterizes the relationship between ion concentration and electric potential, the Nernst-Planck equation, which describes ion transport, and the Navier-Stokes equation for solution volumetric flow rate^[87-88]. However, it is worth noting that although this model is more accurate, its computational solution process is overly complex and cumbersome; therefore, the TMS model is more commonly used in practical applications.

Wang et al.^[89] proposed a new model, namely the electrostatic and steric hindrance (ES) model, by considering electrostatic and pore sieving effects to describe the transport phenomena of charged solutes through charged porous membranes. The ES model assumes that the NF membrane is a bundle of uniform charged tubes. The structural parameters affecting membrane separation performance include membrane pore size, the ratio of porosity to membrane thickness, and the total charge on the membrane surface. Ding et al.^[90] simulated the structural parameters of modified PES composite NF membranes and pure NF membranes. Results indicate that structural parameters of the membrane such as fixed charge density predicted by the ES model, as well as its prediction of retention and separation performance of amino acids under identical conditions, are superior to those predicted by the previous two models across various conditions. This is because the ES model combines features of both the pore model and the TMS model, thereby better simulating the separation mechanism of NF membranes^[90].

Based on the Hybrid Model (HM) proposed in 1996^[91], Bowen et al.^[92-93] considered the steric exclusion at the membrane interface and proposed the Donnan-steric pore model (DSPM) to predict the concentration and flux of permeates. Derived from the extended Nernst-Planck equation, this model describes the diffusion and migration of ions across the membrane, revealing the mass transfer mechanisms in two- or three-component electrolyte solutions^[94]. Furthermore, the DSPM model can characterize the membrane separation performance using three key parameters: pore radius (r_p), equivalent membrane thickness (Δx/A_k), and effective volumetric charge density within the membrane pores (X_d)^[95-96]. Yang et al.^[97] applied a simplified version of the DSPM model for process prediction in the Mg²⁺/Li⁺ system and conducted a concise evaluation of its sieving performance. As shown in Fig. 10a, the DSPM model fitting curve shows good consistency with the experimental data, indicating its applicability for extended predictions and subsequent experimental analyses. Dutta et al.^[98] conducted streaming potential measurements in binary electrolyte mixtures containing symmetric (NaCl) and asymmetric electrolytes (Na₂SO₄) and plotted the membrane pore charge density values estimated from the DSPM model using experimental data on salt rejection and permeate flux. As observed in Fig. 10b, except for a few data points, the calculated X_d values based on the empirical equation fall within ±20% of those calculated using the DSPM model. This indicates that the proposed correlation can predict X_d values quite accurately for any NaCl/Na₂SO₄ ratio, pH, and total salt concentration, making it a very useful tool for calculating X_d without requiring extensive experiments. It should be noted, however, that although both this model and the ES model are based on studies of pore sieving and Donnan effects, the DSPM model more comprehensively considers the effects of charge and steric hindrance on ion partitioning at the membrane surface, thereby offering a broader range of applications compared to the ES model.

The Donnan-Steric Pore Model with Dielectric Exclusion (DSPM-DE) was proposed by Bowen and Welfoot on the basis of the DSPM model^[99-100], taking into account the influence of dielectric exclusion effects on ion rejection. This model employs the Nernst-Planck equation to describe solute transport through the membrane, thereby providing information on the various transport mechanisms within the membrane, namely diffusion (movement of solutes along the concentration gradient), convection (transport of solutes by bulk fluid motion), and electromigration (ion movement caused by the membrane potential gradient)^[101-102]. According to the above theory, Figure 11 illustrates the transport of species through the membrane's active layer, along with qualitative concentration (c), potential distribution (ψ), and solvent flux (J_v). It can be observed that within the membrane, the driving forces generated by the diffusive flux and the potential gradient cause electromigration of charged ions, leading to a gradual decrease in concentration from the feed side to the permeate side, forming a concentration gradient.

This model has been widely applied in the study of ion sieving by NF membranes since its proposal and has shown good performance in fitting data for feed solutions with single-electrolyte and mixed-electrolyte systems. Cevallos-Cueva et al.^[103] established a connection between PIP and TMC concentrations and the ion transport and rejection mechanisms responsible for the membrane's nitrate separation capability using a new fitting scheme with the DSPM-DE model, successfully simulating the Na⁺/NO₃^-/SO₄^2- system and revealing the potential of NF membranes with a PA selective layer for effectively separating nitrate/sulfate. Roy et al.^[101] used the DSPM-DE model to simulate the rejection of ions such as Na⁺, Mg²⁺, and Ca²⁺ by NF membranes at different temperatures. Their results indicated that the rejection of monovalent ions significantly decreases at higher temperatures, while that of divalent ions remains relatively unchanged. This finding provides insights for adjusting temperature to enhance the separation of mono- and divalent ions.

Since the composition of practical multi-component inorganic salt solutions is more complex, and the aforementioned models are mostly used for simulating separations involving three or fewer ions, Wang et al.^[104-105] established two models based on extensive experiments to evaluate the separation performance of NF membranes in mixed salt solutions (with more than three ions, n kinds of cations and m kinds of anions). In these two models, ion permeability is applied to express the separation performance of the NF membrane in mixed salt solutions, which is related to the total concentration, equivalent fraction, and type of each ion in the mixed salt solution. Based on the aforementioned studies, Bi et al.^[106] introduced a modified equation through extensive experiments to revise the adjustment coefficient, establishing an improved semi-empirical model to evaluate the separation performance of NF membranes in salt lake brine. For most ions, the calculated values deviate from the experimental data by less than 20%, with deviations for K⁺ and Li⁺ being less than 15%. However, larger deviations occur for Cl⁻ under high-concentration conditions, indicating further optimization is still required. Since semi-empirical models are derived based on extensive experiments and validations conducted by researchers, their parameters lack specific physical meanings. Therefore, they currently have no widespread application or in-depth research.

Therefore, in the field of NF targeting mono-/divalent ion sieving, various models play significant roles. These models are interrelated and complementary, collectively advancing NF technology in mono-/divalent ion sieving and providing a solid theoretical foundation for related scientific research and practical applications. The non-equilibrium thermodynamic model based on the phenomenological equations is often used to describe the relationship between driving forces and fluxes, neglecting the internal structure of the membrane and considering the ion mass transfer mechanism from a thermodynamic perspective; this model provides mathematical and theoretical methods for the subsequent development of structural models. Meanwhile, the pore model, which considers the pore size sieving effect on the separation mechanism of molecules with different sizes, and the charge model, which is based on the Donnan effect and centers on the membrane's charge characteristics to describe the separation of substances with different charges, together form the basis of structural models and provide ideas for the development of subsequent models. The ES model combines electrostatic interactions and pore size sieving effects to analyze multiple interactions during the separation process, while the DSPM model integrates the Donnan effect and pore size sieving effect to explain the separation performance; the parameters involved in these two models are highly similar, but they differ in steric parameters and the membrane-solution interfacial partitioning equation. The DSPM-DE model, which considers dielectric repulsion, further improves the explanation of the NF membrane separation process on the basis of the previous models. The semi-empirical models based on experimental results and the interactions between ions and NF membranes follow a different construction approach compared to structural models. As shown in Figure 12, these models complement each other and jointly provide a theoretical basis for understanding the separation mechanisms of NF membranes. Among them, the DSPM-DE model, due to its comprehensiveness, holds promise in the field of Li⁺/Mg²⁺ sieving. However, studies applying this model to guide Li⁺/Mg²⁺ separation remain limited, and future research could benefit from more extensive simulations to reduce experimental efforts and save costs. In addition, integrating various models with computer simulations can significantly reduce human research costs. Table 2 summarizes the models in terms of their assumptions, composition conditions, applicable ranges, and existing drawbacks, offering insights for guiding the design of NF systems and optimizing the Li⁺/Mg²⁺ separation process in the future.