Performance metrics for nanofiltration-based selective separation for resource extraction and recovery

Membrane filtration has been widely adopted in various water treatment applications, but its use in selective solute separation for resource extraction and recovery is an emerging research area. When a membrane process is applied for solute–solute separation to extract solutes as the product, the performance metrics and process optimization strategies should differ from a membrane process for water production because the separation goals are fundamentally different. Here we used lithium (Li) magnesium (Mg) separation as a representative solute–solute separation to illustrate the deficiency of existing performance evaluation framework developed for water–solute separation using nanofiltration (NF). We performed coupon- and module-scale analyses of mass transfer to elucidate how membrane properties and operating conditions affect the performance of Li/Mg separation in NF. Notably, we identified an important operational trade-off between Li/Mg selectivity and Li recovery, which is critical for process optimization. We also established a new framework for evaluating membrane performance based on the success criteria of Li purity and recovery and further extended this framework to separation with the target ions in the brine. This analysis lays the theoretical foundation for performance evaluation and process optimization for NF-based selective solute separation. Membranes can be used not only for water filtration but also for solute–solute separation. Using the separation between lithium and magnesium, this analysis provides a platform for evaluating the performance of nanofiltration-based selective solute separation.

where R A (R B ), J A (J B ) and c f,A (c f,B ) are the apparent rejection, solute flux and feed concentration of solute A (or B), respectively. S A/B is also called separation factor. The solute flux and feed concentration can be based separation relies on NF that differentiates the rejections of solutes on the basis of their physicochemical properties (Fig. 1b). In general, NF-based solute-solute separation can be classified into two major categories (Fig. 1c). In the first category, the primary product is water, and the role of solute-solute separation is to improve the NF-based water treatment processes. For instance, in NF-based water softening, hardness ions (Ca 2+ and Mg 2+ ) are rejected whereas monovalent ions (for example, Na + and K + ) can readily pass through 9,10 . NF has also been used to selectively remove micropollutants without removing benign mineral ions 11,12 . The ability to achieve selective solute-solute separation in these contexts can lead to a desired product water quality (for example, reserving nutrient ions for fertigation), prevention of mineral scaling in subsequent desalination processes, and/or energy saving via reducing transmembrane osmotic pressure difference.
The second category of NF-based selective solute-solute separation aims at enabling the extraction of target solutes as the primary product. For example, when strong acid or base is used to recover cationic or anionic adsorbates from polymeric or mineral adsorbents, NF can be applied to concentrate the adsorbates (in the retentate) and recover acid or base (in the permeate) for re-use. A similar application of this type is dye recovery from textile wastewater, where dye molecules are retained and concentrated as the target solutes 13 . One potentially prominent NF application of the second category is lithium (Li) extraction from brines rich in magnesium (Mg) 14 .
The conventional method for Li production from brine is based on evaporation and chemical precipitation, which typically requires that   on either mass or mole as long as the concentrations are consistent within the equation. In the following discussion, we will use mass-based definitions as adopted by most literature, although mole-based definitions are mechanistically more meaningful.
In this Analysis, we will show that S A/B alone is insufficient for evaluating an NF membrane or process for selective solute-solute separations for resource recovery. While the principle should be generally applicable, we focus the current analysis on the specific application of Li/Mg separation to provide a concrete illustration. We start our analysis by evaluating the success criteria for Li/Mg separation and provide a critical analysis of literature data. We then perform coupon-and module-scale analysis to elucidate important operating and material considerations in NF-based Li/Mg separation. Finally, we introduce and discuss two important trade-offs that will guide future process optimization and membrane development to achieve high-performance Li/Mg separation.

Why is selectivity not a sufficient metric?
Assessing the adequacy of the metric S A/B requires first defining a successful Li/Mg separation. As the purpose of the separation is to extract Li from a Li/Mg mixture, the success criteria should have two aspects: purity and recovery (Fig. 1e). Considering a simplified scenario with only Li + and Mg 2+ cations, the permeate Li purity, η Li , is defined as the mass fraction of cations in the permeate that are Li + : where c p, Li The second important success criterion is Li recovery, defined as the mass fraction of Li + in the feed that is eventually recovered in the permeate. Specifically, Li recovery, LiR, can be quantified as where Q p and Q f are the volumetric permeate flowrate and influent flowrate of the feed stream, respectively; c p,Li and c f,Li are the Li concentrations in the permeate and feed influent, respectively; WR is water recovery; and R Li is Li rejection. Both WR and R Li are module-scale performance metrics. As we will show shortly, using R Li evaluated with membrane coupons for module-scale analysis can lead to inaccurate or even unphysical results. With the definitions of Li purity and recovery, it becomes apparent that a successful Li/Mg separation should recover the majority of Li from the feed solution and at the same time produce a permeate with a high Li purity (Fig. 1e). In other words, attaining only high Li recovery or high Li purity alone is undesirable for the purpose of Li extraction (Fig. 1f). We summarize and analyse literature data on the performance of NF membranes for Li/Mg separation. We also tested the performance of several commercial membranes (NFX, NF90 and NF270; Supplementary Table 1) Table 2). The feed MLR spans a wide range from 5:1 to 120:1, and the Mg 2+ concentrations vary by nearly two orders of magnitude (Fig. 2a). The feed composition is critical as it affects Li/Mg selectivity and directly impacts Li purity via equation (3).
The rejections of Li + and Mg 2+ span a wide range of values (Fig. 2b). The rejections of Mg 2+ are typically higher than 70% and can even reach 99.9%. The Li + rejection (R Li ) varies from −140% to 87%. Negative rejection of highly permeable ions (Li + ) is a result of maintaining Donnan equilibrium and is common in NF when the feed solution mixture has an abundance of strongly rejected co-ions (Mg 2+ ) and counter-ions that can easily permeate through the membrane (Cl − ) (refs. 10,26,27 ). The permeation of Cl − promotes the transport of the highly permeable cation, Li + , to maintain charge neutrality in the permeate solution, thereby resulting in a permeate with even higher Li + concentration than that of the feed.
The Li/Mg selectivity, S Li/Mg , is strongly sensitive to Mg 2+ rejection, especially when Mg 2+ rejection is high (Fig. 2c). This dependence is also obvious from the definition of S Li/Mg (equation (1)) in which the denominator is 1 − R Mg . The Li/Mg selectivity and the feed MLR together determine the permeate Li purity, which ranges from below 10% to over 90% (Fig. 2d). The high sensitivity of S Li/Mg to R Mg suggests that a very high S Li/Mg can be achieved even if Li + ions are well rejected, provided that Mg 2+ rejection is near perfect. This property of S Li/Mg renders it an insufficient performance metric as it overlooks the factor of Li recovery.
To illustrate the inadequacy of selectivity as a performance metric, a heuristic comparison between two scenarios with the exact same Li/Mg selectivity (50) is provided in Table 1. Two different separations with the same selectivity fall on the same Li/Mg selectivity line in Fig. 2b. The R Li and R Mg are −80% and 96.4%, respectively, in the first scenario, and 95% and 99.9% in the second scenario. Li recovery, LiR, is estimated by equation (4) to be 90% for the first scenario but only 2.5% for second scenario when WR is 50%. The extreme difference of LiR for the two separations with the same Li/Mg selectivity clearly demonstrates why selectivity is an inadequate metric. Because of the high sensitivity of Li/Mg selectivity to R Mg , especially when R Mg approaches 100%, a very high Li/Mg selectivity can be achieved even when R Li is unacceptably high for any Li recovery.
Notably, applying equation (4) with a WR of 80% in the first scenario predicts an unphysical LiR of 144%. The emergence of this unphysical prediction is attributable to the implicit use of R Li measured using coupon-scale experiments in an equation (equation (4)) that should use R Li of module-scale processes. While an R Li of −80% is not uncommon in literature (Fig. 2b), those reported R Li values were measured using membrane coupons (that is, WR is nearly zero) with a certain feed solution composition. To achieve a WR of 80% with membrane modules, however, the feed composition varies along the module due to the selective transport of water and ions. As we will show, R Li of a modulescale process cannot be highly negative. In other words, an LiR >100% should not emerge in a module-scale analysis that correctly captures the mass transfer behaviour, which is the focus of the next section.

Module-scale analysis of NF-based Li/Mg separation
Performing module-scale analysis requires a model to describe the local mass transfer in a differential element of the module. Such a model outputs the local fluxes of water and ions using applied pressure and local feed composition as the inputs. The module behaviour can then be modelled via finite difference method to relate mass transfer in differential elements (for details, see Methods). In this analysis, we employ the solution-diffusion-electromigration (SDEM) model due to its simplicity and ability to model fluxes of multiple components. The SDEM model assumes that any point inside the membrane is in thermodynamic equilibrium with a virtual bulk electrolyte solution that is charge neutral [27][28][29] . The virtual solution treatment is equivalent to applying a modified Nernst-Planck equation with the ion diffusion coefficient replaced by the ion permeability, which is the product of the Analysis https://doi.org/10.1038/s44221-023-00037-0 partition and diffusion coefficients (Supplementary Text 1) 29 . The ion flux for species i, J i , in the SDEM model is described using the modified Nernst-Planck equation: where P i is the ion permeability, c i is the ion concentration in the virtual solution, x is the transmembrane coordinate normalized by the membrane thickness, z i is the valence of species i, and φ is the local electrical potential in the virtual solution. Solving equation (5) yields the transmembrane distributions of ion concentrations, electrical potential and electrical field ( Fig. 3a as an illustration), which enables calculating rejections in a local differential element.
The SDEM model is semi-empirical because P i is not constant but has a rather complex dependence on the feed composition. While more mechanistic models are capable of describing multi-component transport 30,31 , they usually contain questionable assumptions and many fitting parameters. For simplicity, we employ a linear correlation to relate P i to feed composition:  (1)) values presented in dashed lines (vary from 5 to 100). Data points on the same dash line have the same Li + /Mg 2+ selectivity. c, Li + /Mg 2+ selectivity as a function of Mg 2+ rejection for three commercial NF membranes tested in this study and membranes reported in the literature. The selectivity has a strong dependence on Mg 2+ rejection, especially when Mg 2+ rejection is high. d, Li + /Mg 2+ selectivity as a function of MLR, with the permeate Li purity (defined by equation (2)) presented in dashed lines (varies from 10% to 99%). Data points on the same dash line have the same purity. For all panels, circles represent data from literature studies, including polyamide membranes (legend: PA (lit.)) and polyelectrolyte membranes fabricated by layer-by-layer deposition (legend: LbL (lit.)); diamonds represent data collected in this study.  where c ′ f,Li and c ′ f,Mg are the local interfacial feed concentrations of Li + and Mg 2+ , and α j are fitting coefficients. We note that the linear correlation works well in this analysis but requires further validation before applying to a more complex mixture feed solution. The local interfacial concentrations relate to the local bulk concentrations via concentration polarization (CP): where J w is local water flux and k i is the mass transfer coefficient of species i. J w can be estimated using where P w is the water permeability, and ΔP and Δπ m are the transmembrane difference of hydrostatic pressure and osmotic pressure, respectively. For relatively dilute solutions, the van't Hoff equation can be applied to relate Δπ m to transmembrane concentration differences (Methods). The local mass transfer can be determined by solving equations (5-8) simultaneously with charge neutrality and steady-state conditions. We use the data of water flux and ion rejections of a polyelectrolyte membrane coupon measured with different feed compositions (Fig. 3b) as reported by He et al. 24 to extract the permeability of Li + (P Li ) and Mg 2+ (P Mg ) and determine the correlation coefficients in equation (6). The polyelectrolyte membrane (named LbL in Fig. 3b,c) was fabricated using layer-by-layer (LbL) deposition of poly sodium (4-styrenesulfonate) and poly(allylamine) hydrochloride 24 . Additionally, we measured the performance of commercial NF membranes using the conditions (Methods and Supplementary Fig. 1). The correlations of ion permeability for these membranes are summarized in Supplementary  Table 3. In general, P Li is one to two orders of magnitude higher than P Mg , and the linear correlations can provide reasonable predictions of the permeabilities extracted from the SDEM model using experimental data (Fig. 3c).
With a model to evaluate the local mass transfer in a differential element, we can now extend the analysis to module scale by numerically integrating the governing differential equations for species conservation over a finite membrane area (Methods). Here we based our illustrative analysis on the polyelectrolyte membrane and generate representative results to describe the module-scale behaviours in NF-based Li/Mg separation. Intuitively, the module behaviour can be described as a spatial distribution of solution properties and separation performance along the direction of the feed flow. However, a more universal representation is to replace the position in the module with WR (up to that position) because WR increases as feed water flows past more membrane area.
As more water is recovered, the Mg 2+ concentration in the retentate (that is, the solution remaining in the feed channel after partial water recovery) increases dramatically, whereas the retentate Li + concentration first increases and then decreases but overall remains low (Fig. 4a). The permeate concentrations of Li + and Mg 2+ consistently increase with increasing WR (Fig. 4b). Here we distinguish between the local and cumulative permeate concentrations: the local concentrations are what could have been measured using a membrane coupon with the local feed composition, whereas the cumulative concentrations consider the cumulative ion and water permeation preceding the position corresponding to the current WR (that is, ∫ J i dS/ ∫ J w dS, where dS is the differential membrane area). Correspondingly, the Li + and Mg 2+ rejections can also be defined locally and cumulatively (Fig. 4c). While the local Li + rejection can become strongly negative (as observed with some membrane coupons), the cumulative Li + rejection cannot, thereby preventing the erroneous inference of over 100% LiR presented in Table 1.
The non-monotonic dependence of retentate Li + concentration on WR (Fig. 4a inset) is a direct result of local Li + rejection transitioning  (1). Ion permeabilities were fitted with the SDEM model. c, Fitted permeability extracted from the SDEM model using experimental data versus the predicted permeability obtained using the empirical correlation presented in equation (6) for three commercial NF membranes tested in this study and the LbL polyelectrolyte membrane in He et al.'s study 24 . CP was accounted for by an assumed mass transfer coefficient of 100 l m −2 h −1 for both LiCl and MgCl 2 .
Analysis https://doi.org/10.1038/s44221-023-00037-0 from positive to negative as WR increases (Fig. 4c). Despite the low (but positive) R Li at low WR, the Li + in the retentate is still concentrated with more water recovered, until R Li becomes negative at high WR. The strongly negative rejection of Li + at high WR is a result of both high local MLR ratio (Fig. 4d) and low local water flux due to diminishing driving force with increasing retentate osmotic pressure ( Supplementary  Fig. 2). Notably, both the cumulative selectivity and local selectivity drop with increasing WR (Fig. 4e) despite the progressively more favourable Li + permeation at higher WR (Fig. 4c), which can be explained by the noticeable reduction of R Mg with increasing WR (Fig. 4c inset) and the high sensitivity of S Li/Mg to R Mg (equation (1)). The drop in local Li/Mg selectivity is a result of both the varying retentate composition (Fig. 4a) and water flux, as selectivity could be substantially compromised when the water flux is too low (Supplementary Text 2 and Supplementary Fig. 3). Lastly, Li recovery, LiR, increases monotonically as more water is recovered (Fig. 4f).

Performance trade-off in NF operation
From an operational perspective, the module-scale analysis reveals an intrinsic trade-off between the (cumulative) selectivity, S Li/Mg , and Li recovery, LiR. For a single-stage NF process with a given applied pressure and influent feed flowrate, a higher WR can be achieved by providing more membrane area. Increasing WR increases LiR (Fig. 4f) but at the cost of reduced S Li/Mg (Fig. 4e), resulting in the tradeoff between S Li/Mg and LiR (Fig. 5a).
The characteristic curve quantifying the trade-off between S Li/Mg and LiR, namely the operational trade-off curve, depends on the applied pressure, ΔP, which affects the water flux. At a low ΔP, water permeates through the membrane at a lower rate. However, the ion fluxes are not affected proportionally due to the negligible advective ion transport in NF. Therefore, operating NF at lower ΔP enhances the relative Li + permeation as compared with water permeation, which results in a higher LiR at the same WR and thereby shifts the tradeoff curve towards the right (see dash curves in Fig. 5a). This effect of enhanced LiR at the same WR is more prominent at a higher WR. Notably, the maximum WR achievable with unlimited membrane area is also dependent on ΔP, as water permeation stops when Δπ m reaches ΔP. In the extreme case of applying only 1 bar, the maximum attainable WR and LiR are ~10% and ~22%, respectively.
In the range of low WR, S Li/Mg increases considerably as ΔP decreases from 8 bar to 4 bar, that is, reducing ΔP and water flux in this range also shifts the trade-off curves up (Fig. 5a). However, further reducing ΔP below 4 bar compromises S Li/Mg (see reflection of dash curves in Fig. 5a). With a ΔP of 1 bar, S Li/Mg becomes very low. While the cumulative selectivity, S Li/Mg , has a complex dependence on multiple factors (for example, water flux and feed composition) that varies along the module, the non-monotonic dependence of S Li/Mg can be explained by the flux dependence of local selectivity as shown in Fig. 5b.
In the water flux regime typical of NF (grey region in Fig. 5b), local Li/Mg selectivity decreases monotonically with increasing water flux due to CP. Specifically, because Mg 2+ ions are far better rejected than Li + ions, the accumulation of Mg 2+ near the membrane surface is more severe than Li + , which results in a higher interfacial MLR at a higher water flux ( Supplementary Fig. 4). A higher interfacial MLR is detrimental to local Li/Mg selectivity, which is strongly sensitive to Mg 2+ rejection, because a heightened interfacial Mg 2+ concentration compromises Mg 2+ rejection ( Supplementary Fig. 5). In the very low water flux regime (yellow region in Fig. 5b, untypical of NF), local Li/Mg selectivity drops dramatically with decreasing water flux due to the substantially reduced rejections of all ions because of the weakened 'dilution effect'. As S Li/Mg is much more sensitive to Mg 2+ rejection than to Li + rejection, reducing the rejections of all ions leads to a dramatic drop in S Li/Mg . Analysis https://doi.org/10.1038/s44221-023-00037-0 As selectivity is related to the permeate Li purity, η Li , via equation (3), the trade-off between S Li/Mg and LiR presented in Fig. 5a can be directly converted to a trade-off between η Li and LiR for a given feed MLR (Supplementary Fig. 6). At a given applied pressure, the trade-off between the two important performance metrics in Li/Mg separation suggests that recovering more Li + by using a larger membrane area will inevitably yield a permeate stream with a lower η Li . Within the typical range of NF flux, S Li/Mg and LiR can be simultaneously improved by operating NF at a lower pressure to reduce water flux. Additionally, the trade-off curve can also be shifted towards a more favourable direction when a better membrane is used, with the definition of 'better' to be discussed below.

Performance metrics of NF membrane for Li/Mg separation
What exactly is a 'better membrane' in the context of Li/Mg separation is an important question to the vibrant and growing community for developing high-performance NF membranes for Li/Mg separation. While many previous papers in this field compare membrane performance in a plot of Li/Mg selectivity versus water permeability (S Li/Mg versus P w ; Table 1), we have demonstrated why S Li/Mg is an insufficient metric. We have also shown that a high water flux is detrimental to both Li selectivity (or purity) and recovery (Fig. 5). Therefore, a high P w seems at odds with the success criteria for Li/Mg separation.
Selectivity is defined on the basis of rejections, which are less intrinsic than permeabilities, whereas membrane performance is more commonly quantified on the basis of permeabilities. Although one may rightfully argue that permeabilities are also not entirely intrinsic properties, they are more intrinsic than rejections and are adopted in the most widely used framework for evaluating membrane performance for water-solute separation. Selectivity in water-solute separation is defined on the basis of the ratio between water permeability (P w ) and solute permeability (P s ). The water/solute selectivity (P w /P s ) is usually plotted against the water permeability (P w ) to illustrate the perm-selectivity of membranes 7,32,33 . However, such a performance evaluation framework based on P w /P s versus P w is clearly inappropriate for Li/Mg separation with very different success criteria compared with water-solute separation.
Not only should an ideal NF membrane for Li/Mg separation enable fast Li + transport and slow Mg 2+ transport so that selective permeation of Li + over Mg 2+ can be achieved to maximize Li purity, but it should also favour Li + permeation over water permeation to promote Li recovery. If water permeation is very fast yet Li + permeation is very slow, only a small fraction of Li + in the feed solution will end up in the permeate. Given these considerations, we propose that the performance of an NF membrane for Li/Mg separation should be evaluated on the basis of two permeability ratios, P Li /P Mg and P Li /P w (Fig. 6a). A high P Li /P Mg favours Li/Mg selectivity and Li purity, whereas a high P Li /P w favours Li recovery. These two permeability ratios directly correspond to the two success criteria in NF-based Li/Mg separation.
Like water-solute separation, a membrane with a higher P w can reduce the energy consumption and/or membrane area for a given feed flowrate, thereby reducing overall cost of the separation 3,8 . In Li/Mg separation, however, a higher P w is beneficial only if it does not compromise P Li /P w , because LiR is probably a more important performance metric than water flux or volume-specific energy consumption. In a bubble plot of P Li /P Mg versus P Li /P w , where P w may be quantified by the size of the 'bubbles' (Fig. 6a), an ideal membrane is a 'big bubble' on the upper right of the plot. The concept of a performance upper bound commonly employed for perm-selectivity in water-solute separation also applies here to describe the trade-off between Li purity and recovery. Future studies on developing high-performance NF membranes should aim to populate the bubble plot beyond the current upper bound.
The P Li /P Mg versus P Li /P w bubble plot should be used with caution when comparing membrane performance. Ideally, all data points in this plot should be obtained using the same feed composition and operating conditions, which is not necessarily the case across different studies. These testing conditions impact membrane performance, which is evident from the mild scattering of performance for a given membrane tested in different conditions ( Fig. 6a and Supplementary Table 4). Future studies on membrane development should converge to a unified testing protocol for performance comparison on the P Li /P Mg versus P Li /P w bubble plot.

Extending performance metrics of NF membrane for recovering ions in the brine
While the current analysis primarily focuses on Li extraction where the ions enriched in the permeate (that is, Li + ) are the product, the analysis framework can readily be extended to other selective solute separation where the ions retained in brine are the product (for example, rare Analysis https://doi.org/10.1038/s44221-023-00037-0 earth metal recovery) 34 . Without collecting new sets of data for such applications, here we analyse the same dataset for Li/Mg separation but for a hypothetical scenario where Mg 2+ ions are the product (Fig. 6b). The success criteria for such an application are clearly Mg purity and recovery. Analogous to how we evaluate membrane performance for Li extraction, here the P Li /P Mg ratio remains important as it determines the Mg purity in the brine stream. Unlike Li extraction, however, the relevant membrane property to Mg recovery is P w /P Mg . A high water permeability, P w , not only improves performance from an energy or kinetic perspective as in conventional water-solute separation, but also benefits Mg recovery by ensuring that only a small fraction of Mg 2+ ions in the feed stream will end up in the permeate stream when most water permeates through the membrane. These two membrane performance metrics for Mg recovery using NF-based Li/Mg separation, P Li /P Mg and P w /P Mg , positively correlate with each other on the basis of data evaluated from literature using the SDEM model (Fig. 6b). In other words, there is no trade-off at the membrane level when NF-based Li/Mg separation is used towards Mg recovery.

Perspectives and outlook
Our analysis demonstrates that the existing framework for evaluating NF performance in water treatment is inadequate for quantifying NF performance for selective solute-solute separation. The performance metrics in the existing framework mismatch the success criteria for selective solute-solute separation when the goal is to extract a target solute as the desired product. Li/Mg separation is chosen as an example for illustrating such a mismatch and for developing a suitable framework for evaluating process and membrane performance. In NF-based Li/Mg separation, the key performance metrics at the process level should be Li/Mg selectivity (or Li purity) and Li recovery, not water permeability as currently used. The consideration of these two metrics results in important trade-off relations for operation optimization and membrane development.
From an operation perspective, process optimization of NF for Li/Mg separation should focus on Li purity and recovery, which are constrained by a trade-off relation (Fig. 5). Factors that are critical in water treatment, such as energy consumption and membrane cost, are probably less important in Li/Mg separation due to Li being a commodity with a much higher economic value than water. From membrane development perspective, NF membranes for Li/Mg separation should be evaluated using the P Li /P Mg versus P Li /P w bubble plot (Fig. 6), which captures membrane properties most relevant to the success criteria of Li/Mg separation.
We also demonstrate how this analysis framework can readily extend to another general category of selective solute separation where ions in the brine are the target ions of extraction. We believe this general analysis framework featuring target ion purity and recovery will guide future endeavours in process innovation and optimization and the development of high-performance NF membranes for selective solute-solute separation.

Modelling module-scale performance with the SDEM model
The SDEM model describes local ion transport across the membrane with the modified Nernst-Planck equation (equation (5) in the main text). With a given feed solution composition and a set of ion permeabilities, permeate composition can be solved as a function of permeate flux with the charge neutrality (equation (9)) and steady-state (equation (10)) conditions: where c i is the concentration of ion i in the virtual solution or the external feed and permeate, and c p,i is permeate concentration of ion i. Yaroshchuk and Bruening provided an analytical solution of the SDEM model for a ternary electrolyte system 29 . Thus, a pair of ion permeabilities can be fitted to describe the coupon-scale ion transport behaviour after accounting for CP (equation (7) in the main text), given experimental results of Li + and Mg 2+ rejections at a certain permeate flux. A MATLAB application is provided as Supplementary Code for fitting ion permeabilities. The module behaviour was then modelled via a finite difference method to relate mass transfer in differential elements. Each differential element has either the same membrane area or the same increment of water recovery (WR). Since WR increases as feed water flows past more membrane area, the module-scale behaviours can be equivalently modelled as a function of WR instead of a function of and Mg permeability (P Li /P Mg ) versus ratio between Li permeability and water permeability (P Li /P w ). This evaluation framework applies to extracting Li, or more generally, the ions enriched in the permeate. Increasing P Li /P w improves Li recovery, whereas increasing P Li /P Mg enhances permeate Li purity. b, Ratio between Li permeability and Mg permeability (P Li /P Mg ) versus ratio between water permeability and Mg permeability (P w /P Mg ). This evaluation framework applies to extracting Mg, or more generally, the ions enriched in the brine. The size of the data points quantifies the water permeability, P w (see legend). The permeabilities are extracted using the SDEM model with an assumed mass transfer coefficient of 100 l m −2 h −1 for both LiCl and MgCl 2 . Legends 'PA (lit.)' and 'LbL (lit.)' refer to literature data obtained using polyamide membranes and polyelectrolyte membranes fabricated by layer-by-layer deposition.
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