Dynamic Adsorption of Ions into Like-Charged Nanospace: A Dynamic

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Dynamic Adsorption of Ions into Like-Charged Nanospace: A Dynamic Density Functional Theory Study Leying Qing, Yu Li, Weiqiang Tang, Duo Zhang, Yongsheng Han, and Shuangliang Zhao Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00088 • Publication Date (Web): 06 Mar 2019 Downloaded from http://pubs.acs.org on March 14, 2019

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Dynamic Adsorption of Ions into Like-Charged Nano-space: A Dynamic Density Functional Theory Study Leying Qing1, Yu Li1, Weiqiang Tang1, Duo Zhang2, Yongsheng Han3,4, and Shuangliang Zhao1,* 1

State Key Laboratory of Chemical Engineering and School of Chemical Engineering, East

China University of Science and Technology,130 Meilong Road, Shanghai, 200237, China 2

Ecole Nationale Supérieure des Ingénieurs en Arts Chimiques et Technologiques de Toulouse, 31030, France

3

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, 100190, Beijing, China 4

School of Chemical Engineering, University of Chinese Academy of Sciences, 100049, Beijing, China

*To

whom correspondence should be addressed. Email: [email protected] 1

ACS Paragon Plus Environment

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Abstract The adsorption processes of ions into charged nano-space are associated with many practical applications. Whereas a large number of microporous materials have been prepared towards efficient adsorption of ions from solutions, theoretical models that allow for capturing the characteristics of ion dynamic adsorption into like-charged nanopores are still few. The difficulty originates from the overlapping of electric potentials inside the pores. Herein, a theoretical model is proposed by incorporating dynamic density functional theory (DDFT) with modified Poisson equation for investigating the dynamic adsorption of ions into like-charged nanoslits. This model is rationalized by comparing the theoretical predictions with corresponding simulation results. Afterwards, by analyzing the adsorption dynamics we show that the overlapping effect is associated with the pore size, ion bulk concentration and surface charge density, and it plays a dominant role in the coupling between the total adsorption amount of ions and total adsorption time. Specifically, with weak overlapping effect the total adsorption amount is intuitively proportional to the total adsorption time; however, when the overlapping effect is strong the total adsorption amount may be inversely proportional to the total adsorption time, indicating both high adsorption amount and short adsorption time can be achieved simultaneously. This work provides meaningful insight towards the rational design and optimization of microporous materials for efficient ion adsorption.

Keywords: ion adsorption; overlapping effect; like-charged pore; dynamic density functional theory

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Introduction Dynamic adsorption of ions into porous materials is associated with many practical processes

1

including nano-fluidic transport through carbon nanotubes

technology for water desalination

4-5,

2-3,

membrane-based

energy extraction from the salinity gradients

electrical energy storages such as super-capacitors

8-9

6-7,

and

etc. For promoting the adsorption

amount and efficiency, electrical fields are applied by connecting the porous material with external powers, which essentially causes strong electrostatic adsorption between the ions and the porous materials. 10-13 Along this line, numerous experiments have been conducted.14 For example, it has been demonstrated that the water desalination efficiency could be much improved by imposing external fields.10-11 Porada et al.12 enhanced the charge efficiencies for various carbon materials by imposing an external cell voltage. Sales et al.13 reported that the cycle energy efficiency in the membrane-based energy extraction could be increased from 22% to 56% when increasing the max voltages of cell from 29 mV to 98 mV. Whereas most of these experimental studies provide successful examples for promoting desired efficiency, it is difficult to explore the effects of individual factors, such as pore size distribution, pore inner surface wettability,15 ion size or charge valence etc., on the ion adsorption amount and adsorption efficiency. In contrast, theoretical study provides an ideal platform for identifying the influences of individual parameters, upon which the ion-adsorption processes can usually be rationally designed and optimized.16 Current theoretical approaches for studying ion adsorption in porous materials can be generally divided into two categories. The one is computer simulation including grand canonical Monte Carlo (GCMC) simulation17 and molecular dynamics (MD) simulation.18 The methods in this category are literally exact as long as the involved force fields are reliable. With the help of computer simulations, the effects of ionic 3


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size and charge valence on the selective electrosorption of ions inside nanopores were identified19, and the effects of pore size and solvent type on the capacitance of porous electrodes were also examined.20 Away from these equilibrium studies, non-equilibrium MD simulations have been carried to explore the dynamic adsorption of ions into nanopores. 21 The other category assembles the theories of electrolyte solution such as Gouy-Chapman-Stern (GCS) theory, Debye-H ü ckel (DH) theory, Poisson Boltzmann (PB) theory and its variants,22 and classical density functional theory (cDFT)23 etc. These theoretical methods have proven to be very helpful for identifying the effects of various factors.24-25 For example, the classical PB theory and Debye-Hückel (DH) theory were utilized to investigate the effects of surface potential and surface charge density on the amount of adsorbed ions in electrical double layers.26 However, during the development of these electrolyte theories various approximations apply. For instance, it has been well recognized that classical PB is applicable only for dilute electrolytes as the ionic size effect and electrostatic correlation effect were not included, and these effects turn to be very important in concentrated electrolytes. Consequently, a number of modified PB theories have been proposed by additionally considering the ionic size27-28 through, e.g., a stepwise function for dielectric constant inside and outside the ions.29-30 Nevertheless, these variants are generally less rigorous.30 By taking into account the electrostatic correlation contribution and incorporating with fundamental measure theory that accurately describing the ionic size effect, cDFT grows to be the most comprehensive and accurate electrolyte theory31-34, on which both intuitive and extensive reviews can be found.35-36 Derived from cDFT, dynamic density functional theory (DDFT) that enables us to study the ionic dynamic adsorption has attracted more and more attentions.37-38 In recent years, DDFT has been widely employed to investigate the diffusion process of ions near charged substrates in electrochemical systems.39-40 Most of these studies focused on the ion diffusion 4


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dynamics in charged pores, and assumed that the pore width is much larger than the Debye length so that the overlapping of the electrostatic potentials from both pore walls can be avoided.32-33,

41

This assumption is overall valid for the mesopores (5-50 nm) in porous

electrodes. However, it has been showed that the a pore size distribution in the range of 2-5 nm could significantly promote the energy density and the power capability in activated carbon electrodes.42 Indeed, it has been observed that anomalously high capacitances can be achieved by using fine-tuned carbon electrodes contains a large numbers of small micropores (1-2 nm).43-45 In addition, these micropores are virtually embedded in a single electrode as cathode (or anode) of a supercapacitor, which is directly connected with an external power supply, i.e., the entire porous electrode with numerous micropores is negatively (or positively) charged. In other word, the pore walls of every single micropore are like-charged. Although the ion adsorption in like-charged nanopores plays a pivotal role in understanding the microscopic mechanisms of anomalous ion adsorption behaviors, theoretical model that allows for capturing the characteristics of ion dynamic adsorption into like-charged nanopores is rare. The difficulty not only originates from the pronounced ionic size effect, but also comes from the overlapping effects of mean electrostatic potentials inside nanopores.30 Herein, we extend the DDFT by incorporating with a modified Poisson equation to investigate the ion dynamic adsorption into like-charged nanoslits. This process is common in many technologies such as membraned-based water desalination and supercapacitors. Specifically, we investigate the overlapping effect on the adsorption dynamics. The extended DDFT is rationalized by comparing the predicted ionic density profiles with those from GCMC simulations. Thereafter, the pore size, surface voltage, bulk ionic concentration and system temperature effects on the ion total adsorption amount and adsorption rate are carefully examined and discussed. 5


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Modeling and Theory Theoretical model To recapitulate the dynamic adsorption of ions into like-charged nanoslits, a theoretical model is constructed and depicted in Figure 1. Both walls of a nanoslit with separation H are either positively charged or negatively charged. Namely, both walls carry the same type of charge. Indeed, this model has been previously proposed for studying the capacitances of a single electrode.46-47 The electrolyte is described with restricted primitive model (RPM), in which both cation and anion are modeled as charged hard spheres with same ionic size, and the solvent is treated as dielectric continuum. RPM model is widely used for describing prototypical electrolytes.36, 48 At initial state ( t =0), the ions distribute in the bulk electrolyte outside the nanoslit with a given bulk ionic concentration, and there are free of ions inside the like-charged nanoslit, mimicking the fresh situation of a charged membrane immersed into salt water, as displayed in Figure 1 (a). Driven by the gradients of chemical potentials, both cation and anion transfer from the bulk electrolyte into the nanoslit through the permeable pore walls, and diffuse inside the pore before reaching the final equilibrium state, as displayed in Figure 1 (b). This process is much different from the ion diffusion in double layer capacitors 46, because ions are adsorbed from outer bulk solution into nanoslits through permeable pore walls in the present work, and the total amount of ions inside the pore changes over time. The thickness of the pore walls is ignored in this work. Since in PRM model, the adsorption characteristics of monovalent ions in positively and negatively charged nanoslits are very similar. For simplicity, here and below we only focus on the dynamic adsorption of ions into negatively charged nanoslits. Mean electrostatic potential plays a significant role in ion dynamic adsorption process, which have been employed to modulate ionic density, diffusion process and transport process of ions.49 When the electrostatic potentials from both walls of a narrow pore are not overlapped, the mean electrostatic 6


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potential in the middle of the pore at equilibrium state is zero, which was confirmed experimentally.50 Otherwise, it significantly deviates from zero51-54, resulting from the overlapping of mean electrostatic potential, as depicted in Figure 1 (c).

Figure 1 Schematic representation of ion dynamic adsorption into a negatively charged nanoslit with separation H . Green arrows indicate the diffusion direction of ions. Cations and anions are represented with red and blue spheres, respectively. (a) At initial time the nanoslit is free of ions, and (b) after a dynamic adsorption process, an equilibrium adsorption state of ions inside the nanoslit occurs, and (c) the schematic overlapping of the mean electrostatic potential within the pore. DDFT framework and boundary conditions Within the framework of DDFT, the evolution of ionic density profile inside a nanoslit,

i (r, t ) , is driven by the chemical potential gradient39, 55: i (r, t ) D     i (r, t )i (r, t )  , t k BT

(1)

where D is the diffusion coefficient, k B stands for the Boltzmann constant, and T is the system temperature. i (r, t ) represents the local chemical potential of ionic species i at position r and time t . Generally, the local chemical potential can be decomposed into an ideal gas contribution, an excess one owing to the intermolecular interaction and external electrostatic 7


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potential, and an external one accounting for non-electrostatic external potential. Since in PRM the ions interact with each other through hard-sphere repulsion and electrostatic potential, the excess chemical potential can be further decomposed into three contributions. Namely, the local chemical potential follows. 56-57

i (r, t )  k BT ln[ i (r, t ) 3i ]  Vi 0 (r )  iHS (r, t )  iEL (r, t )  eZ i (r, t ) ,

(2)

where  i denotes an effective thermal wavelength, which is immaterial to the DDFT calculations. Vi 0 (r ) represents the non-electric external potential to the anions and cations inside the nanoslit, and generally it doesn’t vary with time. The remaining three terms on the right-hand side of eq.(2) constitute the excess chemical potential. iHS (r, t ) accounts for the hard-sphere (HS) repulsions and can be evaluated by using the modified fundamental measure theory (MFMT)58; iEL (r, t ) accounts for the electrostatic correlations arising from the ion-ion interaction, and it can be calculated by a second-order functional expansion of the excess Helmholtz free energy functional with respect to the corresponding bulk system. 59 e is a unit charge, and Z i is the charge valence.  (r, t ) is the local mean electrostatic potential, and it follows the Poisson equation:  2 (r, t )  

e

 0 r

Here  0 is vacuum permittivity, and  r

 Z  (r, t ) . i

i

(3)

i

is the relative permittivity of the dielectric

continuum. Due to the slab symmetry, the Poisson equation can be simplified as  2 ( z , t )  

e

 0 r

 Z  ( z, t ) i

i

with z being the normal direction perpendicular to the flat

i

wall. To solve the Poisson equation, two boundary conditions are necessitated. Typically, the local mean electrostatic potential and its derivative are set as zero at the distance sufficiently far away from the charged surface.32-33, 41 These boundary conditions don’t apply any longer 8


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when the pore size is in nanoscale since the electrostatic potentials from both surfaces overlap within the pore. Nevertheless, owning to the minor symmetry in like-charged nanoslit, we employ the following boundary conditions:  d ( z , t ) 0  ,  dz zH /2  (0, t )   ( H , t )  V S 

(4)

where VS is the surface voltage on the wall. Eq.(4) together with eq.(3) constitute the modified PB equation. Note that the modified PB equation applies for symmetrically charged systems, including those with or without overlapping of the mean electrostatic potential inside the pores. The nonlinear DDFT equation in eq.(1) can be numerically solved by combining with the time and geometrical boundary conditions. At the initial time, since the nanoslit is free of ions, the ionic density inside the nanoslit and the corresponding electrochemical potential are zero. Namely:

  ( z , t  0)  0 , 0 zH .    ( z , t  0)  0

(5)

Geometrically, the density of ionic component i outside the nanoslit recovers to its bulk density ib , and consequently the chemical potential is the bulk chemical potential ib , i.e., b  i (z , t )  i ( H  z , t )  i ,  b  i (z , t )  i ( H  z , t )  i

(6)

where z represents the step length in our calculation, and in principle it is an infinitesimal. Whereas the ion concentration near the outer surface of pore should be different with the bulk ion concentration due to the ion-surface interaction, in eq.(6) we take the bulk ion concentration value as the boundary condition owning to the assumption of pore wall without thickness.

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Numerical calculation To solve DDFT equation numerically, the differential equations in eq.(1) is discretized with central difference method:

 i  z j 1 , t   i  z j , t  i  z j 1 , t   i  z j , t      i  z j , t  D   2  z    / z t k BT  i  z j , t   i  z j 1 , t  i  z j , t   i  z j 1 , t   .    2 z D = U i  i  z j , t  , t   k BT 



(7)



Here the subscript j represents the grid number in space, and particularly, the ionic density and electrochemical potential satisfy eq.(6) when j  1 and j  H z . The local chemical potential can be determined as the local ionic densities are provided.32-33, 41 With the numerical value of U i [{i (z j , t )}, t ] , the local ionic densities in next time step can be obtained by integrating both sides of eq.(7) with Adams-Moulton (AM) and Adams-Bashforth (AB) algorithm.39 To ensure the highly nonlinear equation can converge at every time step, the relaxation factor 

is used during each Picard iteration.60 The

numerical calculation details are presented in the Appendix. Results and Discussion Comparison with GCMC simulations After sufficiently long time, the evolution of ionic density profiles inside the nanoslit governed by DDFT should reach final equilibrium. This provides a criterion to rationalize the DDFT calculation. The final equilibrium density profiles of ions predicted by DDFT are compared with the results from grand canonical Mote Carlo (GCMC) simulation,61-62 as presented in Figure 2. In this comparison study, the dielectric constant is set as  r = 78.5 and the temperature T =298 K. Cation and anion are treated as charged hard spheres with equal diameter  = 0.425 nm and bulk ionic concentration 1.0 M, and the corresponding charge 10


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valences are +1 and -1, respectively. The intrinsic connection between surface charge density and surface voltage is determined from the condition of overall charge neutrality. In other words, following the charge neutrality condition, the surface charge density is determined by the ion density distribution inside the pore, while the latter is relevant with the surface voltage.63 Two typical surface charge densities ( Q =-0.05 C/m2 and -0.1 C/m2) and pore widths ( H =1.0 nm and 2.0 nm) are considered. Figure 2 plots the normalized concentrations of ions inside the negatively charged slit pores. The normalized concentration is defined as the ratio of local ionic density to the bulk one. Therefore, the absolute value of ionic concentration can be straightforwardly calculated by multiplying the bulk concentration. It can be found that our extended DDFT gives very satisfactory predictions of the ionic density profiles under various sets of conditions. Due to the electrostatic attraction, the cation adsorbed in the negatively charged slit pores presents higher local density than the anion. Specifically, when the surface charge density is larger, the local density of cation becomes overall higher, resulting in a higher adsorption amount of cation. Oppositely, owing to the electrostatic repulsion, the local density of anion is almost zero as the surface charge density is large. This trend becomes more evident as the pore width is smaller, owing to the overlapping of the mean electrostatic potentials from both walls in small slit pores.

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Figure 2 Comparison of normalized concentration profiles predicted by DDFT and the corresponding GCMC simulations under various sets of conditions: (a) surface charge density Q =-0.1 C/m2 and pore width H = 1 nm; (b) Q =-0.1 C/m2 and H = 2 nm; (c) Q =-0.05 C/m2

and H = 1 nm; (d) Q =-0.05 C/m2 and H = 2 nm. Weak overlapping and strong overlapping Figure 3 plots the time-evolutions of local mean electrostatic potentials inside four negatively charged nanoslits. The dielectric constant is set as  r = 1.0 for representing vacuum and the other parameters are the same as those in Figure 2. It should be noted that, here and below the dielectric constant is chosen to be 1.0, although another value, i.e., 78.5, is employed in Figure 2 solely for comparison study. The relaxation time is given in the unit of reduced time  D   2 D , where D represents ion diffusivity, and this constant is determined by the types of electrolyte solutions. Typically it takes D =10-9 m2/s for the ion diffusivity in aqueous solution.64-65, and thus  D =0.1806 ns. The strength of overlapping is evaluated by the value of mean electrostatic potential in the middle of a nanoslit at equilibrium state (designated as  c ). For convenience, we consider the overlapping effect is overall weak as the reduced mean electrostatic potential in 12


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the middle of a nanoslit, i.e.,  c*   e c , is within (-1, 1), and otherwise the overlapping is strong. This criterion is proposed by following the conventional way of scaling energy with the unit of k BT . For strong overlapping, three zones (I, II, and III) are introduced as illustrated in Figure 3 (a)-(c), respectively. Zone I:  c*  1 and no oscillation for the local mean electrostatic potential profile; Zone II:  c*  1 and oscillation for the profile; Zone III:

 c*  1 . Note that in the zone III, the mean electrostatic potential always displays an oscillation distribution owing to the negatively charged surface. For comparison, the weak overlapping of the mean electrostatic potential is plotted in Figure 3 (d).

Figure 3 Time-evolution of reduced local mean electrostatic potential  * = e inside the nanoslits. (a) pore width H /  =1.5, bulk ionic concentration cb =0.1 M, and the surface

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voltage Vs =-0.15 V; (b) H /  =4.5, cb =1.0 M and Vs =-0.15 V; (c) H /  =3.0, cb =1.0 M and Vs =-0.15 V; (d) H /  =3.0, cb =0.1 M and Vs =-0.01 V. Figure 4 (a)-(b) plot the contour map of  c* in terms of pore width and surface voltage at two representative bulk ionic concentrations cb = 0.01 M and 0.1M. It can be found that the strength of overlapping is associated with the pore width, surface voltage and bulk ionic concentration, and in addition the decrease of pore width and the increase of surface voltage generally enhance the overlapping effect. Figure 4 (c)-(d) plot the contour map of  c* in terms of pore width and bulk ionic concentrations subject to two representative surface voltages Vs =-0.1V and -0.15V. Interestingly, the dependence of  c*

on bulk ionic

concentration is not monotonous, and this is likely due to non-trivial relation between the correlation length of confined electrolyte solution and the system setting parameters including surface voltage, bulk ionic concentration, pore width and ionic size etc. It should be noted that the correlation length is the distance of the range of inter-ion electrostatic interaction, and its maximum value is associated with the system setting parameters (e.g., ion concentration, charge valence, temperature etc.).66-68 Typically, it is about 1.29 nm for water with the reduced temperature Tr  T / Tc =1.05.69 Therefore, the overlapping effect should be considered at the pore size less than approximately twice the correlation length.

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Figure 4 contour of local mean electrostatic potential  e c in the middle of nanoslits: (a) in terms of pore width H / s and surface voltage Vs at bulk ionic concentration cb =0.01 M; (b) in terms of H / s and Vs at cb =0.1 M; (c) in terms of H / s and cb at Vs =-0.10 V; (d) in terms of H / s and cb at Vs =-0.15 V. Figure 5 plots the representative profiles of local cation density and local anion density inside the negatively like-charged nanoslits during the dynamic adsorption. Owing to the slab symmetry, the ionic density profiles evolve symmetrically. Driven by the chemical potential difference, ions transfer through the permeable pore walls, and consequently the local density of cation and anion inside the nanoslit increases gradually, which in return reduces the difference of the chemical potentials. This dynamic adsorption process continues until the difference of the chemical potentials for both cation and anion vanishes. When a final thermodynamic equilibrium occurs, the density profiles of cation and anion inside the nanoslit do not vary any longer. Figure 5 (a)-(b) present the strong overlapping effect ( VS =-0.15 V and H /  =1.5; zone 15


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I), Figure 5 (c)-(d) show the strong overlapping effect ( VS =-0.15 V and H /  =3.0; zone III) and Figure 5 (e)-(f) display the weak overlapping effect ( VS =-0.10 V and H /  =5.5). Firstly, the local densities of cation (in the right panel) gradually increase over time during the dynamic adsorption process. However, the adsorption of anion is firstly promoted and then gradually suppressed when the overlapping is strong and in zone I; differently, when the overlapping is strong and in the other zone (or it’s weak), the adsorption of anion is always gradually promoted over time. This abnormal behavior in Figure 5(b) is due to the strong overlapping effect. In zone I, the adsorptions of both cations and anions in the negatively charged nanoslit result in the increase of mean electrostatic potential. The enhancement of this positive mean electrostatic potential (see Figure 3a) gradually suppresses the adsorption of anion, finally leading to a decrease of the anion concentration. In addition, the local density profiles of both cation and anion at equilibrium state display fascinating oscillation distribution (equivalently, layering behavior) in large pore as shown in Figure 5 (e)-(f). This oscillation in density profile is due to the ionic size effect, which is well captured in our DDFT. Furthermore, we point out that the predicted ionic density profiles in different charged pores are significantly different, owning to combined effects between external potential and intrinsic internal potential.

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Figure 5 Local densities inside nanoslits with strong overlapping effect (top panel) and weak overlapping effect (bottom panel). The bulk ionic concentration of cb =0.1 M. (a)-(b): pore width H /  = 1.5 and surface voltage Vs =-0.15V; (c)-(d): pore width H /  = 3.0 and surface voltage Vs =-0.15V; (e)-(f): H /  = 5.5 and Vs =-0.10V. The other parameters are the same as those in Figure 3.

Overlapping effect on Adsorption capacity and adsorption rate The dynamic adsorption of ions inside nanoslit as a function of pore width H /  can be characterized by the adsorption amount of both cation and anion:

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 (t ) 2  

H /

0

  ( z /  , t ) i

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3

dz /  .

(8)

i

Here  (t ) is the adsorption amount at time t , and the subscript i specifies the type of cation or anion. Figure 6 plots the time-evolutions of adsorption amounts in various nanoslits ( H /  =1.5, 2.0, 3.0 and 5.5) at four different sets of bulk ionic concentrations and surface voltages. Generally, the adsorption amount increases quickly over time, and then reaches a plateau, in accordance with the measured adsorption behaviors in various porous materials.11-12,

24, 70

Herein, the equilibrium adsorption amount  eq 2 is designated as the final equilibrium value, and it’s the total adsorption amount. When the instantaneous adsorption amount reaches 99.99% of  eq 2 , the corresponding time is defined as the equilibrium adsorption time teq  D , representing the total adsorption time. The anomalous increase of the equilibrium adsorption amount is observed in Figure 6 (a) and (b) when the strong overlapping is in Zone I, showing that there is no monotonic increase in the equilibrium adsorption amount as increasing the pore width, surface voltage, and bulk ionic concentration. However, such a monotonic increase can be found when the overlapping is weak, as depicted in Figure 6 (c) and (d).

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2 Figure 6 Time-evolutions of total adsorption amounts,  , versus reduced time t  D in

various nanoslits ( H  =1.5, 2, 3, and 5.5): (a) bulk ionic concentration cb =0.1M and surface voltage VS = -0.10V; (b) cb =0.01M and

VS =-0.10V, and (c) cb =0.1M and

VS

=-0.01V; (d) cb =0.01M and VS =-0.01V. All conditions are the same as those discussed in Figure 3.

Figure 7 plots the equilibrium adsorption time and corresponding equilibrium adsorption amount as a function of pore width. The bulk ionic concentration is 0.01 M, and two surface voltages, -0.10 V and -0.15 V, are considered. From Figure 4, we note that the overlapping is strong when the pore width is less than 2.5  , and it becomes weak if the pore width exceeds 2.5  . The strong overlapping effect from Zone I is shown in the left panel. As shown in in Figure 7 (a), when the pore width is less than 1.6  , the equilibrium adsorption time with surface voltage -0.15 V is less than its counterpart with surface voltage -0.10 V; However, 19


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when the pore width exceeds 1.6  , the opposite phenomenon is observed. The corresponding adsorption amount in terms of pore width is depicted in Figure 7 (c). The equilibrium adsorption amount first increases and then decreases as increasing the pore width, and an anomalous peak is observed when the pore width H /  =1.3. This tendency agrees well the reported experimental findings from Gogotsi and Simon et al.

42-44

They

found an anomalous increase of capacitance in supercapacitor when the pore size less than 1 nanometer and the maximum capacitance was produced at a pore size around a single ion per pore. The tendency also agrees qualitatively with the theoretical study by Jiang et al., in which an oscillation of the capacitance in terms of pore width is reported.71 With weak overlapping as shown in the right panel of Figure 7, both equilibrium adsorption time and corresponding equilibrium adsorption amount are promoted when enhancing surface voltage, and in addition they overall increase monotonically when further increasing the pore width. With the strong overlapping from Zone II and Zone III, the dependence of equilibrium adsorption time and the corresponding equilibrium adsorption amount on pore width is similar to situation with weak overlapping, and thus the Figures are omitted here.

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Figure 7 Equilibrium adsorption amount and corresponding adsorption time as a function of pore width. All conditions are the same as those discussed in Figure 3.

To further interpret the difference among the situations with different overlapping effects, the coupling between the equilibrium adsorption amount and the corresponding adsorption rate is further investigated. Here the adsorption rate is introduced as the reciprocal of equilibrium adsorption time, i.e., 1/(teq /τ D ) . With strong overlapping from Zone I as shown in Figure 8 (a), the equilibrium adsorption amount increases as the adsorption rate increases, especially when the adsorption rate exceeds 0.5. This shows that both high equilibrium adsorption amount and short adsorption time can be obtained simultaneously. However, for the strong overlapping in zone III as shown in Figure 8 (b), and for the weak overlapping effect shown in Figure 8 (c), the equilibrium adsorption amount always decreases as increasing the adsorption rate, indicating that high equilibrium adsorption amount can be obtained with the compensation of long adsorption time. 21


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Figure 8 Equilibrium adsorption amount versus the reciprocal of the equilibrium adsorption time, 1/(teq /τ D ) . The parameters are the same as in Figure 3.

Conclusions An extended DDFT is developed by incorporating with a modified PB equation for investigating the ion dynamic adsorption in like-charged nanoslits. The extended DDFT enables us to address the overlapping effect of mean electrostatic potential on the ion adsorption amount and adsorption rate. The overlapping effect is associated with the pore width, surface voltage and bulk ionic concentration. The overlapping effect emerges when the pore width is small, and this effect is enhanced when increasing the surface voltage. The bulk ionic concentration has profound influence on the overlapping strength, and no trivial relation in-between can be found. The overlapping of the mean electrostatic potentials plays a dominant role in the coupling between the equilibrium adsorption amount (equivalently, the total adsorption amount) and total adsorption time. When the overlapping is weak, the total adsorption amount is intuitively proportional to the adsorption time. Namely, long adsorption time is required generally for large adsorption amount of ions. When the overlapping is strong, three zones are introduced for identifying the difference. For zone I where  c*  1 and no oscillation for the local mean electrostatic 22


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potential profile, the total adsorption amount is inversely proportional to the adsorption time. In other word, both large adsorption amount and short adsorption time can be simultaneously achieved. For zone II and III, similar coupling relation has been found as in weak overlapping case. The effect of different sized ions is not discussed in this work for simplicity. We note that such an effect is not significant when ionic density is low. However, when the ionic concentration is high, this effect becomes significant, and in that circumstance both anions and cations are densely packed near the surfaces of charged pore, preventing the adsorption of counterions to the charged surfaces, the asymmetric sizes of cation and cation then alter greatly the density distributions of ions inside the pore, and the adsorption dynamics as well. Although this certainly deserves a comprehensive study, we expect that overlapping effect remains. Whereas only like-charged nanoslits are considered in the present work, the extension to other like-charged nanopores with different geometries is straightforward. This theoretical work provides helpful guidance towards the design and optimization of microporous materials for efficient ion adsorption and separation. Acknowledgement This work is supported by National Natural Science Foundation of China (Nos. 21878078, U1707602, and 91534123), the National Natural Science Foundation of China for Innovative Research Groups (No. 51621002), and the 111 Project of China (No. B08021). SZ acknowledges the support of Fok Ying Tong Education Foundation (151069). References 1. Roque-Malherbe, R. M. A., Adsorption and diffusion in nanoporous materials. CRC Press: 2018. 2. De Volder, M. F.; Tawfick, S. H.; Baughman, R. H.; Hart, A. J., Carbon nanotubes: present and future commercial applications. Science 2013, 339 (6119), 535-539. 3. Holt, J. K.; Park, H. G.; Wang, Y.; Stadermann, M.; Artyukhin, A. B.; 23


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Grigoropoulos, C. P.; Noy, A.; Bakajin, O., Fast mass transport through sub-2-nanometer carbon nanotubes. Science 2006, 312 (5776), 1034-1037. 4. Porada, S.; Zhao, R.; Wal, A. V. D.; Presser, V.; Biesheuvel, P. M., Review on the science and technology of water desalination by capacitive deionization. Progress in Materials Science 2013, 58 (8), 1388-1442. 5. Shannon, M. A.; Bohn, P. W.; Elimelech, M.; Georgiadis, J. G.; Mariñas, B. J.; Mayes, A. M., Science and technology for water purification in the coming decades. Nature 2008, 452 (7185), 301-311. 6. Logan, B. E.; Elimelech, M., Membrane-based processes for sustainable power generation using water. Nature 2012, 488 (7411), 313-319. 7. Jia, Z.; Wang, B.; Song, S.; Fan, Y., Blue energy: Current technologies for sustainable power generation from water salinity gradient. Renewable & Sustainable Energy Reviews 2014, 31 (1), 91-100. 8. Simon, P.; Gogotsi, Y., Capacitive Energy Storage in Nanostructured Carbon– Electrolyte Systems. Acc Chem Res 2013, 46 (5), 1094-1103. 9. Kondrat, S.; Wu, P.; Qiao, R.; Kornyshev, A. A., Accelerating charging dynamics in subnanometre pores. Nature Materials 2014, 13 (4), 387-393. 10. Zhao, R.; Biesheuvel, P. M.; Wal, A. V. D., Energy consumption and constant current operation in membrane capacitive deionization. Energy & Environmental Science 2012, 5 (11), 9520-9527. 11. Zhao, R.; Biesheuvel, P. M.; Miedema, H.; Bruning, H.; Wal, A. V. D., Charge efficiency: A functional tool to probe the double-layer structure inside of porous electrodes and application in the modeling of capacitive deionization. Journal of Physical Chemistry Letters 2010, 1 (1), 205-210. 12. Porada, S.; Weinstein, L.; Dash, R.; Van, d. W. A.; Bryjak, M.; Gogotsi, Y.; Biesheuvel, P. M., Water desalination using capacitive deionization with microporous carbon electrodes. Acs Applied Materials & Interfaces 2012, 4 (3), 1194-1199. 13. Sales, B. B.; Saakes, M.; Post, J. W.; Buisman, C. J.; Biesheuvel, P. M.; Hamelers, H. V., Direct power production from a water salinity difference in a membrane-modified supercapacitor flow cell. Environmental Science & Technology 2010, 44 (14), 5661-5665. 14. Beguin, F.; Presser, V.; Balducci, A.; Frackowiak, E., Carbons and Electrolytes for Advanced Supercapacitors. Adv. Mater. 2014, 26 (14), 2219-2251. 15. Zhao, S.; Hu, Y.; Yu, X.; Liu, Y.; Bai, Z. S.; Liu, H., Surface wettability effect on fluid transport in nanoscale slit pores. Aiche Journal 2017, 63 (5), 1704-1714. 16. Burt, R.; Birkett, G.; Zhao, X. S., A review of molecular modelling of electric double layer capacitors. Phys. Chem. Chem. Phys. 2014, 16 (14), 6519-6538. 17. Yang, K. L.; Yiacoumi, S.; Tsouris, C., Canonical Monte Carlo simulations of the fluctuating-charge molecular water between charged surfaces. Journal of Chemical Physics 2002, 117 (1), 337-345. 18. Ciccotti;, G.; Kapral;, R.; Sergi;, A., Non-equilibrium molecular dynamics. In Handbook of materials modeling, 2005; Vol. 1, pp 745-761. 19. Hou, C. H.; Taboada-Serrano, P.; Yiacoumi, S.; Tsouris, C., Electrosorption selectivity of ions from mixtures of electrolytes inside nanopores. Journal of Chemical Physics 2008, 129 (22), 224703. 20. Kiyohara, K.; Sugino, T.; Asaka, K., Electrolytes in porous electrodes: Effects of the pore size and the dielectric constant of the medium. Journal of Chemical Physics 2010, 132 (14), 144705. 21. Yuk Wai Tang; István Szalai, a.; KwongYu Chan, Diffusivity and Conductivity of a Solvent Primitive Model Electrolyte in a Nanopore by Equilibrium and Nonequilibrium Molecular Dynamics Simulations. Journal of Physical Chemistry A 2001, 105 (41), 24


Page 24 of 28

Page 25 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

9616-9623. 22. B. Hribar; V. Vlachy; L. B. Bhuiyan, a.; Outhwaite, C. W., Ion Distributions in a Cylindrical Capillary as Seen by the Modified Poisson−Boltzmann Theory and Monte Carlo Simulations. Journal of Physical Chemistry B 2000, 104 (48), 11522-11527. 23. Wu, J.; Li, Z., Density-functional theory for complex fluids. Annual Review of Physical Chemistry 2007, 58 (1), 85-112. 24. Zhao, R.; Satpradit, O.; Rijnaarts, H. H.; Biesheuvel, P. M.; Van, d. W. A., Optimization of salt adsorption rate in membrane capacitive deionization. Water Research 2013, 47 (5), 1941-1952. 25. Pilon, L.; Wang, H.; D'Entremont, A., Recent advances in continuum modeling of interfacial and transport phenomena in electric double layer capacitors. Journal of the Electrochemical Society 2015, 162 (5), A5158-A5178. 26. Trefalt, G.; Behrens, S. H.; Borkovec, M., Charge regulation in the electrical double layer: Ion adsorption and surface interactions. Langmuir 2015, 32 (2), 380-400. 27. Borukhov, I.; Andelman, D.; Orland, H., Adsorption of large ions from an electrolyte solution: a modified Poisson–Boltzmann equation. Electrochimica Acta 2000, 46 (2–3), 221-229. 28. Liu, B.; Liu, P.; Xu, Z.; Zhou, S., Ionic Size Effects: Generalized Boltzmann Distributions, Counterion Stratification, and Modified Debye Length. Nonlinearity 2013, 26 (10), 2899-2922. 29. Xu, Z.; Ma, M.; Liu, P., Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches. Physical Review E Statistical Nonlinear & Soft Matter Physics 2014, 90 (1), 013307. 30. Ma, M.; Zhao, S.; Liu, H.; Xu, Z., Microscopic insights into the efficiency of capacitive mixing process. Aiche Journal 2017, 63 (6), 1785-1791. 31. Yu, Y. X.; Wu, J.; Gao, G. H., Density-functional theory of spherical electric double layers and ζ potentials of colloidal particles in restricted-primitive-model electrolyte solutions. Journal of Chemical Physics 2004, 120 (15), 7223-7233. 32. Jiang, D. E.; Meng, D.; Wu, J., Density functional theory for differential capacitance of planar electric double layers in ionic liquids. Chemical Physics Letters 2011, 504 (4–6), 153-158. 33. Jiang, J.; Cao, D.; Henderson, D.; Wu, J., Revisiting density functionals for the primitive model of electric double layers. Journal of Chemical Physics 2014, 140 (4), 211-233. 34. Jiang, J.; Ginzburg, V.; Wang, Z.-G., Density Functional Theory for Charged Fluids. Soft Matter 2018, 14 (28), 5878-5887. 35. Zhao, S.; Liu, Y.; Chen, X.; Lu, Y.; Liu, H.; Hu, Y., Unified framework of multiscale density functional theories and its recent applications. In Advances in Chemical Engineering, 2015; Vol. 47, pp 1-83. 36. Zhan, C.; Lian, C.; Zhang, Y.; Thompson, M. W.; Xie, Y.; Wu, J.; Kent, P. R. C.; Cummings, P. T.; Jiang, D. E.; Wesolowski, D. J., Computational Insights into Materials and Interfaces for Capacitive Energy Storage. Advanced Science 2017, 4 (7), 1700059. 37. Goddard, B. D.; Nold, A.; Savva, N.; Pavliotis, G. A.; Kalliadasis, S., General Dynamical Density Functional Theory for Classical Fluids. Physical Review Letters 2012, 109 (12), 120603. 38. Löwen, H., Density functional theory: from statics to dynamics. Journal of Physics-Condensed Matter 2003, 15 (6), V1-V3. 39. Jiang, J.; Cao, D.; Jiang, D. E.; Wu, J., Time-dependent density functional theory for ion diffusion in electrochemical systems. J Phys Condens Matter 2014, 26 (28), 284102. 40. Jiang, J.; Cao, D.; Jiang, D. E.; Wu, J., Kinetic charging inversion in ionic liquid 25


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

electric double layers. Journal of Physical Chemistry Letters 2014, 5 (13), 2195-2200. 41. Wu, J.; Jiang, T.; Jiang, D. E.; Jin, Z.; Henderson, D., A classical density functional theory for interfacial layering of ionic liquids. Soft Matter 2011, 7 (23), 11222-11231. 42. Simon, P.; Gogotsi, Y., Materials for electrochemical capacitors. Nature Materials 2008, 7 (11), 845-854. 43. Chmiola, J.; Yushin, G.; Gogotsi, Y.; Portet, C.; Simon, P.; Taberna, P. L., Anomalous Increase in Carbon Capacitance at Pore Sizes Less than 1 Nanometer. Science 2006, 313 (5794), 1760-1763. 44. Largeot, C.; Portet, C.; Chmiola, J.; Taberna, P. L.; Gogotsi, Y.; Simon, P., Relation between the ion size and pore size for an electric double-layer capacitor. Journal of the American Chemical Society 2008, 130 (9), 2730-2731. 45. Raymundo-Pinero, E.; Kierzek, K.; Machnikowski, J.; Beguin, F., Relationship between the nanoporous texture of activated carbons and their capacitance properties in different electrolytes. Carbon 2006, 44 (12), 2498-2507. 46. Jiang, D.-e.; Jin, Z.; Wu, J., Oscillation of capacitance inside nanopores. Nano letters 2011, 11 (12), 5373-5377. 47. Jiang, D.-e.; Jin, Z.; Henderson, D.; Wu, J., Solvent effect on the pore-size dependence of an organic electrolyte supercapacitor. The journal of physical chemistry letters 2012, 3 (13), 1727-1731. 48. Schoch, R. B.; Han, J.; Renaud, P., Transport phenomena in nanofluidics. Reviews of Modern Physics 2008, 80 (3), 839-883. 49. Karnik, R.; Castelino, K.; Majumdar, A., Field-effect control of protein transport in a nanofluidic transistor circuit. Applied Physics Letters 2006, 88 (12), 123114. 50. Karnik, R.; Fan, R.; Yue, M.; Li, D. Y.; Yang, P. D.; Majumdar, A., Electrostatic control of ions and molecules in nanofluidic transistors. Nano Letters 2005, 5 (5), 943-948. 51. Qu, W.; Li, D., A model for overlapped EDL fields. Journal of Colloid & Interface Science 2000, 224 (2), 397-407. 52. Baldessari, F.; Santiago, J. G., Electrokinetics in nanochannels. Part I. Electric double layer overlap and channel-to-well equilibrium. Journal of Colloid & Interface Science 2008, 325 (2), 526–538. 53. Chakraborty, S.; Srivastava, A. K., Generalized model for time periodic electroosmotic flows with overlapping electrical double layers. Langmuir 2007, 23 (24), 12421-12428. 54. Luo, Z. X.; Xing, Y. Z.; Ling, Y. C.; Kleinhammes, A.; Wu, Y., Electroneutrality breakdown and specific ion effects in nanoconfined aqueous electrolytes observed by NMR. Nature Communications 2015, 6, 6358. 55. Cheng, L.; Zhao, S.; Liu, H.; Wu, J., Time-dependent density functional theory for the charging kinetics of electric double layer containing room-temperature ionic liquids. Journal of Chemical Physics 2016, 145 (20), 204707. 56. Henderson, D.; Lamperski, S.; Jin, Z.; Wu, J., Density functional study of the electric double layer formed by a high density electrolyte. Journal of Physical Chemistry B 2011, 115 (44), 12911-12914. 57. Liu, Y.; Zhao, S.; Wu, J., A Site Density Functional Theory for Water: Application to Solvation of Amino Acid Side Chains. Journal of Chemical Theory & Computation 2013, 9 (4), 1896-1908. 58. Yu, Y.-X.; Wu, J., Structures of hard-sphere fluids from a modified fundamental-measure theory. The Journal of chemical physics 2002, 117 (22), 10156-10164. 59. Ren, F.; Pearton, S. J., Semiconductor device-based sensors for gas, chemical, and biomedical applications. CRC Press Boca Raton, USA: 2011. 60. Berinde, V., Picard iteration converges faster than Mann iteration for a class of 26


Page 26 of 28

Page 27 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

quasi-contractive operators. Fixed Point Theory & Applications 2004, 2004 (2), 716359. 61. Hou, C. H.; Taboada-Serrano, P.; Yiacoumi, S.; Tsouris, C., Monte Carlo simulation of electrical double-layer formation from mixtures of electrolytes inside nanopores. Journal of Chemical Physics 2008, 128 (4), 044705. 62. Yang, K. L.; Yiacoumi, S.; Tsouris, C., Monte Carlo simulations of electrical double-layer formation in nanopores. Journal of Chemical Physics 2002, 117 (18), 8499-8507. 63. Wu, J.; Jiang, T.; Jiang, D.-e.; Jin, Z.; Henderson, D., A classical density functional theory for interfacial layering of ionic liquids. Soft Matter 2011, 7 (23), 11222-11231. 64. Bernard, O.; Cartailler, T.; Turq, P.; Bium, L., Mutual diffusion coefficients in electrolyte solutions. Journal of Molecular Liquids 1997, s 73–74 (97), 403–411. 65. Dufrêche, J. F.; Bernard, O.; Turq, P., Mutual diffusion coefficients in concentrated electrolyte solutions. Journal of Molecular Liquids 2002, 96 (01), 123-133. 66. Hansen, J.-P.; McDonald, I. R., Theory of simple liquids. Elsevier: 1990. 67. Tanaka, H.; Kawasaki, T.; Shintani, H.; Watanabe, K., Critical-like behaviour of glass-forming liquids. Nature materials 2010, 9 (4), 324. 68. Jacob, J.; Kumar, A.; Anisimov, M.; Povodyrev, A.; Sengers, J., Crossover from Ising to mean-field critical behavior in an aqueous electrolyte solution. Physical Review E 1998, 58 (2), 2188. 69. Cummings, P.; Cochran, H.; Simonson, J.; Mesmer, R.; Karaborni, S., Simulation of supercritical water and of supercritical aqueous solutions. The Journal of chemical physics 1991, 94 (8), 5606-5621. 70. Biesheuvel, P. M.; Zhao, R.; Porada, S.; Van, d. W. A., Theory of membrane capacitive deionization including the effect of the electrode pore space. J Colloid Interface Sci 2011, 360 (1), 239-248. 71. Jiang, D. E.; Jin, Z.; Wu, J., Oscillation of capacitance inside nanopores. Nano Letters 2011, 11 (12), 5373.

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Dynamic Adsorption of Ions into Like-Charged Nanospace: A Dynamic

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