Direct prediction of the desalination performance of porous carbon electrodes for capacitive deionization †

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Introduction
Providing access to affordable and clean water is one of the key technological, social, and economical challenges of the 21 st century. [1][2][3] For the desalination of water, commercially available methods include distillation, 4 reverse osmosis, 5 and electrodialysis. 6 Novel approaches include ion concentration polarization in microporous media, 7 systems based on batteries, 8,9 forward osmosis, 10 and capacitive deionization (CDI). [11][12][13][14][15] CDI is based on an electrochemical cell consisting of an open-meshed channel for water ow, in contact with sheets of porous electrodes on both sides. Upon applying a cell voltage between the two electrodes, ions become immobilized by an electrosorption process, that is, cations move into the cathode (the electrode into which negative electrical charge is transferred), while anions move into the anode (Fig. 1). Aer some time, when the electrodes reach their adsorption capacity (which depends on cell voltage), a discharge cycle is initiated by reducing or reversing the cell voltage, thereby releasing the salt as a concentrated stream. In the discharging step of the cell, energy recovery is possible. 16,17 Salt immobilization by CDI is considered an energy-efficient method for the desalination of water. 15,18 Though typically applied to the desalination of brackish water sources, seawater can also be desalinated by CDI. 19 In combination with ionselective membrane layers placed in front of the electrodes, CDI can be used to selectively remove a certain ionic species from a mixture of salts or to harvest compounds such as acetic acid, sulphuric acid, insulin, and boron. [20][21][22][23][24][25][26] Such separation processes may nd use in the treatment of wastewater from agriculture (mining), industry, and hospitals.
Various congurations for the design, stacking, and water management of CDI cells are possible. Most studies consider a design where the salt water is directed parallel to two equal electrodes, while a constant cell voltage is maintained, see Fig. 1. 13,27,28 However, stacks of electrodes do not necessarily have to consist of symmetrical cells and, instead, varying the carbon mass between the two electrodes provides the possibility to optimize the usable voltage window. 29 Another approach utilizes carbon rods (called wires) which are sequentially dipped and taken out of the water, instead of using lm electrodes forming a stack through which the water ows. 30 Instead of using bare carbon electrodes, improved energy efficiency has been reported for membrane-CDI (MCDI), where ion-exchange membranes are placed in front of one or both of the electrodes. 26,[31][32][33][34] Further modications are the use of constant current operation, 33,35 directing the water ow straight through the electrodes, 13,28 or the use of owable electrode suspensions. 19 Recently, CDI electrodes have also been used to produce energy from the controlled mixing of river and seawater, based on a reversal of the CDI process. [36][37][38][39][40][41][42] Electrosorption of ions is an interfacial process and in order to have a maximum contact area between the electrode and the water, CDI employs high surface area porous carbon electrode materials. At the water-carbon interface, electrical double layers (EDLs) are formed in which ions are electrosorbed. It has been stated that for optimum performance, pores should be large enough to have only a weak EDL-overlap, that is, mesopores are to be preferred over micropores. [43][44][45] However, some microporous carbons, such as activated carbons 14,46 and carbide-derived carbons 47 actually outperform mesoporous carbons. Recently, Porada et al. 47 reported that CDI desalination capacity positively correlates with the volume of pores in the range below 1 nm, while obtaining a negative correlation with the total pore volume, or with the BET specic surface area (BET SSA, ref. 48). The importance of pores <1 nm has also been demonstrated for the capacitance of EDL-capacitor electrodes, 49,50 for H 2 gas storage, 51 and for CO 2 gas removal capacity. 52 These results relate to equilibrium conditions, and micropores (<2 nm) and especially ultramicropores (<0.8 nm) can pose severe limitations to ion transport in CDI ow cells. Thus, porous electrodes that combine a large micropore volume (for a high deionization capacity) with a network of mesopores (between 2 and 50 nm) and macropores (>50 nm) may yield a highly efficient deionization process. 43,53,54 For an optimum performance, the design of the various components of the CDI system must be tuned to achieve both high salt electrosorption capacity and fast kinetics at the same time. Desalination by porous electrodes is by nature a non-linear phenomenon. Classical transmission-line models applied to CDI are unsatisfactory as they predict zero salt electrosorption and assume a constant ionic resistivity in the electrode. 15,55 Instead, when ions are being electrosorbed in the EDLs formed in intraparticle pores (within carbon particles), the interparticle pores (the pores in between the carbon particles) are subjected to ion starvation and the ionic conductivity will drop dramatically during desalination. This phenomenon results in an internal ionic electrode resistance that is much higher than expected on the basis of the performance derived for high salinity electrolytes, as common in EDLC research. EDLCs are specically designed to operate at large salt concentrations to have a high ionic conductivity and maximum capacity. Such a free choice of electrolyte is obviously not possible for water desalination. Note that the effect of ion starvation and the temporal increase in local resistivity to ion transport in the interparticle pores is included in the porous electrode theory of our paper. Upon applying a cell voltage between the two electrodes, anions and cations are electrosorbed within highly porous carbon electrodes to counterbalance the electrical charge. This immobilization of ions decreases the salt concentration in the flow channel, and results in the production of freshwater.
A variety of carbon materials, including activated carbons, carbon aerogels, carbon xerogels, and carbon nanotubes, have been studied for desalination by CDI. 15,18 New developments of advanced CDI electrode materials include asymmetric electrodes made of activated carbon coated with alumina and silica nanoparticles, 56 reduced graphene oxide and activated carbon composites, 57 graphene electrodes prepared by exfoliation and reduction of graphite oxide, 58 carbon nanotubes with polyacrylic acid, 59 carbon ber webs obtained from electrospinning, 60,61 and mesoporous activated carbons. 62 Templated carbons, although they require a more elaborate synthesis, are of particular interest as they provide additional means to precisely tailor the pore network, in order to combine a high electrosorption capacity with fast salt removal rates. A particularly high level of pore size control has been documented for carbons synthesized by selective etching of metal carbides with chlorine gas, called carbide-derived carbons, or CDCs. 63 Lately, templated CDCs have been reported 60,61 that combine a large micropore volume with hierarchic mesopores. Compared to conventional CDCs, templated ordered mesoporous CDCs (OM CDCs) show signicantly larger specic surface areas ($3000 m 2 g À1 ) and total pore volumes ($2 mL g À1 ). 64 Furthermore, using foam-like CDCs, synthesized by the "high internal phase emulsion" (HIPE) approach, it is possible to obtain control over macropores. The resulting material has both high surface areas of up to 2300 m 2 g À1 and extremely large pore volumes of up to $9 mL g À1 . 65 Despite many studies on various kinds of porous carbons, describing both equilibrium salt adsorption and the dynamics of the process, tools are not yet available to directly predict the performance of a certain carbon material and CDI design. The present work is aimed to be a rst step towards a method for direct prediction of desalination performance in CDI. Our approach consists of two main routes.
(1) To extract data of the equilibrium salt adsorption and kinetics of CDI for carbon materials with precisely tailored and designed pore architectures. With these data we demonstrate how we can directly predict the desalination performance of a carbon material based on its pore size distribution (PSD).
(2) To use a two-dimensional porous electrode CDI transport model to predict the actual salt electrosorption kinetics. This model demonstrates how desalination kinetics depend not only on the intraparticle pore morphology, but also on the electrode thickness and interparticle porosity.
In the next sections we briey describe the porous electrode transport theory, and discuss the synthesis of carbon materials and electrode architecture. We describe the salt adsorption performance in terms of equilibrium adsorption and kinetics, present a method to correlate equilibrium adsorption with PSD, and compare the dynamics of ion adsorption with theoretical predictions.

Theoretical section
To describe salt electrosorption and electrical current in porous carbon electrodes forming a CDI cell, we extend existing onedimensional porous electrode theory to two dimensions, to consider both the ow direction of the aqueous solution through the spacer channel, and the movement of salt in and out of the electrodes. Within the electrodes, we consider simultaneously ion transport through the space between the carbon particles, that is, the large transport pathways across the electrode (interparticle pore volume), and the electrosorption of ions inside carbon particles (intraparticle pore volume). To describe the latter, a powerful and elegant approach is to assume that the EDLs inside the intraparticle pore volume are strongly overlapping and, therefore, that the potential in these pores does not vary with position in the pore. This is the common "Donnan" approach for charged porous materials. The electrical potential in the intraparticle pore volume is different from that in the interparticle pore volume (the transport pathways) by a value Df d . The direct Donnan approach is modied 29,31,47 to consider the Stern layer located in between the electronic and ionic charge, and to include a chemical Fig. 2 Schematic view of the time-dependent two-dimensional porous electrode model, combining a sequence of sub-cells in the flow direction, with ion fluxes into the electrode. A symmetric CDI geometry is assumed, thus only half of a cell is depicted. The electrode contains an electrolyte-filled volume allowing for ion transport, and carbon material in which ions and charge are stored. Electrical current (denoted by "+") flows through the conductive carbon material. attraction energy for the ion when it transfers into the intraparticle pores, described by a term m att . 66 The modied Donnan (mD) model equals the limit situation of the Gouy-Chapman-Stern (GCS) theory when approaching full EDL overlap in micropores where the Debye length is of the order or larger than the pore size. In addition to GCS theory it includes the nonelectrostatic adsorption energy m att that reects that also uncharged carbons adsorb some salt. A difference between the mD and GCS model is that in the mD model, EDL properties are described per unit pore volume, whereas in the GCS model charge and salt adsorption are described as functions of pore area. Numbers in either denition can be converted when the pore area/volume ratio is known.
To describe the dynamics of ion transport and charge formation, we set up a two-dimensional porous electrode theory for a CDI cell consisting of two porous electrodes placed in parallel, with a at planar slit, or transport channel, or spacer, in between. In the direction of ow, this transport channel is mathematically divided into M subsequent sub-cells, see Fig. 2. 34 In the porous electrode, two coupled partial differential equations describe the salt concentration in the interparticle pores, the electrostatic potential there, f, the charge density, and the salt electrosorption in the intraparticle pores as a function of time and depth in the electrode. The porous electrode transport theory requires various geometrical measures as inputs (thickness, porosity) that can be calculated from electrode dimensions. Besides, it requires an estimate of the diffusion coefficient of the ions in the macropores, which may be lower than the corresponding value in free solution. There are no other tting functions. The present model neglects a transport resistance between interparticle pores and intraparticle pores, which can be incorporated, but requires an additional transport coefficient. Further details of the mD and transport model are provided in section 5 of the ESI. †

Experimental section
Electrodes from three different CDCs were prepared and compared to establish a basis of reference materials for further analysis of the salt electrosorption capacity. Details of material synthesis, electrode manufacturing and CDI testing are given in the ESI. † The synthesis methods are summarized in Fig. 3 for titanium carbide-derived carbon (TiC-CDC, Fig. 3A), ordered mesoporous silicon carbide derived carbon (OM SiC-CDC, Fig. 3B), and HIPE SiC-CDC (Fig. 3C).
Electrodes were prepared from these powders following the procedure outlined in ref. 47. A carbon slurry was prepared by mixing 85 mass% of CDC, 5 mass% of carbon black (Vulcan XC72R, Cabot Corp., Boston, MA), and 10 mass% of polyvinylidene uoride (Kynar HSV 900, Arkema Inc., Philadelphia, PA); the latter was previously dissolved in N-methyl-2-pyrrolidone. Thus, the nal electrode contains 85 mass% of porous CDC carbon. Electrodes were prepared by painting of the carbon slurry directly on one or both sides of a graphite current collector, taking care that approximately the same mass was coated on each side. Results for thickness and total electrode mass density are provided in Table S5. † Together with openmeshed porous spacer materials (thickness d sp ¼ 350 mm) the current collector/electrode layers are stacked together forming three parallel cells (i.e., one stack). 29,47 The ow of salt solution through the stack is kept constant, owing rst into a housing around the stack, entering the spacer layers from all four sides, and leaving via a centrally placed outlet to ow along a conductivity meter placed in-line.
An array of activated carbons and other carbon materials (see ESI †) were investigated along with the CDC materials for comparison. These materials were not painted, but prepared by a wet-casting technique following the procedure explained in ref. 47. In addition, carbon onions were tested as a representative of the class of fully graphitic, dense carbon nanoparticles (Fig. S9, ESI †) with no intraparticle porosity. The synthesis of carbon onions is based on the vacuum treatment of nanodiamonds at 1750 C as outlined in more detail in the ESI. † Ion electrosorption occurs when applying a cell voltage V cell to each of the three cells, dened as the voltage difference between the positively and negatively polarized electrodes. At the end of the salt electrosorption step, the cell voltage is reduced to zero and ion desorption begins. The electrical current running from the cathode to the anode is measured and is integrated over time to provide a measure for the total charge transferred between the electrodes. This total charge is divided by the total electrode mass in the stack, m tot , to obtain the charge expressed in C g À1 , see Fig. 5A, 7A and 8A. From the conductivity of the effluent solution, the salt concentration is calculated and, thus, by integrating over time, the salt electrosorbed, G salt , is calculated, see ref. 15 and 47. For each new experiment, the salt electrosorption/desorption cycle was repeated several times until the differences between cycles became negligible. We like to stress that in this work, the salt removal data are not obtained from the rst cycle aer a new condition has been applied, but instead are obtained when the system has reached the limit cycle, also called dynamic equilibrium (DE). This is the situation that the same amount of salt is electrosorbed during the adsorption step as is being removed in the desorption step of the cycle, as will be typical during practical long-term operation of a CDI system. All experiments were done using a c N ¼ 5 mM NaCl-solution (290 ppm, 550 mS cm À1 ).

Structure of the porous carbons
The CDC materials used for this study are produced from selective etching of silicon or titanium atoms out of a carbide precursor (SiC or TiC), a procedure which results in a material with a high BET SSA which, in the case of OM SiC-CDC, is as high as 2720 m 2 g À1 (Table 1). Fig. 4 displays the cumulative pore volume of these materials, together with the salt adsorption capacity, or z(s)-curve, which is discussed below.
All CDCs investigated in this study are predominantly amorphous, as evidenced by the broad D-and G-bands observed in Raman spectroscopy (see ESI †). TiC-CDC (Fig. 3A) powders are composed of anisometric particles with a size distribution ranging from approximately 1 to 10 mm and an average size of $5 mm. Compared to that, the structures of OM SiC-CDC and HIPE SiC-CDC differ in many aspects. HIPE SiC-CDC has a cellular pore structure as can be seen from Fig. 3C. Owing to the HIPE synthesis route, the material exhibits 2 to 4 mm sized cages that are interconnected by 300 to 500 nm sized windows. The walls are highly porous, but yet in the nanometer range. Thus, this material exhibits a hierarchical pore structure consisting of macro-, meso-, and micropores. For the other materials used in this study, macropores are only present in the form of large pores between carbon particles, but not within the porous particles themselves. OM SiC-CDC was synthesized as a powder of strand-like particles (Fig. 3B) having an average strand diameter of approximately 1 mm. These strands are built from nanorods which are arranged in a hexagonal ordered fashion and have very narrowly distributed mesopores located in between (Fig. 3B). The narrow distribution in the mesopore size is due to the method of nanocasting which employs ordered mesoporous silica templates as conformally corresponding exotemplates for the resulting CDC. 64,[67][68][69][70][71] Besides the ordered mesopores, micropores are also present in OM SiC-CDC. As a consequence, this material has a hierarchy of micro-and mesopores but no internal macropores. 68 The data for cumulative pore volume, see Fig. 4, show a hierarchical pore size distribution (PSD) with contributions from micro-and mesopores for HIPE and OM SiC-CDC, while TiC-CDC is predominantly microporous: more than 90 vol% of the pores is smaller than 2 nm (see Table S1 in the ESI †). HIPE SiC-CDC shows a total percentage of 37 vol% of mesopores and for OM SiC-CDC the majority of the total pore volume is associated with mesopores ($75 vol%). In that regard, HIPE SiC-CDC has Table 1 Pore volume, specific surface area (SSA; calculated with the BET equation 48 and quenched solid density functional theory, QSDFT 72 ), average pore size and local pore size maxima of the three CDC-materials. The average pore size is the volumetric average, i.e., half of the total pore volume is associated with pores larger or smaller than this value and not reflect, for example, the bimodal pore size distribution in OM SiC-CDC the largest total micropore volume (0.72 mL g À1 ) of the CDCmaterials. The hierarchic porosity of OM SiC-CDC is exemplied by its two narrow distribution maxima at approximately 1 nm and 4 nm. HIPE SiC-CDC does not show such a strongly pronounced bimodality, but it still exhibits two pore size distribution maxima at around 1 nm and another one at 2.4 nm.

Equilibrium desalination performance
Equilibrium data for salt adsorption and charge are presented in Fig. 5, based on underlying data for the desalination cycle for which examples given in Fig. S5 and S6 of the ESI, † using a symmetric CDI cell. Fig. 5A and B present data for salt adsorption and charge per gram of both electrodes, as functions of cell voltage. In Fig. 5B and C, the salt adsorption is presented relative to that at zero cell voltage in a two-electrode CDI cell. Fig. 5D presents the calculated total ion concentration in the pores (relative to an uncharged electrode), per mL intraparticle pore volume (for all pores below a size of 30 nm), as a function of the charge, also expressed per mL of intraparticle pores. The data for pore volume are given in Table 1 and Fig. 4. Fig. 5A and B show that the material with the highest capacitance (22.3 F g À1 at 5 mM NaCl, low-voltage limit), OM SiC-CDC, also has the highest salt adsorption capacity, 12.8 mg g À1 at a cell voltage of V cell ¼ 1.2 V. Per gram of carbon (not total electrode) the adsorption is 15.1 mg g À1 at 1.2 V.
Figs. 5A and B clearly show how with increasing cell voltage both charge and salt adsorption increase non-linearly. This is different from typical results for EDL capacitors where the charge increases linearly with voltage (see ref. 15). Fig. 5C plots salt adsorption vs. charge (both expressed in mol g À1 ; for salt by dividing the data of Fig. 5B by M w,NaCl and for charge by dividing the results of Fig. 5A by Faraday's number), which is a novel representation, which shows how all three datasets overlap. Fig. 5C also shows that the total ion adsorption is always somewhat less than the charge, i.e., the charge efficiency (L ¼ salt adsorption/charge) is below unity. 73 The high suitability of the materials tested for CDI can be deduced from the fact how close the measured charge efficiency is to unity, with measured values of L generally beyond 0.85. Indeed, Fig. 5C shows how close the data points are to the "100% charge efficiency line", the ideal limit where for each electron transferred one full salt molecule is removed. Interestingly, beyond the rst data points (charge density $0.1 mmol g À1 ) the data run parallel to the "100% charge efficiency line" which demonstrate that in this range, for each additional electron transferred, a full salt molecule is adsorbed, i.e., the differential salt efficiency is unity. 15,74 Fig. 5C clearly makes the point that a strong correlation exists between the capacitance of a material (how much charge can be stored for a given cell voltage, typically evaluated under conditions of use for EDL capacitors) and desalination performance in CDI. 47 Evaluating the data in Fig. 5B per unit pore volume (all pores <30 nm), one can calculate a salt adsorption of 0.39 M for TiC-CDC, 0.20 M for HIPE SiC-CDC, and 0.13 M for OM SiC-CDC, at a cell voltage of 1.2 V. This salt adsorption, SA, having dimension M, just as the z-function that will be discussed shortly, see Fig. 4, is equal to half the total ion concentration in the intraparticle pores (<30 nm) as given in Fig. 5D (relative to salt adsorption at zero voltage), evaluated for a symmetric twoelectrode cell. Clearly, per unit pore volume the performance is decreasing in this order, which is opposite to the order when the more common metric of mg g À1 is used (as plotted in Fig. 5B). Careful assessment of the inuence of pore size increments on desalination performance is required, and care must be taken in dening what is the "best" material, which may relate to electrode mass or volume, dependent on the nal application.
Next, we dene the performance ratio of a material, or PR. As we will take HIPE SiC-CDC as the reference, for HIPE this value is unity, PR ¼ 1. For TiC-CDC, which has twice the desalination per unit pore volume compared to HIPE at the reference conditions, PR ¼ 2. Likewise for OM SiC-CDC the value is PR ¼ 0. 66.
In a later section we will discuss how well the value of PR correlates with known data of the material's total pore volume, BET SSA, and full pore size distribution. If here a correlation can be found, this would allow one to estimate the PR of a new material when only the PSD is known, without having data of CDI experiments available. From the PR-value, desalination performance at the reference conditions of a symmetric CDI cell operating at V cell ¼ 1.2 V and for a salinity level of 5 mM NaCl can then be calculated. But in addition, knowing PR it will also be possible to calculate desalination at any other condition (different salinity, voltage, cell design), with the aid of the mDmodel, which requires knowledge of the three parameters m att , C St,vol,0 and a that are used in the mD-model. Thus, we rst address, when the value of PR is known, how we can calculate appropriate values to be used in the mD-model, which then predicts desalination, not only under the reference conditions as dened above, but also at other voltages (see Fig. 5B), other salinities, and very different CDI cell designs. The procedure that we propose is that relative to the reference material (HIPE SiC-CDC), for which the three parameters in the mD-model, being m att , C St,vol,0 and a, are determined as explained below, for materials with a different PR, the following rescalings are used: to m att is added a term ln(PR), C St,vol,0 is multiplied by PR, and a is divided by PR. This procedure is based on the nding that rescaling the total pore volume by PR gave a perfect match to the data. However, to avoid introducing the concept of a theoretical volume different from the actual one, for which there is no physical basis, the above procedure is proposed. In this way, once the value of PR of a new material is calculated (from the PSD data and using the z(s)-curve), then by correlating to the known performance of HIPE SiC-CDC, its CDI performance can be directly predicted.
The values for m att , C St,vol,0 , and a for HIPE SiC-CDC are calculated as follows. The novel representation in Fig. 5D is the starting point to derive by a structured method the parameters in the mD-model. Moreover the data in Fig. 5D can be tted only by adjusting the value of m att , without any inuence of Stern layer properties on this t, see eqn (S3) and (S4) in the ESI. † For HIPE an optimum value of m att ¼ 2.0 kT is found, in line with values used in previous work. 29,30 Next, for HIPE the full data of Fig. 5A and B must be tted by optimizing C St,vol,0 and a, for which only one combination ts the curves well (namely, C St,vol,0 ¼ 72 MF m À3 and a ¼ 50 F m 3 mol À2 ). Having established all of these values, the curves for the other two materials in Fig. 5A, B, and D automatically follow, and a very satisfactory t is obtained. Using a constant Stern layer capacity does not t the data well, see Fig. S5 in the ESI † for a comparison with a calculation with a ¼ 0.

Direct prediction of the desalination performance based on porosity analysis
We aim to nd a method to correlate desalination performance in CDI to the porosity analysis of the carbon material. This is a hotly debated topic and the claim is oen made that for CDI pores must be mesoporous (i.e., above 2 nm), 43,44 or even beyond 20 nm (ref. 45) to avoid overlap of electrical double layers in the pores, an effect that is claimed to be deleterious for CDI. However, electrodes made of microporous AC and CDC powders showed very high performance in CDI, higher than electrodes based on mesoporous carbon aerogels. 15,47,75 Also for the materials tested in this work, the predominantly microporous carbons (TiC-CDC, and HIPE SiC-CDC) show a general trend of higher salt adsorption per unit pore volume than the predominantly mesoporous OM SiC-CDC.
One question remains: what porosity metrics are most suitable to predict CDI performance? In agreement with ref. 47, we nd that desalination is positively correlated, even proportional, with the volume of pores smaller than 1 nm (see Fig. S3A and Table S1 in the ESI †), but only for materials that are mainly microporous. However, when including in the correlation materials with a signicant portion of mesopores, such as HIPE SiC-CDC and even more so for OM SiC-CDC, for these materials a signicant deviation from this proportionality (between salt adsorption and pore volume in pores <1 nm) is observed, with a much higher salt electrosorption than predicted based on this correlation, which can be explained by the contribution of mesopores to the ion immobilization. This contribution is not as high, per unit volume, as for micropores, but mesopores nevertheless also contribute to the ion electrosorption capacity. Thus, this measure of pore volume <1 nm cannot be the input parameter for a reliable predictive method. This situation is quite different from that in ref. 47 where it was demonstrated that for microporous carbons (AC and TiC-CDC), a positive correlation between the volume of pores smaller than 1 nm and the CDI performance could be established with a negative correlation of salt adsorption with BET SSA and with the total pore volume.
An appropriate metric based on PSD is not just correlated with salt adsorption, but ideally is proportional with desalination. Proportionality implies that the metric is a true measure of desalination, with an increase in this metric by a factor 2, resulting also in a two-times increased desalination. Such a metric is more likely to have a chemical-physical basis than a metric that is merely correlated with desalination. In Fig. S3 of ESI † we show four metrics based on the PSD and their proportionality with the salt adsorption performance: micropore volume <1 nm, <2 nm, total pore volume, and BET SSA. However, a satisfying t is not observed in either case. Thus, we cannot establish a clear and unambiguous proportionality between the salt electrosorption capacity and either of these metrics (see Fig. S3 in the ESI †).
Still, porosity measurements present a very facile method to characterize porous carbons and it remains very attractive to base a predictive CDI performance method on porosity data. We, thus, propose a new approach to predict the CDI performance, based on considering the relevance to salt adsorption of each pore size increment, which we call "salt adsorption capacity analysis" (or, z-analysis), which determines the relevance of each size increment to the measured desalination under one reference condition (V cell ¼ 1.2 V, c N ¼ 5 mM NaCl, symmetric cell). The function z is a property with dimension M and describes for the reference condition the contribution to desalination by a CDI cell of a certain pore size, s, per unit pore volume (within the carbon in one electrode). Deriving the z(s)-function is done by the simultaneous t of the experimentally available PSD of a set of materials to their desalination performance. This analysis quanties the fact that salt electrosorption depends not only on the total pore volume, but also on the pore size distribution: the volume associated with some pores contributes more to the total sorption capacity than other pores.
Mathematically, the aim of the analysis is to nd the z(s)-function, see Fig. 4, by which the salt adsorptions (SAs) of a set of materials (at the reference condition) predicted by eqn (1), t as closely as possible the measured values of SA. Note that the ratio of this SA to the SA of our reference sample (HIPE SiC-CDC) is the performance ratio, PR. In the z(s)-analysis, the total salt adsorption in mg g À1 of a symmetric two-electrode cell is given by eqn (1) where M w,NaCl is the molar mass of NaCl (58.44 g mol À1 ) and V is the pore volume (we will consider in all cases the pore size distribution up to a size of 30 nm) in mL g À1 , see Table 1. In the z(s)-analysis it is assumed that each material will have a different PSD, see Fig. 4, but that only one common function for z(s) is allowed. Note that eqn (1) describes the salt adsorption not per gram of electrode material, but per gram of carbon, which in all of our experiments is 85% of the electrode mass.
To nd the optimum z(s)-function we have used various methods, using e.g. predened functions, but in the end we decided to use a "function-free" approach in which the value of z is adjusted separately for each increment in size s, with the only imposed constraint that z must be decreasing with size s.
Assuming, instead, as a rst approximation z to be invariant with pore size s, we obtain the parity plot of Fig. 6A, where for the three CDC-materials, and also for twelve other materials (listed in Table 2) we show the correlation between the predicted value of SA and the measured value. As can be observed in Fig. 6A, there is a large deviation between the measured and predicted salt adsorption when assuming z to be constant at z ¼ 0.21 M and not varying with pore size.
Next we discuss our results of using a modied z(s)-function. The optimized z(s)-function is found by a least-square tting procedure of the difference of predicted (see eqn (1) above) and measured desalination. Several a priori constraints are imposed: (1) The full PSD curve is divided in short size ranges of 0.1 nm, for each of which the value of z can be adjusted by the optimization routine, independently of the others. (2) With increasing size s, z is not allowed to increase, but only to stay constant or decrease. Thus, we impose the rather stringent condition that the z(s)-curve must monotonically decrease and the smallest pore size will have the highest z.
(3) We assume that beyond a certain size, when EDL overlapping starts to become minor, and desalination must be proportional with area, that desalination per unit volume must be inversely proportional with pore size. We impose this condition from a rather arbitrarily chosen point of a size of s ¼ 6 nm.
We apply this analysis method to the three CDC-materials discussed before, and we arrive at the z(s)-curve as sketched in Fig. 4, where for a size s from 1.1 to 6.0 nm a constant z is predicted of z ¼ 0.11 M, from a size s 0.7-1.1 nm we have z ¼ 0.28 M and below s ¼ 0.7 nm z ¼ 0.51 M. (Note that the computer routine predicts tiny variations within each "block", and we removed these slight changes manually giving the z(s)curve plotted in Fig. 4, which was used as input in Fig. 6B). Next the optimized z(s)-correlation function is validated by applying it to twelve different materials, see Fig. 6B. As can be observed, for the three CDC-materials the t is now perfect, while also for the other materials, the t between predicted desalination (x-axis) and actual desalination (y-axis) has improved substantially.
This analysis demonstrates that pores smaller than 1.1 nm contribute more substantially to desalination than larger pores. The nding of a very high value of the electrosorption capacity associated with these micropores is in good agreement with our previous study on a comparison of CDC and AC materials 47 and is also in line with the data presented in Table 1 and Fig. S3A (see ESI). † It is closely related to the reported phenomenon of the anomalous increase in capacitance in EDL-capacitors in subnanometer-sized pores. 49,76 In conclusion, the z(s)-analysis gives the possibility to predict the CDI performance for both common and specialized carbons, purely based on easy-to-access cumulative PSD data. In contrast to this accuracy, our results ( Fig. 6A and S3 †) also underline that a convoluted, single value of pore analysis such as average pore size, total specic surface area, or total pore volume is not suited for direct prediction of the salt electrosorption capacity. Clearly, the complexity of carbon porosity must be appreciated and PSD data must be combined with consideration of the desalination efficiency of each pore size increment.
A spreadsheet le for the z-analysis based on arbitrary PSDdata is provided as ESI. †

Kinetics of salt electrosorption and charge transfer
Besides equilibrium electrosorption, the dynamics of ion sorption is of great importance for the practical application of CDI devices, and for a comprehensive understanding of differences between different porous carbon materials. In this section we apply for the rst time a rigorous procedure based on a twodimensional porous electrode theory that predicts the dynamical CDI behavior of a porous carbon electrode, see the ESI, † based on ion electrodiffusion through the interparticle pores in the electrodes, and ion electrosorption in the intraparticle pores. Electrosorption is described by the modied Donnan (mD) model for which appropriate parameter values for m att , C St,vol,0 , and a were derived in Section 4.2 (see Fig. 5). The mD model not only predicts desalination at the reference condition of V cell ¼ 1.2 V and for c N ¼ 5 mM NaCl, but also for other conditions, and in addition, also describes electrosorption in a dynamic calculation during which the salt concentration in the interparticle pores becomes signicantly different from c N , to drop for a short period during desalination, while increasing sharply, again only for a short period, during ion release. 31 The only dynamic tting parameter is the ion diffusion coefficient.
As depicted in Fig. 7, the porous electrode theory describes the rate of salt electrosorption and charge accumulation in CDI electrodes very well for the materials with a fair amount of mesopores (HIPE SiC-CDC, and OM SiC-CDC). The only difference in the input values for these calculations is the electrode thickness and inter-and intraparticle porosity, all calculated from geometrical measurements (see also Tables 2 and S6 in the ESI †), and the parameters for the mD model obtained from the equilibrium analysis of Section 4.2. The dynamics are described by the ion diffusion coefficient, for which a value of D ¼ 1.34 10 À9 m 2 s À1 is used for all materials (see ESI †). While hierarchic porous carbons, that is, OM SiC-CDC and HIPE SiC-CDC, yield an excellent agreement between the measured and calculated dynamic behavior, for TiC-CDC which has practically no mesopores, and is much denser (Table S5 in ESI †), we do obtain a good t of the salt electrosorption rate, but at the same time the charge accumulation rate (current) is underestimated initially. This possibly relates to a transport resistance from the interparticle space to the intraparticle space, see Section 5.3 in the ESI. †

Effect of electrode thickness on salt electrosorption and charge transfer
As we have seen, ion transport is strongly inuenced by the structure of the pore network, with a very good description of the dynamics of desalination performance for the hierarchical materials, as shown in Fig. 7. To further validate the twodimensional porous electrode theory for these hierarchical materials, electrodes characterized by the same mass density  but different thicknesses were prepared from OM SiC-CDC. The choice of this material was motivated by the excellent correlation between data and model shown in Fig. 7. As shown in Fig. 8A and B, there is a very strong inuence of the electrode thickness on both the salt electrosorption and the charge transfer rate. By increasing the thickness of the electrodes, the rate by which the maximum desalination is reached in the CDI process slows down, while as expected the nal equilibrium values (dened per gram of material) remain exactly the same. We see that this applies for both the charge accumulation rate and the salt electrosorption rate. Fig. 8C and D analyze theoretically the effect of higher (or lower) electrode packing density (the overall electrode mass density, as presented in column 2 of Table S5 †), by reducing the interparticle volume while keeping the mass and total intraparticle volume the same. Fig. 8D shows the interesting effect that a reduction of the interparticle volume is at rst advantageous, with the time to reach 50% of the maximum desalination rst decreasing (reaching a minimum value in the range of porosities between 30 and 60%), with this time increasing again for even lower porosities. The positive effect of a higher packing density of the electrode is that the length of the pathways for ions to traverse across the electrode goes down (the deepest regions of the electrode are more quickly reached), while the opposite effect at low porosity is because the transport pathways are being squeezed out of the electrode, with the apparent resistance for ion transport increasing (i.e., simply no transport pathways remain). These observations have a number of important implications, because optimized kinetics are very important for actual CDI application. As Fig. 8 demonstrates, faster ion electrosorption (per unit mass of electrode) can be achieved by decreasing the electrode thickness and by optimizing the electrode porosity. Thus, our study demonstrates that it is not only important to appreciate the micro-and mesopores present inside a carbon particle, but also to understand the porous carbon electrode in its entirety. The latter also entails the pores in between the carbon particles, and the total thickness of an electrode.

Conclusions
We have studied capacitive deionization of water using three carbide-derived porous carbon materials with strongly varying contributions to the total pore volume originating from microand mesopores, and compared performance with various reference materials. We have demonstrated that there is no direct relationship between salt electrosorption capacity and typical pore metrics such as BET SSA and the total volume of pores. However, we have demonstrated that the salt electrosorption capacity can be predicted by analysis of the pore size distribution and the pore volume correlated with incremental pore size ranges, considering that differently sized pores exhibit a different electrosorption capacity for the removal of salt ions. This analysis has been validated by comparison to literature data and other carbon materials and we were able to quite reliably predict the CDI performance of a range of carbons used for CDI.
Modeling is an important part of CDI performance analysis, not only to access information on the equilibrium salt removal capacity but also to gain understanding of the ion electrosorption process. Using the diffusion coefficient as the only dynamic t parameter, two-dimensional porous electrode theory is capable of predicting the dynamics of charge accumulation and the resulting process of salt electrosorption for the CDC-materials with sufficient amounts of mesopores. For materials without pores in this size range, the theory underestimates the initial current. Although CDI is a complex process depending on various parameters, such as pore volume, pore size distribution and process parameters, our work demonstrates that prediction of the CDI dynamic equilibrium salt adsorption capacity and the kinetics for ow-by electrodes is feasible. These results will facilitate the rational development of carbon electrode designs for CDI. An important next step will be to adapt our model to more advanced CDI techniques, such as ow-through CDI, 13,82 CDI using wires, 30 or CDI using owing electrodes. 19