Dynamic Langmuir Model: A Simpler Approach to Modeling

We conclude that the model could accurately describe a wide range of key features while being a simpler approach than the commonly applied theories fo...
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Article Cite This: J. Phys. Chem. C 2019, 123, 16479−16485

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Dynamic Langmuir Model: A Simpler Approach to Modeling Capacitive Deionization Johan Nordstrand and Joydeep Dutta* Functional Materials, Applied Physics Department, SCI School, KTH Royal Institute of Technology, Isafjordsgatan 22, SE-16440 Kista, Stockholm, Sweden

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S Supporting Information *

ABSTRACT: Capacitive deionization (CDI) is emerging as an environmentfriendly and energy-efficient water desalination option for meeting the growing global demand for drinking water. It is important to develop models that can predict and optimize the performance of CDI systems with respect to key operational parameters in a simple way. Such models could open up modeling studies to a wider audience by making modeling more accessible to researchers. We have developed the dynamic Langmuir model that can describe CDI in terms of a few fundamental macroscopic properties. Through extensive comparisons with data from the literature, it is shown that the model could describe and predict charge storage, ion adsorption, and charge efficiency for varying input ion concentrations, applied voltages, electrolyte compositions, electrode asymmetries, and electrode precharges in the equilibrium state. We conclude that the model could accurately describe a wide range of key features while being a simpler approach than the commonly applied theories for modeling CDI.

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INTRODUCTION

While water on earth is abundant, most of it is saline and not suitable for human consumption. Seawater and brackish water constitute 97.5% of the available water supply, while glaciers constitute 2% and the fresh water that can be found in groundwaters, rivers, and lakes only constitutes around 0.5% of the water of earth.1,2 Scarcity of fresh water is a rising issue affecting large populations on earth,3 making efficient desalination of water critical for addressing this shortage.4 Capacitive deionization (CDI)4−6 is emerging as an environment-friendly option7 with low energy consumption for desalinating low salinity brackish water.8,9 The CDI technique is based on using a cell comprising of two large-surface-area (porous) electrodes separated by a spacer.10 During operation, voltage is applied to the electrodes while water is flowing through, which removes the salt ions from the saline water4 (Figure 1). The performance of the CDI operation depends strongly on structural,11−13 material,14 and operational parameters such as the water flow rate,15 the inlet ion concentration,16 the applied voltage,17 and type of ion species being removed.18,19 Because of this strong dependence on a wide range of parameters, appropriate models can be important tools for understanding and optimizing device performances. Various models have been developed to describe CDI processes such as circuit-based models (simple RC20 or transmission line21) or models considering how the ions are adsorbed in electric double layers (EDL) on the electrodes’ surfaces.22 Examples of the latter include Gouy−Chapman− © 2019 American Chemical Society

Figure 1. Illustration of water desalination by flow-between CDI. A CDI cell is constructed from two porous carbon electrodes separated by a spacer. When a voltage is applied to the cell, the ions are removed from the water and adsorbed in the electrodes, resulting in the production of fresh water from brackish water.

Stern theory,23 microporous electrode theory,24 the Nernst− Planck and Poisson equations,25 the modified Donnan (mD) model,10,26,27 and improved modified Donnan (i-mD).16 While complex models might be able to describe detailed system properties that might be difficult to investigate experimentally, such as the spatial variation of the concentration within the cell, simpler models could provide a more easily accessible tool to understand and optimize a system by describing it in terms of a few macroscopic properties, thus requiring less theoretical knowledge to interpret the results. This could be especially important, considering the wide range of parameters influencing the CDI processes. However, few models exist Received: May 3, 2019 Revised: June 5, 2019 Published: June 7, 2019 16479

DOI: 10.1021/acs.jpcc.9b04198 J. Phys. Chem. C 2019, 123, 16479−16485

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Charge Efficiency. In real systems, not all charges stored in the cell will lead to adsorption of ions. Thus, charge efficiency Λ can be calculated as the fraction of charges to adsorbed ions (eq 5). Here, F is the Faraday constant, Mv is the salt molar weight, Σ is the stored charge per electrode mass, and Γ is the mass of adsorbed salt per electrode mass. Although several mechanisms can be identified depending on the conditions,40 mainly co-ion expulsion will be considered here, that is, a net charge difference at the surface is created either because ions of the correct charge are brought to the surface or because ions of the opposite charge are removed.

that are both simple to implement and understand, while still being able to cover the key aspects of the CDI dynamics. One simple model that has been used successfully to describe ion-adsorption characteristics in CDI28−36 is the Langmuir isotherm,37 which describes how the saturated state of the electrode varies with the ion concentration of the input water. The Langmuir isotherm assumes that adsorption and desorption are simultaneous processes where the adsorption rate is limited by the number of free adsorption sites.38 Despite the successes in modeling concentration dependence, the Langmuir isotherm does not account for other system properties such as charge efficiency. However, as the model works well for describing the device characteristic with varying input ion concentrations, an adapted version to CDI might provide a pathway to describe a range of key features of a CDI operation in a simpler fashion. In this work, we have developed a model extending the Langmuir isotherm to describe CDI processes. The model could describe and predict both electrode charges and ionadsorption capacities for a wide range of key features such as the voltage and the electrolyte composition while being simpler and easier to visualize than other models developed to date.

Λ=

THEORY In this section, the dynamic Langmuir (DL) model is derived based on the classic Langmuir isotherm for describing the equilibrium state of adsorption and desorption of gases on surfaces.37,38 The Langmuir theory suggests that adsorption and desorption are simultaneous processes in which adsorption is limited by the number of free and active adsorption sites on the surface. Thus, for ideal gas adsorption and desorption on a perfectly flat surface, eqs 1 and 2 describe the isotherm and the dynamics of the system, respectively. Here, θ is the fractional coverage, pA is the adsorbent partial pressure, and KL is the equilibrium constant, while kads, kdes are constants related to adsorption and desorption processes.

dcads = kadsc(S − β0 − β1c0−cadsz)−kdescads dt

dθ = R ads−R des = kadspA (1 − θ )−kdesθ dt

(1)

(2)

Ideal Conditions. The Langmuir isotherm has been applied to liquids by substituting pA with the adsorbent concentration c.38 For the DL model, it is assumed that rather than passive adsorption, the ions adsorb onto the electrodes because of the application of voltage. Therefore, the sites considered are “voltage-induced sites”, S, determining the maximum charge adsorption. Charge storage in supercapacitors is proportional to the applied voltage,39 and thus S should also be proportional to the voltage (S = S0V/V0 for some base level S0,V0). Exchanging pA for the concentration of charges σ [M] thus leads to expressions for the adsorption/ desorption process (eq 3). Using σ = cz, where z is the ion valence, the corresponding equation for ideal salt adsorption can then be derived (eq 4). dσads = kadsσ(S − σads)−kdesσads dt

(3)

dcads = kadsc(S−cadsz)−kdescads dt

(4)

(6)

Setting the derivatives to zero and solving the equation yields the new isotherms for charge storage and ion adsorption (eqs 7 and 8). Here, the parameters A, B, and D are constant at a given voltage. Also, ρ is the ratio of electrode mass to cell volume. Note also that the proportionality relationship between S and V makes A′ and A″ increase linearly with voltage. Dividing the adsorption by the charge yields the charge efficiency. Notably, the linear blockage of sites implies that the charge efficiency is linear with concentration (eq 9. Parameters λ0 and λ1 are constants).

KLpA 1 + KLpA

(5)

In the DL model, charge efficiency is accounted for by assuming that a fixed number of sites are blocked off from storing extra ions because the charge-storage capacity has been used for removing co-ions instead (S is reduced by a term that is constant at given experimental conditions). It will be assumed that this blockage of sites arises from two sources. First, the passive presence of ions close to the pore wall should increase proportionally to the concentration in the bulk solution. Second, even for net uncharged electrodes at low ion concentrations, there will always be some ions present in the pores, which is often attributed to the presence of various charged chemical groups attracting ions.18 Incorporating the co-ion expulsion into eq 4 yields eq 6 for salt adsorption, where the linear factor is assumed to be the only source of unideal charge efficiency.

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θe =

FΓ MvΣ

Σ [C/g] = =

F [sA/mol] ρ [g/L]

F Ac0 ρ 1 + Bc0

A′c0 1 + Bc0

Γ [mg/g] =

(7)

M v [mg/mol] ρ [g/L]

ce [mol/L] =

Λ=

σe [mol/L] =

A ″ c 0 − D″ c 0 2 M v Ac 0 − Dc0 2 = ρ 1 + Bc0 1 + Bc0

λ + λ1c0 FΓ =1− 0 MvΣ V

(8)

(9)

Multicomponent Electrolytes. In the DL model, a competitive environment between ions of the same charges could be incorporated by considering each ion type separately, with different adsorption and desorption constants. However, 16480

DOI: 10.1021/acs.jpcc.9b04198 J. Phys. Chem. C 2019, 123, 16479−16485

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The Journal of Physical Chemistry C one should also account for the fact that the ions fill the same S-sites (eq 10).

the last step comes from the capacitance being proportional to the charge.

(i) dcads (i) (i) (i) (i) = kads c0 (S − β0(i) − β1(i)c0(i) − Σiz(i)c(i))−kdes cads dt

1 1 1 k 2k = + → C tot = C → Q k = Q1 C tot kC C k+1 k+1 (10)

(12)

A simpler expression can be derived regarding the equilibrium state in a competitive environment, that is, a solution containing different ion species of the same charge sign. The relative adsorption of two species at equilibrium can be calculated by setting the derivatives to zero and taking eq 10 for one species divided by the corresponding equation for the other. Using the simplifying assumption that the charge efficiency is the same for both species, the factors in the parenthesis are the same for both species as well and cancel. This leads to a simplified expression in eq 11, showing that the relative adsorption of two species should be proportional to their relative concentrations. Here, it is interesting to note that no assumption has been made about species (i) and (j) being the only species in the solution.

Implementation. For the saturated state, all features of interest such as adsorption and charge efficiency have been expressed in the form of isotherms, that is, expressions that can be evaluated directly, given the values for the fitting parameters. Thus, the model can be fitted to experiments directly, which in this work has been performed using the leastsquares (lsqcurvefit) function in MATLAB. Notably, because the expressions are decoupled, it is also possible to investigate specific trends such as the relative variation with asymmetry (eq 12) without the experimental data required to fit and describe other parameters such as the absolute charge and ionadsorption characteristics.

(i) cads (j) cads

=

(i) (i) kads /kdes (j) (j) kads /kdes

ce(i) ce(j)

=

c(i) α e(j) ce

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EXPERIMENTS The DL model predictions were extensively compared to experimental data, extracted from prior reports in the literature. The WebPlotDigitizer software was used to extract the data and convert graphs in manuscripts to data files.41 The reports were chosen based on which showed the dependency of the CDI process on the parameter of interest (such as concentration) and had extensive data in a wide parameter range (such as concentrations ranging from drinkable to the upper operating conditions in CDI of around 200 mM). Preferably, they should have had data about the same trend of interest under various other factors (such as displaying the trend corresponding to increasing electrode asymmetry for different voltages and inlet concentrations). Where the same trend is demonstrated for multiple conditions in the report from which the data were extracted, one representative sample is shown in the main text, and the rest have been added to the Supporting Information for completeness.

(11)

The concentration on the right-hand side of the equation mentioned above is the equilibrium concentration in the cell. For continuous-mode operation of a CDI device, this is the same as the input ionic concentration, while for batch-mode operation, the equilibrium concentration can be exchanged for the initial reservoir concentration if the batch is large (the fraction of ions removed is small) or the adsorption is similar between the species (α ≈ 1) (derivation in eq S1). Note also that because only the fraction of ion adsorptions and the fraction of initial concentrations are considered in eq 11, the units of concentration can be expressed as either concentration per volume of the cell (or batch) or in units of mg/g adsorption-to-electrode-mass ratio, without affecting the results. Electrode Asymmetry. Electrode asymmetry in terms of unequal electrode masses can affect the total adsorption capacity of a CDI system. Symmetrically scaling a system (such as having two identical cells instead of one) should have no impact of the adsorption per mass of electrodes and thus the state equations remain the same. If only one electrode is enlarged, the total size still increases. Seemingly, this would imply that more sites for adsorption are created on one side. However, because the effluent water produced should be charge-neutral, the adsorption of charges should be the same in both electrodes. This implies that the voltage will be distributed in such a way that the voltage-induced effective number of surface sites S increases equally on both sides, even if the total surface area is larger on one side. For simplicity, it is therefore assumed that every asymmetric system is equivalent to some symmetric system. To quantitatively describe the effect of asymmetric electrodes, a separate calculation can be made. The two electrodes of an electrolytic capacitor can each be thought of a separate capacitor between the electrolyte and the electrode pore matrix and, therefore, be considered as connected in series. Denoting the capacitance of the electrode of smaller electrode C and taking the larger electrode to be k times as heavy (capacitance kC), the total capacitance of the system can be calculated as in eq 12 as per the standard formula for capacitors in series. Here,

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RESULTS Because of the complexity of the CDI system, there are lots of parameters affecting the process that can be studied. In this section, it is shown using extensive comparisons with the literature that the DL model accurately describes most of the important concepts in CDI, including how the charge storage, charge efficiency, and salt adsorption vary with the inlet concentration, applied voltage (including discharge voltage), electrode precharge, electrolyte composition, and increasing asymmetry of the electrodes. Concentration Dependence. The DL model assumes that storage of charges onto electrosorptive sites is the fundamental process in CDI and that it should follow a standard Langmuir isotherm (eq 7, Figure 2a). In contrast, the salt adsorption is an indirect effect of applying a voltage and can be reduced by co-ion expulsion which causes the charge storage capacity to be used for pushing away ions rather than adsorbing ions in the electrodes (eq 6). These co-ions present on the surface were attributed to both a passive (concentration dependent) presence near the electrodes and a fixed presence due to immobile charged groups, leading to a predicted linear trend between charge efficiency and inlet concentration (eq 9, Figure 2b). Putting charge storage and charge efficiency together, the ion adsorption increases quickly at low 16481

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More comparisons showing that the DL model indeed describes the dependency of both charge, adsorption, and charge efficiency on the charging voltage are provided in the Supporting Information. Figure S1 shows a more extensive dataset from Biesheuvel et al. in ref 16 with the trends for the charge storage, adsorption, and charge efficiency. Also, Figure S2a,c shows the same data as in Figure 3a (also from Kim et al. in ref 42) but for two different ion concentrations of 5 and 80 mM, respectively. Discharging Voltage. In the data shown above, the CDI cell was always discharged at zero applied voltage. However, net adsorption could be calculated between two arbitrary voltages applied to the electrodes by computing the isotherms at each voltage and taking the difference. Notably, for the adsorption of salt, the voltage difference between the electrodes before the CDI process starts is important (Figure 3a) because it determines the number of co-ions present in the vicinity of the electrodes when the process starts. From eq 8, the charge efficiency should be nearly 100% if the discharge voltage is over a threshold voltage Vt, as the ion adsorption increases ideally with the charge storage. However, at discharge voltages below Vt, no additional adsorption should occur. This predicted trend was investigated by calculating Vt from Figure 2a and comparing the predicted trend with the charge and adsorption data from the same reports (by Kim et al. in ref 42). There is a good agreement between the DL model and experimental results (Figure 2c), showing that these assumptions for a CDI system accurately describe what is observed in the experiments. More extensive comparisons between model and experiment to show that the DL model satisfyingly predicts experimental observations can be found in the Supporting Information. Figure S2b,d shows similar results as in Figure 3c, but for 5 and 80 mM ion concentrations (also from Kim et al. in ref 42). Moreover, Figure S3 (data from Avraham et al. in ref 43) shows that the DL model could be fitted to adsorption-overdischarge-voltage data directly by fitting Vt rather than calculating it based on a separate set of experimental data. Precharged Electrodes. Introducing fixed charges on the electrode surface changes the point of zero charge (PZC)44 and can lead to significantly higher charge efficiency in CDI systems.44−46 Without an applied voltage, the charges from the fixed groups are balanced by adsorption of ions and desorption of co-ions.44 Because of the change in the PZC, a potential is generated without an external voltage. Compared with the results from the discharge voltage section mentioned above, having a baseline charge could thus be thought of as raising both the charging and the discharging voltage by a level corresponding to the amount of precharge introduced. For example, Cohen et al. (ref 44) reported the charge efficiency of an uncharged system as 58% with an applied voltage of Vch = 0.9 V and a discharge voltage Vdisch = 0 V. Because the charge efficiency Λ depends on the fraction of Vt to Vch (derived in eq 13, but more easily understood from Figure 3a), the threshold voltage can thus be calculated as Vt = 0.38 V. Cohen et al. (ref 44) used a 100 mM NaCl solution, so this value is reasonable considering that the threshold in the earlier data was below 0.2 V for 5 mM ion concentration as reported by Kim et al. (ref 42) (Figure S2a) and around 0.3 V for 80 mM (Figure S2c).

Figure 2. All data from ref 16 where a voltage of 1.2 V was utilized during operation. (a) Charge and adsorption under varying inlet concentrations. Model lines calculated by first fitting the isotherm for the charge, then fixing those parameters, and fitting only the co-ion expulsion term for adsorption. (b) Charge efficiency as calculated from the data in (a).

concentrations until a steady level is reached but decreases slowly at higher concentrations (Figure 2a). Comparisons with experiments from ref 16 show that all these predicted trends hold for a wide range of inlet ion concentrations (Figure 2). Charging Voltage. In standard capacitors, the amount of charge that can be stored increases linearly with the applied voltage. This is included in the DL model by assuming that the number of voltage-induced sites S increases linearly with the applied voltage (eq 7, Figure 3a using data from Kim et al. in

Figure 3. All data from ref 42. (a) How the charge and adsorption vary with voltage. Model lines fitted using eqs 6 and 7. (b) Because the charge and adsorption is known, the charge efficiency of the system can be calculated (eq 8). (c) Variation with the discharge voltage when the charging voltage was constant at 1.2 V. The adsorption and charge were calculated as the net increase between the charge/discharge voltage points in (a).

ref 42). Also, the difference between the number of charges stored and the amount of adsorbed salt was attributed to a fixed blockage of surface sites (eq 6). This suggests that no significant net adsorption of ions should take place up to a fixed threshold voltage Vt required to overcome the blockage, after which the adsorption increases ideally with an increase in stored charges (Figure 3a). The trend for the charge efficiency as predicted by the DL model (eq 9) can be understood by considering that the number of co-ions that are present before the voltage is applied is independent of the applied voltage. Therefore, the fraction of blocked sites to total sites should be inversely proportional to the applied voltage, which agrees very well with the experimental results extracted from the report by Kim et al. in ref 42 (Figure 3b). 16482

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0·Vt + 1·(Vch − Vt) → Vt = Vch(1 − Λ) = 0.38 V Vch (13)

With the precharged electrodes, it was reported that the electrodes were at −0.25 and 0.25 V with respect to a reference electrode, without an externally applied voltage. This sums up to 0.5 V. Thus, the applied voltage 0−0.9 V could effectively be considered between 0.5 and 1.4 V. Because the lower level is above the threshold of zero adsorption, the calculations presented here imply that the charge efficiency should rise to nearly 100%. Indeed, Cohen et al. (ref 44) have reported that the experimentally measured charge efficiency for the precharged system rose to 93%. Multicomponent Electrolytes. While the classic Langmuir isotherm has been used solely for describing the adsorption of salt rather than ions, the DL model incorporates competition between ions of the same charge sign by considering their adsorption and desorption strengths separately, although the electrosorptive sites are shared. This view leads to an expression for the relative adsorption of ion species that can be easily tested with experiments, that is, that their relative adsorption should be proportional to their relative concentration (cads/c′ads = αc/c′) (eq 11). Prior experiments from the literature reveal that this holds both for anions (Figure 4a, from Tang et al.47) and cations

Figure 5. (a) Variation in charge efficiency depending on the mass ratio. Model lines were fitted by assuming a constant Λ. Data were taken from ref 16 where a voltage of 1 V was used. (b,c) Charge and adsorption under varying asymmetry. Data were taken from ref 49. The y-axis shows charge/adsorption as a fraction of the total mass (for both electrodes) of the original 1:1 mass ratio system. The model lines were fitted following eq 12.

effect as some symmetric scaling of electrode mass. In the Theory section, it was shown that this scaling could be calculated as Qk = Q12k/(k + 1) for a cell where the larger electrode has k times the mass of the smaller electrode (eq 12). This relationship holds very well for both adsorption and charge storage for different voltages (Figure 5b with data from Porada et al. in ref 49). Porada et al. studied two concentrations, 5 and 20 mM. In the Supporting Information, the same data as in Figure 5b but for 20 mM ion concentration are shown, indicating that it holds true for varying concentrations as well as voltages (Figure S4).

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DISCUSSION It is interesting to note that proponents of earlier theories have argued that isotherm-based models cannot be used for CDI4,5 because such models could not be used to describe key system properties in CDI, namely, how charge storage and ion adsorption depend on concentration, applied voltage, system asymmetry, and electrolyte composition (see Figure 12 in ref 5). It was argued that for reasons such as that the Langmuir isotherm is derived for the adsorption of gases (which is not governed by electric forces)5 and does not consider anions and cations separately,4 these important concepts could not be integrated into an isotherm-based model. However, in this article, we have demonstrated that the isotherm-based DL model could indeed be used to describe all the listed properties and more. Simplicity is an important virtue for models.50,51 While both the DL model and previous models could be used to describe the wide range of phenomena presented here, the DL model is simple and transparent as it is based on the clear concepts of adsorption and desorption strengths and requires few fitting parameters. For comparison, the data used in this work were taken from manuscripts where the mD/i-mD models were used (except ref 48 on electrolyte composition that did not apply any model). Despite the i-mD being praised as simpler than other common models,16 it still requires up to four fitting parameters to be accurate.42 Additionally, it requires more theoretical knowledge to understand as well as empirical

Figure 4. (a) Low concentration of F (FCl) ions, 20 mg, was mixed with a higher concentration of NaCl, {0.5, 1, 1.5, 2, 3} g, yielding a range of relative initial concentrations. Data were taken from ref 47. (b,c) For the three data points corresponding to higher values on the x-axis, the electrolyte contained a constant 2 mM of Ca and K, while the Na concentration was varied as 2, 4, and 6 mM. For the data point corresponding to the lowest point on the x-axis, the concentrations of K+, Na+, Ca2+, and Mg2+ were 0.26, 10.57, 1.45, and 2.41 mM, respectively. Data were taken from ref 48, Table 2. Model lines were based on eq 11.

(Figure 4b, from Hou et al.48), in solutions where one species is much more diluted than the other (Figure 4a) as well as solutions where the concentrations are similar (Figure 4b). Notably, it also holds for the competition between monovalent and divalent ions as described by Chen et al. (Figure 4c, from ref 48) and for solutions containing more than two ion species (Figure 4b,c were taken from a solution containing K+, Na+, Mg2+, and Ca2+). Asymmetric Electrodes. In a cell with uncharged electrodes, asymmetry can be created by making one electrode larger than the other. It is reported in the literature that this type of asymmetry does not affect the charge efficiency of the CDI system (Figure 5a, data from Biesheuvel et al. in ref 16), suggesting that an asymmetric scaling of the mass has the same 16483

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The Journal of Physical Chemistry C correctional factors to be accurate. For instance, an understanding is required of the EDL structure and overlapping double layers in micropores (volumetric stern layer capacitance in the zero-charge limit Cst,vol,0) as well as nonelectrostatic adsorption potential in combination with image forces (micropore ion-correlation energy E).

Notes

CONCLUSIONS In this work, the dynamic Langmuir (DL) model has been derived as an extension of the classical Langmuir isotherm, adapted for CDI processes. It has been shown by comparing extensively with reports from the literature that the model could describe charge storage, ion adsorption, and charge efficiency in the equilibrium state for varying inlet ion concentrations, applied voltages, electrolyte composition, and electrode asymmetry. Using the model is simple because all these results are summarized with decoupled mathematical formulas. Thus, fitting can be done using a least-squares method, and predictions can be made by direct computation. Also, describing specific properties requires a fitting of the parameters related to that property only, meaning that the complexity of the model scales with the task at hand. The model is also transparent, in that the CDI process can be understood directly through the macroscopic properties of adsorption and desorption strengths. In conclusion, the dynamic Langmuir model is simpler and a more transparent modeling tool that can be used to describe a wide range of key CDI properties. It is hoped that it can serve as a way of making CDI modeling more accessible to researchers.

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge funding from the MISTRA Terraclean project (diary no. 2015/31) and the Swedish research council (diary no. 2018-05387).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b04198.

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REFERENCES

Additional comparisons between the DL model and experiments; the first comparison showing trends for ion adsorption, charge storage, and charge efficiency as a function of the charging voltage based on data from the report by Biesheuvel et al. in ref 16; the second showing the trends for ion adsorption, charge storage, and charge efficiency as a function of both charging and discharging voltage based on the data in the report by Kim et al. in ref 42; the third showing the trends for adsorption with variations in the discharge voltage, based on the data from the report by Avraham et al. in ref 43; the fourth showing the trend for adsorption and charge for varying electrode asymmetry based on data from the report by Porada et al. in ref 49; comparisons between the DL model and experiment; and a derivation showing that eq 11 holds true even if the right-hand side of the equation is taken to be the initial batch concentration rather than the equilibrium batch concentration under the conditions that the batch is large or the adsorption is similar between the ion species (PDF)

AUTHOR INFORMATION

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Joydeep Dutta: 0000-0002-0074-3504 16484

DOI: 10.1021/acs.jpcc.9b04198 J. Phys. Chem. C 2019, 123, 16479−16485

Article

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DOI: 10.1021/acs.jpcc.9b04198 J. Phys. Chem. C 2019, 123, 16479−16485