Predicting and Enhancing the Ion Selectivity in Multi-Ion Capacitive Deionization.

Lack of potable water in communities across the globe is a serious humanitarian problem promoting the desalination of saline water (seawater and brackish water) to meet the growing demands of human civilization. Multiple ionic species can be present in natural water sources in addition to sodium chloride, and capacitive deionization (CDI) is an upcoming technology with the potential to address these challenges because of its efficacy in removing charged species from water by electro-adsorption. In this work, we have investigated the effect of device operation on the preferential removal of different ionic species. A dynamic Langmuir (DL) model has been a starting point for deriving the theory, and the model predictions have been validated using data from reports in the literature. Crucially, we derive a simple relationship between the adsorption of different ionic species for short and long adsorption periods. This is leveraged to directly predict and enhance the selective ion removal in CDI. Furthermore, we demonstrate an example of how this selectivity could reduce excess removal of ions to avoid remineralization needs. In conclusion, the method could be valuable for predicting the impact of improved device operation on capacitive deionization with multi-ion compositions prevalent in natural water sources.

Introduction

Water scarcity is a rising issue across the world.[1] Less than 1% of the world’s water supplies are available as freshwater in rivers, lakes, and groundwater.[2,3] With increasing freshwater demand due to population growth and higher water consumption in industries, desalination techniques could be crucial to tap into the abundant saline water to fulfill the requirements.[4−13] However, saline water sources often contain a multitude of ions in addition to sodium chloride (NaCl) that should be addressed when considering desalination.[14−16] Some of the ions that are harmful need to be entirely removed, such as arsenic,[17] while other ions (such as fluoride) can be beneficial in smaller quantities and should ideally be optimally removed.[18] This necessitates the development of technologies that can remove all kinds of ions from multi-ion solutions and, at the same time, preferably remove only the desired amounts of each ion species.

Capacitive deionization (CDI)[19−21] is an environmentally friendly[22] desalination technology that can remove any charged species from water by electro-adsorption. In CDI operations, water is flown between two porous electrodes over which there is an applied potential (Figure ) so that the ions are adsorbed on the electrodes.[23,24] Ion removal in a CDI process is thus strongly affected by material[25−27] and operational parameters,[28−30] and different ions are typically adsorbed in different quantities. Important parameters governing the adsorption of different ions include the electrode pore-size distribution,[15,31] the ion size,[32] and the ion valency.[14] Selective removal of ions using CDI has been reported in the literature, either with carbon electrodes[33,34] or by modification of electrodes with intercalation materials.[35−37]

Figure 1

(Inset) CDI device is based on using a cell comprised of two porous electrodes. During operation, a voltage is applied so that the ions are directed from the water and adsorbed on the electrodes. (Main figure) In a process where the treated water is put back as inlet water in a closed-loop (batch-mode), the concentration decreases linearly at the beginning after the voltage is applied and stabilizes when the electrode is saturated. Typically, CDI is run in cycles wherein the desalination period shown is followed by a regeneration period (without an applied voltage) wherein the ions are released into water, which is discarded. The concentration data displays Cl– ions and was extracted from ref (42).

Because of the differences between ions, studies have investigated and ranked the relative adsorption of different ionic species in mixtures of anions[32] or cations[15] for either usual CDI systems[15,32] or systems with surface-modified electrodes.[38] In this process, it has been found that a degree of ion selectivity can be obtained by changing the device operation, as applying the voltage for a long time favors adsorption of ions with high valencies, such as Ca2+ over Na+.[14][14]

In a recent study, we used a dynamic Langmuir (DL)[39,40] model to develop a metric for ion competitiveness in CDI.[41] A single experiment could be performed to extract a “periodic table” of competitiveness scores for all ions in the calibration solution at equilibrium, and the method made it possible to not only rank the comparative adsorption of different ions but also quantitatively predict it. A fundamental finding that allowed the predictions was that the relative adsorption of ionic species is proportional to their relative concentration.

In the current work, we demonstrate that applying the voltage for a short time leads to the same relation between relative adsorption and relative concentration with a different proportionality constant. Thus, the ion adsorption in both short and long adsorption periods can be quantified for every ionic species in a multi-ion solution, and this makes it possible to quantify and predict the ion selectivity obtained by operating in short or long periods of adsorption.

Theory

Dynamic Langmuir Model

The DL model used in this work is derived from the principles of Langmuir adsorption. In Langmuir adsorption, the equilibrium adsorption of a gas on a plane surface depends on a balance between adsorption and desorption rates (eq ).[43,44] Also, the adsorption rate depends on the number of free sites on the surface (eq ).[43,44] Note that θ denotes partial coverage, pA denotes gas partial pressure, and ka, kdes, and KL are constants.The classical Langmuir adsorption has been adapted for liquids by exchanging the partial pressure of the ion concentration,[44] which has allowed researchers in multiple studies to accurately describe how the basic equilibrium-adsorption performance in CDI varies depending on the ion concentrations.[45−52] As the model works well for this basic operation, it has been found that the classical Langmuir adsorption can be broadly expanded to incorporate a wider range of operational metrics, which means there are fundamental similarities between CDI electro-adsorption and Langmuirian adsorption.

Because the CDI system contains more operational parameters than the plain gas-adsorption case, there are several additional elements present compared to passive adsorption. First, the fundamental process is storing charges rather than adsorbing gas or ions, meaning that the partial pressure or ion concentration should be exchanged for the concentration of charge σ. Second, the adsorption sites in CDI are voltage-induced site S. Previous studies have shown that modeling adsorption using these sites and taking them to be proportional to the applied voltage means that ion adsorption, charge storage, and charge efficiency can be predicted.[39] Implementing these changes results in an expression for the change in charge storage over time (eq ). Note that subscript “ads” denotes the corresponding adsorbed quantity.The storage of charges can be related to the ideal ion adsorption through the valency z as σads = zcads (eq ). However, not all charges contribute to adsorption. Here, co-ion expulsion is mainly considered as a source of this unideal charge efficiency; that is, the applied voltage is used for pushing away co-ions rather than adsorption of counterions, meaning that the effective sites for adsorption decrease. These co-ions are present near the surface partly because charged groups on the electrodes’ surfaces are neutralized[53] (modeled by site reduction β0) and partly because of a passive presence due to the ion concentration in the solution (modeled by site reduction β1c0). Incorporating this leads to eq , describing the adsorption of ions over time.

Multi-Ion Solutions

Interestingly, eq is fundamentally not limited to describing ion adsorption characteristics of salt solutions with only one type of ion. If there are multiple ions in the solution, the ions of similar charges block the same sites (exchange cadsz for ∑z(cads(). Apart from that, every ion can be considered separately[39] (eq ).While this expression is complicated to implement, a simpler result can be derived for the relative adsorption between a pair of ionic species in a multi-ion solution at equilibrium. The equilibrium state can be modeled by setting the derivative to zero, while the relative adsorption can be computed by adding the second term on the right-hand side to both sides and dividing this equation for one ion by the corresponding result for the other ion. Finally, by assuming that the charge efficiencies of the ionic species are similar,[39,54]eq is obtained, showing that the relative adsorption is proportional to the relative equilibrium concentration. If either a continuous-mode experiment is used or a batch-mode experiment is used, where the adsorption is similar between species (α ≈ 1) or the total adsorption is small, the equilibrium concentration can additionally be exchanged for the initial concentration (Supplementary eq S-2).

Crucially, this result is independent of the composition of the solution, meaning that the α values for all ions in a multi-ion solution could be extracted from a single experimental result.Two more results are of importance for implementing eq in real-world systems. For the first result, even in unideal batch-mode operations, where the equilibrium concentration cannot be directly exchanged for the initial concentration, eq can still be expressed in terms of the initial concentration, although it is not linear (eq ). This derivation is done by noting that the concentration can be rescaled to ‘ions per batch volume’, meaning that ce = c0 – cads. This equation can further be rewritten so that the adsorption of one species can be uniquely determined if the initial concentration of both species and the adsorption of the other species are known (eq ).For the second result, if the α value relating (i) to (b) and (j) to (b) are known, (i) to (j) can be calculated (eqs and 11). Thus, a calibration only needs to extract the α values of ions with respect to a common baseline ion to allow predictions of ion-adsorption behavior for all combinations of ions.

Selectivity of Ions

This section will show how a degree of ion selectivity that can be achieved and quantified through the choice of device-operation parameters. Previous studies have shown ion exchanges occur during adsorption in a capacitive device, meaning that different proportions of ions are adsorbed depending on how long the ion adsorption is carried out.[14]

The derivation of eq used the simplifying assumption that the equilibrium state is reached. However, another simplified situation arises when the device is operated for a short adsorption period. Consider eq describing the adsorption over time. Assuming all adsorbed ions are removed during regeneration of the electrodes, the concentration of adsorbed salt on the electrodes, cads, is negligible at the beginning of the adsorption period and thus can be omitted from the equation. Dropping the terms with cads from eq leads to eq below. Also note the added zero in the subscript of the first concentration factor on the right-hand side in eq , which introduces the assumption that the variation in ion concentration during the short operation has a negligible effect on the adsorption rate. This is introduced because the concentration at the beginning of the adsorption period differs negligibly from the initial ion concentration.At a given initial concentration c0(, eq can be solved to yield the adsorption at some short time t after the application of a voltage across the capacitor electrodes (eq ). As in the derivation of eq for equilibrium conditions, eq for one ion (i) can be divided by the corresponding equation for some other ion (j) to derive an expression for short adsorption periods. Again, assuming the charge efficiency is similar between species, the factors containing S are identical so that the integrals cancel, leading to eq , which shows that the relative adsorption is proportional to the relative initial concentration.Equation has the same form as eq , meaning that they share the same underlying trend. However, κ and α values are fundamentally different since α depends on both kads and kdes, while κ only depends on kads. While implementing the model, these will thus be extracted from the same calibrating experiment but independent of each other. Effectively, driving the adsorption process until saturation of the electrodes means the α value governs the final state while interrupting the adsorption process in the linear adsorption region, meaning the relative adsorption is determined by κ. Thus, some degree of ion selectivity could be achieved through the choice of device operation, and the relative adsorption could be predicted for both long and short deionization cycles.

Predicting Adsorption of Trace Ions

Consider a solution with a majority ion (m) being the most concentrated species and a trace ion (i) with a lower concentration. Equation can be rearranged to eq , showing the adsorption (not relative) of the trace ion. The adsorption of the majority ion at a given concentration should not depend on the concentration of the trace ion (since the adsorption of a trace ion is negligible). Thus, a single experiment could determine the adsorption of the majority ion at a given concentration in a salt solution, and this value would be expected to be constant independent of changes in the concentration of the trace ion in the solution. Under these conditions, the adsorption of the trace ion would be proportional to the trace-ion concentration, with a known proportionality constant.

Crucially, the derivation works when starting from either eq or 7 (although c0 is exchanged for ce), meaning that the derived expression in eq can model short adsorption periods (using κ) or longer adsorption periods (using α). This expression is used when validating the model using experimental data obtained from the literature.

Experimental Validation

In the section above, essential mathematical formulations were derived for the relative adsorption of different ions in multi-ion solutions, including how the adsorption changes depending on whether short or long periods for ion adsorption are used. To validate these claims, the model was applied to experimental data from reports in the literature. The data was extracted from figures in the papers using the WebPlotDigitizer software.[55]

Only literature reports that included data for multi-ion adsorption over time were considered for validating the model. Additionally, papers that reported several data sets at similar conditions were preferably considered so that both fittings and predictions could be verified. It was also desirable to find reports using a variety of conditions, such as investigating monovalent and divalent ions, which led to the choice of the data sets considered in this work.

Results

This section will seek to validate the derived results for long and short adsorption periods using experimental data extracted from the literature. Also, a method will be presented for implementing the derived results for predicting the adsorption in real-world multi-ion solutions.

Equilibrium Adsorption

A core concept in this work is that the relative adsorption between ions of the same charge sign in a multi-ion solution is proportional to their relative initial concentration (eqs and 7). This is important because it allows the adsorption in a multi-ion solution to be simply described.

To illustrate the principle, consider the experimental results reported by Tang et al., where NaCl with trace amounts of sodium fluoride (NaF) or sodium nitrate (NaNO3) was used in a batch-mode experiment and adsorption of each ion over time was reported.[42] There is a strong linear relationship between the relative adsorption at equilibrium and the relative initial concentrations for both F– (Figure a) and NO3– (Figure b). For the model line, note that α values are close to unity, meaning that the relative concentration in eq can be taken as the initial concentration instead of the equilibrium concentration. Previous work has further demonstrated that the linear relationship holds both for anions and cations and can be employed even for solutions containing other ionic species than the investigated pair.[39]

Figure 2

(a) Data from Figure 4 in ref (42). A fixed concentration of 20 mg/L NaF was mixed with 0.5, 1, 1.5, 2, and 3 g/L NaCl, respectively, and batch-mode desalination was performed at 1.6 V until saturation. The relative adsorption between F– and Cl– is shown as a function of their relative initial concentrations (eq ). (b) NaNO3 (300 mg/L) was mixed with 1.5, 2, and 3 g/L NaCl, respectively. The relative adsorption between NO3– and Cl– is shown as a function of their relative initial concentrations. The data for NO3– adsorption was taken from Figure 6 in ref (42). Since no specific Cl– adsorption data is provided for this experiment, the data for Cl– adsorption was taken from the experiment with the corresponding concentrations in Figure 4 in ref (42), using the assumption that changing the concentration of trace ions has a negligible impact on the major-ion adsorption. (c) Relative adsorption between Ca2+ and Na+ based on multi-ion competition data from Table 2 in the report by Hou et al.[32]

Figure a,b shows only monovalent anions, but the model is found to describe the data reported by Hou et al.[32] on the equilibrium adsorption for varying concentrations of NaCl, CaCl2, KCl, and MgCl2 very well (Figure c). Thus, the principles verified for monovalent anions apply equally to cations and monovalent/divalent ionic mixtures.[32]

Short Adsorption Times

The crucial result derived in this work is that the relative adsorption of different ions in a multi-ion solution is proportional to their relative concentration when operating in short adsorption periods as well as when operating in saturation. This suggests that adsorption operation over shorter and longer periods of applying the voltage can be described and predicted. Because the proportionality constant is not the same for longer adsorption periods (when quasi-saturation of an electrode occurs) as for shorter adsorption periods, the choice of operation can be used to achieve a degree of ion selectivity.

Consider again the data from the report of Tang et al.[42] Plotting the relative adsorption at 80 s (in the linear adsorption region) versus the relative initial adsorption, a clear linear trend is observed (Figure ). Interestingly, the proportionality constant is about the same in both operations for F– ion adsorption but increases for NO3– from 1.12 to 1.52. This suggests that shorter adsorption periods would preferentially remove NO3– in these experiments.

Figure 3

Data from the same ref (42) experiments as used in Figure . Here, the data points for adsorption correspond to the linear adsorption regions and were determined at 80 s from the start of the deionization period. The relative adsorption between (a) F– and Cl– and (b) NO3– and Cl– is shown as a function of the relative initial concentration. The model lines were fitted using eq .

Predicting Equilibrium Adsorption

The previous sections have shown how to calibrate the model, and having a calibration allows applying eq to generally predict the adsorption of any new ion concentrations. Furthermore, keeping the majority-ion concentrations constant simplifies eqs –15, which means that the adsorption of varying concentrations of trace ions is proportional to the trace-ion concentration.

Tang et al. performed several experiments including varying the majority NaCl concentration (Figure ) at a fixed trace-ion concentration and varying the trace-ion concentration at a fixed 2 g/L NaCl background concentration.[42] As the fitting had been obtained from the first of these experiments, the equilibrium ion adsorption in the second experiment could be accurately predicted for both F– (Figure a) and NO3– (Figure b). This result highlights that a calibrating experiment can be used to accurately predict the adsorption of varying ion-concentration mixtures.

Figure 4

Model lines were predicted using the α parameters extracted in Figure , and the relationship is shown in eq . (a) This equilibrium data is from Figure 3 in ref (42), showing the adsorption of F– for 5.6 mg/L, 10.4 mg/L, 20.9 mg/L, and 27 mg/L initial concentrations when mixed with 2 g/L NaCl. (b) This equilibrium data is from Figure 6a in ref (42), showing the adsorption of NO3– for 100, 300, and 500 mg/L initial concentrations when mixed with 2 g/L NaCl.

Predicting Adsorption for Shorter Time Durations

Since the principles of relative adsorption apply to both long and short operations of the capacitive devices, it should be possible to predict the adsorption for short adsorption periods as well. In the data set from Tang et al.,[42] adsorption after 80 s of operation can be predicted κ using the calibration of κ from Figure . This prediction also yields excellent results for both F– (Figure a) and NO3– (Figure b).

Figure 5

Model lines were predicted using the κ parameters extracted in Figure and the relationship shown in eq . (a) This short-time data (adsorption at 80 s) is from Figure 3 in ref (42), showing the adsorption of F– for 5.6, 10.4, 20.9, and 27 mg/L initial concentrations when mixed with 2 g/L NaCl. (b) This short-time data is from Figure 6a in ref (42), showing the adsorption of NO3– for 100, 300, and 500 mg/L initial concentrations when mixed with 2 g/L NaCl.

Note that the total adsorption is generally lower for short operations (Figure ) compared to longer operation times (Figure ) since the adsorption of all species increases with time. Still, the relative difference between the equilibrium and short-term adsorption demonstrates an increased preference for NO3– adsorption at shorter ion-adsorption periods. This finding that there can be time-varying relative adsorption between monovalent anions agrees with a previous study performed by Chen et al., who observed ion exchange processes between NO3– and Cl–.[56][56]

Ion Selectivity

The previous section demonstrated a slight preference for NO3– adsorption. However, there are cases when substantial differences in ion adsorption can be observed between short and long adsorption periods. Consider the work by Zhao et al., where the time-dependent adsorption of Na+ and Ca2+ was investigated using a continuous-mode operation.[14] Notably, it was reported that high adsorption of both species occurs at the beginning of a deionization process but, over time, Ca2+ increasingly adsorbs at a much faster rate than Na+, leading to preferential adsorption of Ca2+ closer to the saturation of the electrodes, which can be quantified applying the model developed in this work. An interpretation of this is that the higher valency of Ca2+ dominates in the competitive equilibrium state, but the initial adsorption is similar because the baseline adsorption rate and ion diffusivity are more important when there is negligible ion adsorption on the electrodes.

Zhao et al. described a continuous-mode experiment, where the effluent concentration was measured over time while the influent concentration was constant (Figure 3a in ref (14)). When operating the device, they applied the voltage for 1 h, going to a quasi-saturated state of the electrodes. By integrating the difference between influent and effluent concentration up to the time where the lowest effluent concentration is reached, the adsorption of each species for short operation periods can be estimated. This estimation yields a κ value of 1.1, suggesting similar relative adsorption of the ionic species.

While the adsorption for a short adsorption period was similar, the relative adsorption of Ca2+ was much larger than that for Na+ at longer adsorption times because the Ca2+ increasingly replaced Na+ in the electrodes. The total adsorption of Na+ and Ca2+ at the end of the 1 h desalination phase was reported directly in Figure 2c in ref (14). A calculation based on these adsorption values thus reveals that the α value is 7.2, showing a strong preference for calcium adsorption.

Implementation

The previous sections have demonstrated that the relative adsorption between ions can be predicted for shorter and longer ion-adsorption periods, meaning that a degree of ion selectivity can be obtained and predicted. Below, we describe a method for implementing the result in practice for real-world samples that contain multiple ionic species in different concentrations. To calculate the adsorption of each ion in a multi-ion solution, the following six steps are used:

Calibrate α and κ, if unknown.

Measure the initial concentration of each ion for the solution of interest, if unknown.

Pick a priority ion and determine the concentration to be removed.

Calculate the removal of the other ions of the same charge sign using eq .

Determine the total charge removed.

Calculate the removal of the ions of an opposite charge sign by numerically solving the system of equations defining the total charge removed and by eq for each ion compared to a baseline ion of the same charge sign.

For clarification, step (1) need only be conducted once for a device, in principle, using a standard solution with known ions that are of interest (more ions are acceptable as well). α and κ can be calculated by determining ion-adsorption characteristics with short/long electrosorption periods to determine the adsorption trends of each ion relative to a baseline ion, as in eq . For baseline ions, we suggest Na+ for cations and Cl– for anions. Note that previous reports have demonstrated that ion-adsorption characteristics can vary significantly between devices. Therefore, the model should preferably be recalibrated if the device or operational conditions are changed.

Step (3) means that the most important ion (in the sense that the final concentration should be exact) is picked along with a desired final concentration. The electrosorption process should be stopped when the concentration of this ion reaches the desired level. Note that if there are several ions with critical thresholds, the priority ion can be chosen such that the model predicts that when that ion reaches the threshold, the other critical ions reach their threshold as well.

Steps (4)–(6) can be computed using a numerical solver. To make the method presented in this work more accessible to researchers, a MATLAB program implementing these steps is provided in the Supporting Information (Supplementary Script S-1). Thus, a researcher who is interested in implementing the method can enter the relevant values from steps (1)–(3) into the program to obtain the final concentrations.

Implementation Example

This work has demonstrated a straightforward method of predicting ion selectivity achievable in CDI by operating for long or short electrosorption periods. As an illustration of the value of this method, let us consider the following example:

Excessive fluoride intake could be dangerous as it leads to dental fluorosis, and in many places across the world, groundwater contains concentrations above the recommended highest concentration[57] of 1.5 mg/L.[58] On the other hand, small quantities of fluoride can be beneficial, and 0.5 mg/L has been suggested as a lower limit in drinking water.[59] Another ion that is often of interest to remove is calcium, which causes water hardness.[15] However, classical ion removal techniques are not selective and, for example, desalination plants (operating with reverse osmosis or membrane distillation processes) often remove so much that the rejected water becomes corrosive. Often, there is a need to remineralize the desalinated water by adding ions to ensure a calcium concentration of roughly 50 mg/L recommended for drinking water.[60] Thus, both fluoride and calcium have preferred minimum and maximum concentrations, while the removal of one ion influences the removal of the other.

Adhikary et al. reported that the groundwater in New Delhi (India) contained high concentrations of fluoride (0.20–5.12 mg/L).[61] Several other ions were also present in the groundwater, and the average concentration of the most concentrated ion species include Na+: 247 mg/L, Ca2+: 109 mg/L, Mg2+: 63 mg/L, HCO3–: 419 mg/L, and Cl–: 419 mg/L. Consider such a water sample with a fluoride concentration of 3 mg/L that should be brought down to the suggested value of 1.5 mg/L (thus removing half of the fluoride ions). While every ion typically has its own threshold, here we highlight the effects this has for Ca2+. At the same time as F– is removed, it would be preferable to not spend extra energy on removing too much Ca2+ and, instead, obtain the optimal concentration directly without the need for remineralization.

Here, a simplified calculation indicates that long desalination periods would remove 78% of the Ca2+ down to 24 mg/L, while short desalination periods would remove 51% down to 53 mg/L (a detailed derivation is provided in the Supplementary S-3). This suggests that a deionization process using short electrosorption periods would not require Ca2+ remineralization at all. Furthermore, if the calcium concentration had been much higher, it would still have been possible to reduce both calcium and fluoride to desirable levels using longer electrosorption periods instead or a combination of long and short electrosorption periods. Generally, it can be noted that picking the best option means that fewer ions need to be removed than if the other operation had been used. This is valuable since it could lead to a reduction of deionization time and subsequently the energy consumption for reaching the required water standards, although quantifying this gain is outside the scope of the current work.

Discussion

Let us discuss the aspects of the underlying physics that were not elaborated in the theory section including limitations and prospects of the model.

First, previous authors have found that physical parameters such as material texture, ion valency, ion size, ion diffusivity, etc. are important for determining the adsorption. Because they are important, they substantially influence the adsorption/desorption rate constants, meaning they are indirectly reflected in the values for α and κ. Thus, the model can produce predictions for the adsorption without an explicit dependence. In fact, having the implicit dependence simplifies the model structure and allows for compact system descriptions such as eq , and is a key strength of the current work.

The effect of not having a direct dependence of α and κ on material texture, ion valency, ion size, ion mobility, etc. is that the model cannot predict the effects of changing these parameters. But this would hardly be possible to do accurately anyway without some form of additional calibration because of the complex interdependency between ionic properties and detailed material nanostructure. Thus, recalibrating the currently developed and indirectly dependent model works equally well for a typical process and electrode material issues. Ultimately, these findings mean that the α and κ parameters should be viewed as condensed competitiveness ratings for a given ion in a given CDI device.

Second, the adsorption model assumes that the process is driven by electro-adsorption and thus does not account for chemical reactions at the electrode surface, meaning that the modeling accuracy might be lower in the faradaic high-voltage regime.

Third, the dependency of α and κ on the device structure and the ion type may initially seem to imply that a separate calibration is required for every ion and each system. But a single calibrating experiment is sufficient for a given system because eq shows that a single experiment can directly extract the parameter values for all present ionic species for a given device and operational conditions. After that, the model widely predicts adsorption for any new ion concentrations for short and long adsorption periods. Ultimately, this means that the model’s practical value lies in its ability to predict the adsorption for untested combinations of ion concentrations in a given device.

Finally, notice that the various derived equations (eqs and 9) allow the model to address varying complex experimental conditions. Equation suggests that the trace-ion adsorption is proportional to the trace-ion concentration, yielding a linear prediction trend that was verified in Figures and 4. Crucially, this means trace-ion adsorption could be directly predicted through analytical calculations. In contrast, eq additionally allows the model to estimate the final adsorption of arbitrary ionic mixtures by solving a set of nonlinear equations. Because the nonlinear equation is more complex to solve, the computer code provided in the Supplementary S-1 implements this general version of the derived model to make it more accessible to researchers.

Conclusions

We have developed a method for quantifying and predicting the adsorption in multi-ion solutions during electrosorption in a simple way. Crucially, it was shown that using either short electrosorption periods (in the linear adsorption region) or longer periods (to equilibrium), the relative adsorption of different ionic species is proportional to their respective concentrations in the source water. The proportionality constant is different for shorter and longer electrosorption periods, meaning a degree of ion selectivity can be achieved and predicted based on the appropriate choice of electrosorption periods used.

The results for short and long electrosorption periods were validated using data from reports in the literature. It was demonstrated that the method could describe ion adsorption in various multi-ion solutions and accurately predict the adsorption of trace ions based on their initial concentrations. Finally, it was demonstrated that enhancing the ion selectivity through the method described in this work could enable a desalination process to reach the drinking water standard while removing fewer ions, which might, in turn, reduce its time or energy requirements while also reducing the need for additional remineralization.