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Dynamic Adsorption/Desorption Process Model for Capacitive Deionization P. M. Biesheuvel,*,†,‡ B. van Limpt,†,‡ and A. van der Wal§ Department of EnVironmental Technology, Wageningen UniVersity, Bomenweg 2, 6703 HD Wageningen, The Netherlands, Wetsus, Centre of Excellence for Sustainable Water Technology, Agora 1, 8900 CC Leeuwarden, The Netherlands, and Voltea B.V., Wassenaarseweg 72, 2333 AL Leiden, The Netherlands ReceiVed: October 31, 2008; ReVised Manuscript ReceiVed: January 24, 2009
In capacitive deionization (CDI), an electrical potential difference is applied across oppositely placed electrodes, resulting in the adsorption of ions from aqueous solution and a partially ion-depleted product stream. CDI is a dynamic process which operates in a sequential mode; i.e., after a certain ion adsorption capacity has been reached, the applied voltage is reduced, and ions are released back into solution, resulting in a solution concentrated in ions. The energetic input of CDI is very small, while there are no ion-exchange materials involved that need to be replaced regularly. Here we present a dynamic process model for CDI which includes the storage and release of ions in/from the polarization layers of the electrodes. The charge and ion adsorption capacity of the polarization layers is described using the equilibrium Gouy-Chapman-Stern (GCS) model, while the charge transfer rate from bulk solution into the polarization layer is modeled according to Ohm’s law, i.e., depends solely on an electric field term across a mass-transfer layer. An important element in the model is the differential charge efficiency: the effective salt removal rate relative to the electronic current, for which an analytical expression is derived based on the GCS model. We present results for the effluent salt concentration and electron current, both as function of time, based on a process model that assumes ideal mixing in the CDI unit cell. The theoretical results are in very good agreement with an example data set. Introduction In capacitive deionization (CDI), or “electrosorption desalination”, ions are removed from an aqueous stream by applying a potential difference across two juxtaposed electrodes.1-19 The ions that are removed are stored in the diffuse part of the electrostatic double layers (or polarization layers) which develop on both electrodes. When a continuous stream of solution is fed to a CDI stack, the ion concentration of the effluent will be reduced compared to the inflowing solution, which can continue until the polarization layers reach their final ion adsorption capacity. By reducing the applied voltage, ions are released back into the solution, and a product stream concentrated in ions is obtained. To obtain a large ion adsorption capacity, it is important that a large surface area is created and thus typically materials such as porous activated carbons are used that have internal surface areas of the order of 103 m2/g. In principle, ions can be adsorbed both physically in the diffuse part of the double layer as well as due to chemical adsorption. In this paper we will only consider physical adsorption. CDI is a complicated dynamical nonlinear process, and exact dynamical multidimensional numerical mass and charge transport models which include such geometrical aspects as the porous structure of the carbon electrodes and the exact flow profiles in the spacer compartment are not yet available. Therefore, it would be valuable to have a simplified model that can be used to rationalize CDI performance and thus to optimize CDI design and operating conditions. Such an empirical process model describes the dynamics of ion adsorption into the porous electrodes and includes the structure of the polarization layers. * Author to whom correspondence should be addressed. E-mail:
[email protected]. † Wageningen University. ‡ Wetsus, Centre of Excellence for Sustainable Water Technology. § Voltea B. V.
Modeling approaches for the CDI process are presented by Murphy and Caudle2 and Johnson and Newman,3 but they do not take the structure of the polarization layers into account. More general theoretical work based on solving the timedependent Poisson-Nernst-Planck equation for macroscopic systems in contact with blocking electrodes is reviewed and extended in ref 20; see also refs 21 and 22. The objective of the present contribution is to develop a mathematically simplified electrochemical process model for CDI in which we couple a macroscopic description of the CDI process to a nanoscopic model for ion and charge storage in the polarization layers (double layers) that develop on the internal surfaces of the porous electrodes. The double-layer model is used to develop expressions for the (differential) charge efficiency, which describes how many ions are removed from solution for each electron that is transferred from one electrode to the other. The CDI process model predicts the effluent ion concentration, and electron current, as function of time. To keep things simple to start with, we will make several significant assumptions. First of all, we will assume that the bulk volume in a CDI unit cell is ideally mixed, which implies that the ion concentration of the effluent is equal to that within the CDI compartment. Theoretically, one can modify this model and describe the transport process in the compartment by an increasing number of ideally stirred volumes placed in-series, which for high numbers becomes equivalent to the plug-flow regime. In the present note, however, we will only present calculation results for a CDI process model in which the entire solution volume in the compartment is assumed to be ideally mixed. To describe the structure and ion adsorption capacity of the polarization layers that form on the electrodes, we will use the Gouy-Chapman-Stern (GCS) model, which is a twolayer nanoscopic double-layer model which includes the diffuse layer in which the ions (modeled as point charges)
10.1021/jp809644s CCC: $40.75 2009 American Chemical Society Published on Web 03/03/2009
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are stored in the local electric field.20,23 The diffuse layer is separated from the electrode surface by the charge-free Stern layer (also called the inner, or compact, layer). To keep matters as simple as possible in this contribution, we will only consider a planar electrode surface (planar at the dimension of the double layer, which is several nanometers) and will neglect ion volume constraints.24-26 Although we know that the following effects are important in reality, we will also assume that the two electrodes are perfectly non-Faradaic; i.e., no current is assumed to flow via electrochemical reactions from the electrodes into solution, or vice versa, while additionally we will neglect the possibility of direct chemical ion adsorption or desorption on the electrode surface; i.e., we assume the electrode to be uncharged when in contact with water. Therefore, in the model an ionic charge only develops in response to an electric current and is due to the diffuse charge from the excess of counterions and the depletion of co-ions. Without chemical ion adsorption, at equilibrium the diffuse layer charge is exactly compensated by the free electron charge residing at the electrode-solution interface. Important is the realization that this charge does not equal the salt adsorption, because the electron surface charge is compensated in the diffuse layer by an adsorption of counterions as well as by a depletion of co-ions.3,5,10,19 Consequently, the charge efficiency (the ratio of removed salt molecules, over the electronic current) is always below unity. This important effect is included in the model and will be discussed below in more detail. To describe the charge transport from bulk solution to the interface, we use an approximate approach based on the concept of a mass-transfer (boundary) layer (diffusion layer, or film layer), envisioned in front of the electrodes. The charge transport rate through the mass-transfer layer is modeled as the product of the ion concentration in bulk, a mass-transfer coefficient, and an electrostatic driving force, thus neglecting diffusional (concentration gradient) effects. This is analoguous to using Ohm’s law and is an approach that is often very effective in describing ion transport in stagnant media.22,27 Here we apply the same concept to the mass-transfer boundary layer in CDI. Theory We describe a unit cell in the CDI process by assuming that the flow compartment, being the space in between the electrodes, is ideally stirred. This is a significant simplification which can be relaxed in future work. In this case the overall unit cell salt mass balance is given by
V
dc ) Φ(c0 - c) - Φsalt dt
(1)
where V is the volume of the flow compartment, c the salt concentration (equal to the ionic strength, as we will only consider a monovalent salt solution), t time, Φ the volumetric solution flow rate (in volume/time), c0 the ionic strength of the inflowing solution, and Φsalt the rate by which ions are removed from solution and adsorbed at the electrodes. A complication in the description of the ion adsorption process in CDI is the fact that the ion removal is not equivalent to the adsorption of a single component (as in a typical adsorption process) but is the consequence of several processes occurring jointly: on the positively charged electrode the negative ions (anions) are stored, while the positive ions (cations) are expelled from the diffuse part of the double layer and are forced into solution. On the negatively charged electrode,
the exact opposite process occurs. Because the ion adsorption of the counterion (the ion of opposite charge to the electrode charge) is always larger than the expulsion of the co-ion (the ion of the same charge as the electrode), the result is a net ion removal from solution. The full description of this process is complicated, requiring at least the solution of the dynamic Nernst-Planck-Poisson model both in solution as well as in the pores of the electrodes. Here, however, we will take a much simplified approach, one which is analogous to Ohm’s law for charge transport in an electric field, and assume that the major limitation for ion transport is in a thin planar mass-transfer layer next to the electrodes (with area A, the projected area per electrode, which we assume equal for both electrodes). In this approach we neglect ion concentration gradients as a driving force for transport and only consider the effect of the electric field (migration). We will not describe the transport of each ion separately but focus solely on the net charge transport rate from solution into the electrodes, Jcharge, which we will empirically describe as the product of c, the ion concentration in solution (which we assume constant across the mass-transfer layer), a mass-transfer coefficient k, and the electrostatic driving force across the mass-transfer layer, ∆φmtl. This factor is a nondimensional electrostatic driving force, which can be multiplied by the thermal voltage VT ) RT/F to give a voltage difference with dimension V. Thus, we describe Jcharge as
Jcharge ) k · c · ∆φmtl
(2)
Equation 2 is analogous to Ohm’s law for charge transport in an electric field where kL · c represents an electrical conductance (one over an electrical resistance). However, compared to Ohm’s law, here we split this factor in a concentrationindependent term k and the ion concentration c. This has the effect that the resistance for charge transport increases when the ion concentration decreases. Next we discuss how Jcharge relates to the salt removal rate Jsalt, which when multiplied by A, the total projected area of the mass-transfer layer per electrode, results in Φsalt which is required in eq 1. As discussed in ref 19, the ratio between the total amount of ions removed from solution, Γsalt (per unit internal electrode area a) over the surface charge density of the diffusion layer, σ, is described by the integral charge efficiency Λ, a number which is always below unity, because the electron charge σ is for the one part compensated in the diffuse part of the double layer by counterion adsorption from solution, and for the other part by co-ion expulsion back into solution. (If only the first process would take place, Λ would be unity.) For the Gouy-Chapman-Stern (GCS) model, an analytical expression was derived for Λ in ref , given by
Λ)
Γsalt ∆φd ) tanh σ 4
(3)
where ∆φd is the nondimensional electrical potential difference over the diffuse part of the double layer. Equation 3 is based on eq 7 to be discussed later for the surface charge density σ, and on Γsalt ) 8 · c∞/κ · sinh2(∆φd/4) for the salt absorption per unit area, an expression which was already presented in Bazant et al.20 in reduced units. Equation 3 is not only based on the GCS model but additionally assumes that the area of the two opposite electrodes is exactly the same, and that for each
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electrode the Stern and diffuse layer structures are exactly identical, except for the difference in the sign of the potential. Now, the ratio between Jsalt and Jcharge is given by λ, the differential charge efficiency, which can be derived from the expressions for σ and Γsalt, and is given by
λ)
Jsalt dΓsalt ∆φd ) ) tanh Jcharge dσ 2
(4)
Combining these elements we can describe the salt removal rate Φsalt as
Φsalt ) A · Jcharge · λ
(5)
with Jcharge given by eq 2 and λ by eq 4. In the model, the applied nondimensional cell voltage φcell (which is the dimensional cell voltage Vcell divided by the thermal voltage VT) is divided over the two electrodes and is (on each electrode) a summation of the Stern layer voltage difference ∆φSt, the diffuse layer voltage difference ∆φd, and the voltage difference over the mass-transfer layer, ∆φmtl
1 V /V ) ∆φmtl + ∆φd + ∆φSt 2 cell T
(6)
In the GCS model, the electric charge σ is given by
c∞ 1 σ ) 4 · sinh ∆φd κ 2
(7)
where σ is a dimensionless surface charge density (number of charges per unit electrode area), κ is the inverse of the Debye length, given by κ2 ) 8πλBc∞ where λB is the Bjerrum length (λB ) 0.72 nm in water at room temperature), and c∞ is the ionic strength expressed in numbers per volume. The voltage difference over the Stern layer ∆φSt relates directly to σ according to Gauss’ law
CSt · ∆φSt · VT ) σ · e
(8)
where e is the electron charge and CSt the Stern layer capacity. Finally we must relate the charge density on the electrode surface σ to the charge transport rate through the mass-transfer layer, Jcharge, according to
dσ A ) Jcharge dt a
(9)
Results and Discussion In Figure 1 we present both experimental and theoretical results for the ion removal and ion release steps in CDI, as function of time. Results are presented for both the electronic current and the effluent ion concentration. The data are obtained using a stack with four unit cells. Each unit cell consists of a pair of parallel porous carbon electrodes, with each electrode positioned on top of a graphite current collector which is electrically connected to a potentiostat. The electrodes are separated by a glass fiber spacer, with a carbon mass per electrode of 0.42 g. The electrodes are synthesized from Norit
Figure 1. Effluent ion concentration and electron current in a capacitive deionization stack: (a) experimental data, (b) theoretical calculations (parameter settings in text). At t ) 0 s, a potential difference of 1.2 V is applied across the electrodes, which is reduced to zero at t ) 300 s. At t ) 600 s -1.2 V is applied across the electrodes, which are again short-circuited at t ) 900 s.
DLC Super 50 according to an in-house preparation method; see ref 28. The BET surface area of this material is measured as 1224 m2/g carbon. Each unit cell has a square geometry with each electrode of thickness 0.32 mm and area of A ) 33 cm2, with the spacer of 0.32 mm thickness placed in between each pair of electrodes. The electrolyte with salt concentration c0 ) 10 mM was continuously pumped through the CDI stack (flow rate Φ ) 0.38 mL/s per unit cell), flowing radially outward from the center of each unit cell through the planar slit between the parallel electrodes. The total stack volume per unit cell was V ) 17.5 ml. Based on a calibration curve, the measured conductivity of the effluent was recalculated to salt concentration. Note that the salt concentration data as presented in Figure 1a are shifted “to the left” by 10 s compared to the actual data. The lag in the original data is due to the fact that the conductivity device is located a short distance downstream from the CDI stack. At time zero Vcell ) 1.2 V is applied across each pair of electrodes, which is reduced to zero (effectively short-circuiting the system) at t ) 300 s, after which Vcell ) -1.2 V is applied across the electrodes at t ) 600 s, a voltage which is reduced to zero again at t ) 900 s. (It must be noted that in the actual experiment each phase had a duration of 400 s, but for clarity we do not present the data of the last 100 s of each period in Figure 1a.) Analyzing the data, we find that the removed amount of salt in the charging step (equal to the total amount of salt returned to solution in the discharge step) is equal to Γsalt · a ∼ 70 µmol per unit cell. The total charge input equals the charge output and is given by σ · a ∼ 8.8 C per unit cell. Based on these two
Model for Capacitive Deionization numbers, we can calculate that the integral charge efficiency Λ is given by Λ ) 70*10-6*96 485/8.8 ∼ 0.77. This number is lower than calculated values using the GCS model at 10 mM ionic strength (Λ ) 0.881 at a cell voltage of 1.2 V and for CSt ) 0.2 F/m2), also when ion volume effects are included (Λ ) 0.866 using the GCS-Carnahan-Starling model as discussed in ref 19 using an ion hydrated size of 0.7 nm). Several explanations can be forwarded to rationalize this observation, one being the possibility that part of the charge accumulates at the interfaces of pores in the carbon electrodes that are too small for ions to penetrate and therefore do not participate in ion removal. Alternatively, double-layer overlap in the micro- and mesopores might lead to lower salt adsorption.29 The analysis of double-layer models that include the detailed structure of a realistic carbon electrode material will be taken up in future work. Instead, in the present paper we will use the planar GCS model. From the data and the theory, we can furthermore estimate the internal surface area a of the electrodes. Using the GCS model with a Stern capacity of CSt ) 0.2 F/m2, the salt adsorption at 1.2 V is given by Γsalt ) 0.84 µmol/m2 (surface charge density σ ) 0.57 electrons/nm2), and thus we estimate an effective area of a ) Γsalt · a/Γsalt ) 70/0.84 ) 83 m2 per electrode, which is ∼16% of the total BET surface area (83/ (1224*0.42) ) 16%). The experimental data are presented in Figure 1a, which shows a rapid decrease in ion concentration during the ion removal step and the slower return of the effluent ion concentration back to the concentration of the inflowing solution. The increase in the effluent ion concentration is due to the fact that the capacity of the electrodes to store salt is limited, and therefore after some time solution will flow through the CDI stack without any more ions being removed. At t ) 300 s the applied voltage difference of Vcell ) 1.2 V is reduced to zero, and ions are released back into solution. During ion release, the electron current has the opposite sign as during the previous ion removal step. The data show that ion release goes much faster than the ion adsorption process, and the ion concentration profile is much “more peaked”. A second run at opposite voltage shows exactly the same behavior (for the time period 600-1200 s). Results of the theoretical calculations are presented in Figure 1b using as parameter settings a ) 83 m2, V ) 17.5 mL, Φ ) 0.38 mL/s, c0 ) 10 mM, A ) 33 cm2, k ) 0.77 µm/s, Vcell ) 1.2 V, and CSt ) 0.2 F/m2. The predictions of the theoretical model correspond very well to the experimental results for the charging step (ion removal step), with similar magnitudes and profiles both for the effluent ion concentration and the electron current. However, taking these parameter settings for the discharge step, we obtain too low values for the current (especially initially) and the salt concentration (predicted maximum 12.1 mM; results not shown). To correct for this discrepancy, a very simple modification in the model was very effective, namely, to reduce the volume of the flow compartment. Using in the calculation V ) 6.5 mL during the ion desorption step (instead of V ) 17.5 mL as used during the ion adsorption step), the results as presented in Figure 1b are obtained, which are almost in quantitative agreement with the data. Only the current is somewhat lower in the model which is due to the higher value for Λ (Λ ) 0.88 in the theory vs Λ ) 0.77 for the data), requiring in the theory less electronic current for the same salt adsorption capacity. Obviously, the use in the theory of a lower internal volume for the ion desorption step compared to the adsorption step is unphysical and implies that a more detailed model is required for the fluid
J. Phys. Chem. C, Vol. 113, No. 14, 2009 5639 flow profiles in the CDI compartment and for the ion transport rates from solution in and out of the electrode structure. Such a more detailed model can be based on dynamically solving the full Poisson-Nernst-Planck (PNP) theory for ion transport as was done for a one-dimensional geometry in ref 20. For fully laminar flow during CDI and for planar electrodes (for instance, for electrodes coated on the transport channels of a microfluidic device), combination of the PNP theory with the Navier-Stokes equation (for fluid flow) will result in a comprehensive theory that is solvable numerically. For turbulent flow, and for CDI with porous electrodes, also more empirical descriptions will be useful to develop (such as the use of a mass-transfer layer description extended with concentration diffusion besides migration as a driving force). With those developments still in the future, for the moment we can conclude that the presented model is an important step toward obtaining a quantitative understanding of the CDI process. Additionally, because of the relative simplicity of the model, and the observed agreement between the theoretical results and the data, the model can be a useful tool for CDI process modeling and design. Conclusions A dynamic adsorption/desorption process model is presented for capacitive deionization (CDI), based on the Gouy-ChapmanStern model for the structure of the polarization layers, an empirical description for ion transport from the solution bulk phase through a mass-transfer layer to the electrodes, analogous to Ohm’s law for charge transport under the influence of an electric field, and a description of the flow compartment as being ideally stirred. Despite these simplifications, the model compares well to data for the electron current and ion concentration of the effluent during the ion adsorption and desorption steps. However, more accurate nanoscopic models are required which include in better detail the structure of the polarization layers in the porous carbon electrodes and the ion transport toward and into the electrodes. The presented process model can be used to describe the influence of flow rate, compartment volume, applied voltage, ionic strength, Stern capacity, and electrode area on CDI performance and can thus be used for optimization of CDI design and operation. Acknowledgment. Voltea B.V. (Leiden) is gratefully acknowledged for financial support. Part of this work was performed at Wetsus, Centre of Excellence for Sustainable Water Technology. Wetsus is funded by the Ministry of Economic Affairs. References and Notes (1) Arnold, B. B.; Murphy, G. W. J. Phys. Chem. 1961, 65, 135. (2) Murphy, G. W.; Caudle, D. D. Electrochim. Acta 1967, 12, 1655. (3) Johnson, A. M.; Newman, J. J. Electrochem. Soc. 1971, 118, 510. (4) Soffer, A.; Folman, M. Electroanal. Chem. 1972, 38, 25. (5) Oren, Y.; Soffer, A. J. Appl. Electrochem. 1983, 13, 473. (6) Farmer, J. C.; Fix, D. V.; Mack, G. V.; Pekala, R. W.; Poco, J. F. J. Appl. Electrochem. 1996, 26, 1007. (7) Andelman, M. D. Filtr. Separat. 1998, 35, 345. (8) Spiegler, K. S.; El-Sayed, Y. M. Desalination 2001, 134, 109. (9) Gabelich, C. J.; Tran, T. D.; Suffet, I. H. EnViron. Sci. Technol. 2002, 36, 3010. (10) Andelman, M. D. Canadian Patent CA 2444390, 2002. (11) Welgemoed, T. J.; Schutte, C. F. Desalination 2005, 183, 327. (12) Oh, H.-J.; Lee, J.-H.; Ahn, H.-J.; Jeong, Y.; Kim, Y.-J.; Chi, C.-S. Thin Solid Films 2005, 515, 220. (13) Lee, J.-B.; Park, K.-K.; Eum, H.-M.; Lee, C.-W. Desalination 2006, 196, 125. (14) Park, K.-K.; Lee, J.-B.; Park, P.-Y.; Yoon, S.-W.; Moon, J.-S.; Eum, H.-M.; Lee, C.-W. Desalination 2007, 206, 86.
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(15) Jung, H.-H.; Hwang, S.-W.; Hyun, S.-H.; Lee, K.-H.; Kim, G.-T. Desalination 2007, 216, 377. (16) Xu, P.; Drewes, J. E.; Heil, D.; Wang, G. Water Res. 2008, 42, 2605. (17) Zou, L.; Morris, G.; Qi, D. Desalination 2008, 225, 329. (18) Li, H.; Gao, Y.; Pan, L.; Zhang, Y.; Chen, Y.; Sun, Z. Water Res. 2008, 42, 4923. (19) Biesheuvel, P. M. J. Colloid Interface Sci. 2009, 332, 258. (20) Bazant, M. Z.; Thornton, K.; Ajdari, A. Phys. ReV. E 2004, 70, 021506. (21) Chu, K. T.; Bazant, M. Z. Phys. ReV. E 2006, 74, 011501. (22) van Soestbergen, M.; Biesheuvel, P. M.; Rongen, R.; Ernst, L. J.; Zhang, G. Q. J. Electrostatics 2008, 66, 567.
Biesheuvel et al. (23) Stern, O. Z. Elektrochem. 1924, 30, 508. (24) Freise, Z. Z. Elektrochem. 1952, 56, 822. (25) Kilic, M. S.; Bazant, M. Z.; Ajdari, A. Phys. ReV. E 2007, 75, 021502. (26) Biesheuvel, P. M.; van Soestbergen, M. J. Colloid Interface Sci. 2007, 316, 490. (27) Olesen, L. H.; Bruus, H.; Ajdari, A. Phys. ReV. E 2006, 73, 056313. (28) van Limpt, B. Ph.D. Thesis, Wageningen University, Netherlands, 2009. (29) Yang, K.-L.; Ying, T.-Y.; Yiacoumi, S.; Tsouris, C.; Vittoratos, E. S. Langmuir 2001, 17, 1961.
JP809644S
