Non-Negligible Roles of Pore Size Distribution on Electroosmotic Flow

Jun 25, 2019 - ABSTRACT: Electroosmotic flow in nanoporous materials ... cells, and batteries.4 Due to confinement effects, the physics underlying ion...
0 downloads 0 Views 1MB Size
Download PDF
Non-Negligible Roles of Pore Size Distribution on Electroosmotic Flow in Nanoporous Materials Cheng Lian,† Haiping Su,† Chunzhong Li,‡ Honglai Liu,*,† and Jianzhong Wu*,§

Downloaded via UNIV OF SOUTHERN INDIANA on July 22, 2019 at 16:17:48 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

State Key Laboratory of Chemical Engineering, Shanghai Engineering Research Center of Hierarchical Nanomaterials, and School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, P.R. China ‡ Key Laboratory for Ultrafine Materials of Ministry of Education, Shanghai Engineering Research Center of Hierarchical Nanomaterials, School of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, P.R. China § Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521, United States ABSTRACT: Electroosmotic flow in nanoporous materials is of fundamental importance for the design and development of filtration membranes and electrochemical devices such as supercapacitors and batteries. Recent experiments suggest that ion transport in a porous network is substantially different from that in individual nanochannels due to the pore size distribution and pore connectivity. Herein, we report a theoretical framework for ion transport in nanoporous materials by combing the classical density functional theory to describe the electrical double layer (EDL) structure, the Navier−Stokes equation for the fluid flow, and the effective medium approximation to bridge the gap between individual nanopores and the network connectivity. We find that ion conductivity in nanoporous materials is extremely sensitive to the pore size distribution when the average size of micropores is comparable to the EDL thickness. The theoretical predictions provide an explanation of the giant gap between the conductivity of a single pore and that of a porous network and highlight the mechanism of ion transport through nanoporous materials important for numerous practical applications. KEYWORDS: ion transport, nanoporous electrodes, network connectivity, pore size distribution, molecular modeling flow in nanopores.15,16 Previous investigations are mostly focused on individual nanochannels or based on phenomenological equations that ignore the discrete nature of ionic species near the fluid−solid interface. Porous materials are typically represented by a bundle of uniform capillaries with an effective pore radius. The single-pore model does not adequately address the heterogeneity of porous networks like those in amorphous membranes or porous electrodes.17 Although several network models have been proposed to describe fluid transport and reaction in micropores,18 their applicability to ionic transport is questionable because existing models cannot adequately capture thermodynamic non-ideality arising from complex ion−ion and ion−surface interactions. Recently, a number of theoretical models have been proposed to account for complex ion−ion interactions,19−23 electric double layer (EDL) overlap,24 ion−wall interactions,25−28 and surface reactions.29 Although these theoretical models are able to reproduce certain experimental data for

I

onic transport through nanoscale pores is of fundamental importance for the design and development of a wide range of devices and industrial processes, such as nanofiltration and desalination,1 and for energy production and storage including harvesting blue energy,2 production of solar fuels,3 and charging/discharging of supercapacitors, fuel cells, and batteries.4 Due to confinement effects, the physics underlying ion transport in nanopores is fundamentally different from that in bulk systems.5−10 Whereas recent years have witnessed much progress in understanding electroosmotic flow in individual nanochannels, barely exploited is ionic transport through amorphous membranes or porous electrodes that involve pore connectivity and variations in both pore size and geometry.11,12 Many experimental techniques, such as electrochemical quartz crystal microbalance (EQCM), NMR spectroscopy, XRD, and infrared (IR) spectroelectrochemistry, have been used to study the dynamic properties of ionic fluids in confined geometries.13 Ionic dynamics and transport in nanopores have also been investigated extensively with both atomistic and coarse-grained molecular dynamics (MD) simulations.14 Besides, analytical methods are often employed to examine ionic © XXXX American Chemical Society

Received: April 29, 2019 Accepted: June 25, 2019 Published: June 25, 2019 A

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

Article

www.acsnano.org

Cite This: ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano

Figure 1. Schematic of ionic transport through a porous material as represented by a carbon electrode. (a) A macroscopic view of the porous material−electrolyte interface. (b) Representative microscopic structure for the porous material submerged in an electrolyte solution. (c) Ionic flow through a charged nanopore of width d under an electric field. (d) Illustration of a two-dimensional pore network (connectivity coordinate number Z = 4). According to effective medium approximation (EMA), the discrete network of micropores (d) is represented by a homogeneous network of the same pore size (e).

ionic flow in nanopores, still missing is a comprehensive description of electroosmotic flow in nanopores that incorporates all of the important effects in order to capture the entire picture. While MD simulation can describe microscopic details in principle, the classical density functional theory (DFT) is particularly useful to account for the thermodynamic non-ideality of confined ionic systems.30,31 Classical DFT has been routinely utilized to describe the ionic excluded-volume effects and electrostatic correlations neglected in the Poisson−Nernst−Planck (PNP) equations, a conventional microscopic theory for predicting ion transport.32 The non-mean-field effects are especially important for ionic flows in nanochannels due to extreme confinement. In our previous work,33−35 we demonstrated that a combination of classical DFT with the Navier−Stokes (NS) equation is able to predict ionic transport in individual nanopores in quantitative agreement with experimental data. From a numerical perspective, classical DFT is fully consistent with the NS equation for describing the fluid flow through nanochannels. In this article, we report a multiscale framework to describe ionic transport through nanoporous materials by integration of theoretical methods for single-pore modeling with the effective medium approximation (EMA) for network conductivity.36 The adoption of EMA allows us to bridge the gap between ionic transport in individual pores and that in porous networks such as porous electrodes or membranes. The numerical results from the DFT-NS-EMA predictions reveal the significant influence of the pore size distribution (PSD) and network connectivity on ion conductivity in realistic nanoporous materials. Our model system mimics a typical experimental setup for ionic flow through nanoscale porous carbon electrodes in contact with bulk electrolytes at a steady-state condition. Figure 1a shows a schematic representation of the electrochemical system. The porous material is submerged in an electrolyte solution consisting of one type of cations and one type of anions. For simplicity, we assume that cations and anions have the same size and absolute charge but opposite valences. In addition, we assume that the porous material is

ideally polarizable and that the confined ionic system is in equilibrium with a bulk electrolyte solution. To establish a generic framework that captures the PSD and the pore network parameters of realistic porous materials, we adopt the slit-pore model as typically used in the characterization of amorphous porous materials.37 As shown in Figure 1c, we describe each pore in terms of the slit width d and length L, with the latter fixed at L = 900 nm throughout this work. Ionic flow inside the pore is assumed to be induced by an electric field, E0 = V/L, with V being the voltage applied to the direction of pore length L (along the x-direction). A gating potential Vg is imposed at the surface of the nanopores. The electroosmotic flow leads to an electrical current due to ionic motion in the charged nanopore. For simplicity, we do not consider the concentration and pressure gradient across the porous material, albeit such effects could be included in our model. As in the space-charge model,15 the ionic distribution in the z-direction of the pore, that is, in the direction perpendicular to the pore surface, is represented by that at equilibrium. The electrical current and conductance are determined by the ion distributions and the solvent velocity profile inside the pore calculated from classical DFT and the NS equations, respectively. The distributions of cations and anions are assumed identical to those at equilibrium because the time scale for EDL formation inside the pore is on the order of κ −1d

τ1 = D m ∼ 0.5 ns (where κ−1 is the Debye screening length, κ−1 ∼ 1 nm for electrolyte systems considered in this work, dm ∼ 1 nm is the mean pore diameter, and D ∼ 2 × 10−9 m2/s is the diffusion coefficient of cations and anions). The relaxation time for EDL formation is negligible in comparison with the L2

time for the fluid flow τ2 = D ∼ 0.4 ms (where the pore length is L = 900 nm). Therefore, the equilibrium assumption is justified, that is, the EDL structure inside the pore can be assumed the same as that at equilibrium. In general, variation of the ion concentrations in the direct of flow can be calculated from the dynamic DFT (DDFT) or the PNP equation. B

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano

Figure 2. Conductance versus the pore size for ionic transport in individual nanopores at different electrolyte concentrations. Solid lines are predicted from the classical DFT and dashed lines from the PB equation, both in combination with the NS equation for the fluid flow. (a) The surface potential of nanopores is fixed at 0.1 V. (b) The surface potential is fixed 0.5 V.

Figure 3. (a) Ionic density profiles in slit pores of different pore widths. Here the surface potential is fixed at 0.5 V, and the electrolyte concentration in the bulk is 1 M. The ionic density profiles emerge from a concave shape (0.6 nm) to fully developed electric double layers (2.5 nm). (b) The solvent velocity profiles in individual slit nanopores.

small gating potential. However, there are significant differences when the electrolyte concentration or the surface potential increases and the pore size falls. In comparison with classical DFT, the PB equation overpredicts the ionic conductance when the surface potential is large. As expected, the discrepancy between classical DFT and the PB predictions becomes more apparent at high salt concentrations and large surface potentials because of stronger size effects and electrostatic correlations. In particular, the EDL structure near a highly charged surface is extremely sensitive to electrostatic correlations and ionic excluded volume effects. Such effects are faithfully described by classical DFT while ignored in the PB equation.39 As a result, the PB predictions result in a significant overestimation of the ionic conductance of electrolytes across the nanopores, especially for high salt concentrations, high surface potentials, and small pores. Figure 2a shows that the conductance of a nanopore rises as its pore size increases and approaches an asymptotic value when the pore size is larger than a few nanometers. A further increase of the pore width makes no contribution to the ion current. Because of a stronger accumulation of counterions near the surface, the conductance is significantly increased when we raise the gating potential (Figure 2b). In that case, the pore size effect on the conductance is evident even in large pores (d > 10 nm). Because EDL structures are fully established within only a few nanometers (Figure 3a), variation of the conductance with the pore size for larger pores must be affiliated with the flow effect. From the solvent velocity profiles shown in Figure 3b, we see that a larger pore gives a much higher solvent velocity inside the nanochannel, leading to a

Besides, the hydrodynamic effects can be accounted for by solving the DDFT/PNP equation along with the NS equation.38 Because the present work is focused on pore connectivity, we neglect ion concentration and pressure gradients in the direction of the electroosmotic flow and consider ionic transport only by an external electric field. Full technical details for the theoretical model are described in the Model and Theoretical Methods section. Briefly, we use EMA36 to quantify ionic transport in heterogeneous porous materials. As illustrated schematically in Figure 1d,e, EMA describes transport phenomena in a network of heterogeneous pores in terms of that of identical pores with the same connectivity. In previous work,33−35 we have benchmarked the performance of the classical DFT in comparison with the Poisson−Boltzmann (PB)-based method and demonstrated that DFT is able to account for thermodynamic non-ideality due to ion−ion interactions in good agreement with experimental data. The focus of this work is directed at analyzing the effects of PSD and network connectivity on ionic transport.

RESULTS AND DISCUSSION Pore Size Effect on Ionic Transport in Individual Pores. Figure 2 displays theoretical predictions for the ionic conductance in individual nanopores at different electrolyte concentrations and two gating potentials (0.1 and 0.5 V). For comparison, we present results from both the PB equation (dashed lines) and from the classical DFT (solid lines). As reported in our earlier work,35 the classical DFT and PB equations yield similar results at low salt concentrations and a C

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano

Figure 4. Conductance (G) and conductivity (K) of electrolyte (1 M) in porous carbon electrodes with different pore size distributions. Here dm is the average pore size and σ is the standard deviation. In all cases, the surface potential is fixed at 0.5 V.

reduction of the conductivity. The difference can be up to many orders of magnitude when the pore size is extremely small (Figure 4b). Such non-intuitive effects may help explain, for example, why there exists a giant gap between single-poreand membrane-based nanofluidic osmotic power generators.17 With the increase of standard deviation (σ), the pore network with a wide PSD does not exhibit the oscillatory conductivity with the mean pore size, as the wide distribution of pore width reflects only an average pore size effect. Similar to the effect of PSD on ionic conductance, widening the pore distribution reduces the conductivity. Pore Connectivity Effect on Ionic Transport. Figure 5 shows how pore connectivity affects ionic transport in porous

larger conductance. Also included in Figure 3b are the velocity profiles from COMSOL calculations, which are the same as those predicted from the PB equation for the local electric potential. At a low surface potential (0.5 V), the PB equation significantly under predicts the solvent velocity profiles in small pores. As shown in Figure 2, the PB predictions are reasonable when the pore size is large but deteriorate as the surface potential increases. The conductance increases at elevated electrolyte concentrations because more counterions accumulate near the charged surface, in particular, for nanopores with high surface electrical potentials. Influence of the Pore Size Distribution. To illustrate the effect of PSD on ionic conductance, we use the log-normal function to represent the polydispersity of nanopores.40,41 Figure 4a presents the conductance versus the average pore sizes with different standard deviations (σ). Here the bulk electrolyte concentration is set at 1.0 M, and the surface potential is 0.5 V. Surprisingly, the conductance increases with the mean pore size even beyond 10 nm, and its magnitude is drastically reduced with increasing the polydispersity when the mean pore size is comparable to the EDL thickness. In the latter case, the conductance of a porous network is much smaller than that of the single pore of the same average diameter. With a fixed mean pore size dm, an increase of the standard deviation σ means a larger percentage of small pores, thus a smaller conductance. The effect of PSD becomes relatively insignificant when the mean pore size dm is larger than about 4 nm. In comparison to ionic transport in a single pore, the effect of (mean) pore size on conductivity is magnified due to contributions from different pores. Figure 4b presents the theoretical electrical conductivity as a function of the mean pore size when the bulk electrolyte concentration and the surface voltage are fixed at 1.0 M, and 0.5 V, respectively. The conductivity shows a maximum at dm = 1−2 nm, relatively independent of the PSDs. For a porous material with a narrow PSD (σ = 0.01), the conductivity oscillates with the increasing pore size, which is reminiscent of a similar behavior for ionic flow in individual nanopores.35 The peak values coincide with the slit-pore widths that induce the overlapping of EDLs from the left and the right sides of the pores, as shown in Figure 3b. For macroscopic pores, the asymptotic value of conductivity is around 25 S/m, which agrees well with the experimental value.42,43 Interestingly, nanopores with the widths in the range between 0.6 and 5.0 nm have the largest conductivity. When the average pore size is comparable to the EDL thickness (∼1 nm), broadening the PSD may lead to a severe

Figure 5. Conductance of an electrolyte (1 M) in porous carbon electrodes with different pore connectivity parameters when the average pore size and standard deviation are 1.75 nm and 0.2, respectively.

materials. The conductance rises about 10% when the network connectivity parameter Z increases from 3 to 11, suggesting that the connectivity effect is relatively unimportant in comparison to that due to the PSDs. As expected, higher pore connectivity increases the chances for electrolytes to transport across the porous network. However, its effect on the conductance levels off when the pore connectivity is sufficiently high.

CONCLUSIONS In summary, we have developed a microscopic model for ion transport across porous materials by combining EMA for network connectivity, the NS equation for electroosmotic flows, and classical DFT for ionic distributions in nanopores. The theoretical framework is utilized to investigate the effects of mean pore size, PSD, and pore connectivity of simple D

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano

predicted by the PB equation, which is conventionally used in the space-charge model. In eq 1, Vi(z) represents the external potential due to the confinement effect and is given by

electrolytes in nanoporous carbon electrodes on ionic conductance and conductivity. We find that ion conductivity in porous materials can be drastically reduced by increasing the polydispersity of micropores if the pore size is comparable to the EDL thickness. The conductivity shows an oscillatory dependence on the mean pore size only for porous materials with narrow PSDs. While increasing pore connectivity promotes ion transport, its effect becomes less important when the pores become well connected. Interestingly, ion transport is enhanced when the porous electrodes have narrow pore distributions and the average pore sizes are under 2 nm. By integrating continuous models from different scales, we are able to account for PSD and pore connectivity effects on ion transport through amorphous membranes and electrodes. The main focus of this work was to establish a theoretical procedure to describe electroosmotic flow in realistic porous materials. Although the numerical results were not concerned with specific experimental systems, we explained drastic differences between ion transport in a single pore and that in a network of many pores observed in previous experiments. For most amorphous porous electrodes or membranes, direct characterizations of pore shape, PSD, and pore connectivity are experimentally challenging. Meanwhile, simulation of ion transport through such materials is also difficult due to the lack of atomic details. Theoretical calculations based on relatively simple models provide an alternative way to understand ion transport through porous media with complex pore networks, surface charges, and surface chemistries. While only monovalent cations and anions of the same size were considered in this work, a similar procedure can be readily established for more realistic ion models that account for different ion sizes, valences, and even shapes. We expect that this multiscale model can be further applied to various molecular transports in porous materials, including those broadly related to electrochemical applications.

l o o 0, σi/2 < z < H − σi/2 Vi (z) = m o o∞ , otherwise n

where H is the pore width. The electric potential satisfies the Poisson equation:

ρ (z) d2ψ (z) =− e 2 ε0εr dz

(3)

where ε0 is the vacuum dielectric constant and εr is the relative dielectric constant, and we assume the solvent could be treated as a continuum with a single value of dielectric constant equal to the value in bulk water. As demonstrated by Bonthuis and co-workers,46,47 nanoconfinement may reduce the solvent dielectric constant near the electrolyte−solid surface within a thickness in the range of 0.07−0.22 nm. Because the surface and specific ion effects are neglected in this work, our model does not account for the decease of solvent dielectric constant in nanopores. ρe (z) = ∑i Zieρi (z) represents the local charge density. Eqs 1 and 3 can be solved self-consistently with the boundary conditions ψ(0) = ψ(H) = Vg, Vg represents the pore surface potential. The relation between the surface potential and the surface charge density is given by the Gauss law: Q = − ε0εr

∂ψ (z) ∂z

(4)

z=0

Due to the planar symmetry for slit pores,

∂ψ (z) ∂z z = H /2

= 0 , and the

derivative of the electrostatic potential at the pore surface could be obtained by integrating eq 3: − ε0εr

∂ψ (z) ∂z

= ε0εr z=0

∫0

H /2

∂ 2ψ (z) dz = − ∂z 2

∫0

H /2

ρe (z)dz

(5) Eq 5 shows that the charge neutrality is automatically satisfied for individual slit pores:

MODEL AND THEORETICAL METHODS Classical Density Functional Theory for Ionic Profile. For simplicity, we use the restricted primitive model (RPM) to represent all ionic species. Ions are charged hard spheres of equal diameter, and the solvent is a dielectric continuum. The diameters of the hydrated ions are set as 0.5 nm, approximately corresponding to those in an aqueous NaCl solution.15 At room temperature, the dielectric constant εr of liquid water is 78.4. Whereas RPM ignores effects such as specific ion properties, (de)hydration, and size disparity that are important for understanding ion transport in micropores, the simple model allows us to illustrate how the PSD and pore connectivity affect electroosmotic flow in nanoporous materials. As commonly used for the characterization of porous materials, the slit-pore model is adopted to represent the PSD and the geometry of individual micropores. The pore curvature is typically not quantified in experiments, but its effects on the equilibrium properties of EDL can be accounted for by using alternative pore models.44 Similar to our early work,33−35 ionic distributions in nanoscale pores are described by the classical DFT. Within the slit-pore model, the ionic density profile varies only in the z-direction, that is, the direction perpendicular to the wall: ρi (z) = ρi bulk exp[− βZieψ (z) − β Δμiex (z) + Vi (z)]

(2)

Q+

∫0

H /2

ρe (z)dz = 0

(6)

The Navier−Stokes Equation for Ionic Flows. From the ionic density profiles, we can solve for the solvent velocity by using the NS equation: η∇2 u − ρe E0 − ∇P = 0

(7)

where u represents the solvent velocity in the pore length direction (x direction), and η stands the solvent viscosity. The pressure term in the NS equation is not considered in this work because there is no pressure difference at the two sides of the porous material. When the ionic flow is driven by an external electrical field, the pressure gradient inside the pores is extremely small.48 For an electroosmotic flow in slit pores, the boundary conditions for eq 7 are

l o obu′(z) = u(z), at z = 0 m o o o at z = H /2 n u′(z) = 0,

(8) 42

where b is the slip length. According to previous studies, the slip length takes a negative value when water flows near a hydrophilic surface, while it is positive near a hydrophobic surface. Besides, the slip length is affected by ion-wall electro-friction and molecular-wall flow.21−23 In principle, such effects could be incorporated into our calculations but are ignored in this work because it is mainly focused on pore connectivity. By assuming that the slip length is negligible, we can derive an analytical expression for the solvent velocity from eqs 7 and 8:

(1)

is the bulk concentration of ionic species i, T = 298 K is where ρbulk i the absolute temperature, β = (kBT)−1, kB is the Boltzmann constant, Zi = ±1 stands for the ion valence, ψ(z) is the local electric potential, and Δμex i (z) represents deviation of the local excess chemical potential as given in previous publications.45 Without considering the local excess chemical potential Δμex i (z), eq 1 reduces to that E

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

ACS Nano u(z) = ε0εrE0[Vs − ψ (z)] /η

(9)

f (d) =

Eq 9 indicates that the maximum solvent velocity appears at the pore center where the potential gap between the local and the surface potentials is large. When the pore size is on the order of the ionic diameters, the local electrical potential ψ(z) is highly inhomogeneous. Because the pore center has a non-zero electrical potential, the solvent velocity is always smaller than asymptotic value u∞ = ε0εrE0Vs/η for larger pores. The ionic velocities in the nanopore, ui(z), can be calculated from a superposition of the solvent velocity, u(z), and the ionic motion driven by the external electrical field:

ui(z) = u(z) − eZiυiE0

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Cheng Lian: 0000-0002-9016-832X Chunzhong Li: 0000-0001-7897-5850 Honglai Liu: 0000-0002-5682-2295 Jianzhong Wu: 0000-0002-4582-5941

H

i

Notes

The authors declare no competing financial interest.

(11)

ACKNOWLEDGMENTS This work was sponsored by the National Natural Science Foundation of China (91834301, 21808055), State Administration of Foreign Experts Affairs of China (B08021), China Postdoctoral Science Foundation (2019M651416), and Shanghai Sailing Program (18YF1405400, 19YF1411700). J.W. thanks the financial support from the Fluid Interface Reactions, Structures and Transport (FIRST) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Basic Energy Sciences. We also thank Professor Marc-Olivier Coppens for the helpful discussion.

Ionic transport through a nanopore is typically characterized in terms of the conductance, G = I/V, that is, the ionic current I divided by the voltage drop V, or in terms of the electrical conductivity, which is defined as K ≡ GL/(WH). Effective Medium Approximation. The original concepts of EMA stem from theoretical modeling of resistor networks in an electric circuit.36 In this work, EMA is adopted to quantify ion transport in hierarchical porous materials. As illustrated in Figure 1d,e, EMA describes the transport properties of a disordered porous material by replacing the heterogeneous pore network with an imaginary homogeneous network of identical pores. According to EMA, we can estimate the effective conductance Geff for a network of pores from36 d = d max

∑ d = d min

f (d)

Geff − Gd

(

Z 2

)

REFERENCES

=0

− 1 Geff + Gd

(12)

(1) Deshmukh, A.; Boo, C.; Karanikola, V.; Lin, S. H.; Straub, A. P.; Tong, T. Z.; Warsinger, D. M.; Elimelech, M. Membrane Distillation at the Water-Energy Nexus: Limits, Opportunities, and Challenges. Energy Environ. Sci. 2018, 11, 1177−1196. (2) Siria, A.; Bocquet, M. L.; Bocquet, L. New Avenues for the Large-Scale Harvesting of Blue Energy. Nat. Rev. Chem. 2017, 1, 0091. (3) Chabi, S.; Papadantonakis, K. M.; Lewis, N. S.; Freund, M. S. Membranes for Artificial Photosynthesis. Energy Environ. Sci. 2017, 10, 1320−1338. (4) Bazant, M. Z. Theory of Chemical Kinetics and Charge Transfer Based on Nonequilibrium Thermodynamics. Acc. Chem. Res. 2013, 46, 1144−1160. (5) Comtet, J.; Niguès, A.; Kaiser, V.; Coasne, B.; Bocquet, L.; Siria, A. Nanoscale Capillary Freezing of Ionic Liquids Confined between Metallic Interfaces and the Role of Electronic Screening. Nat. Mater. 2017, 16, 634. (6) Sint, K.; Wang, B.; Král, P. Selective Ion Passage through Functionalized Graphene Nanopores. J. Am. Chem. Soc. 2008, 130, 16448−16449. (7) Gopinadhan, K.; Hu, S.; Esfandiar, A.; Lozada-Hidalgo, M.; Wang, F.; Yang, Q.; Tyurnina, A.; Keerthi, A.; Radha, B.; Geim, A. Complete Steric Exclusion of Ions and Proton Transport through Confined Monolayer Water. Science 2019, 363, 145−148. (8) Kamcev, J.; Paul, D. R.; Manning, G. S.; Freeman, B. D. Ion Diffusion Coefficients in Ion Exchange Membranes: Significance of Counterion Condensation. Macromolecules 2018, 51, 5519−5529. (9) Chang Wu, R.; Papadopoulos, K. D. Electroosmotic Flow through Porous Media: Cylindrical and Annular Models. Colloids Surf., A 2000, 161, 469−476.

where Z is the network connectivity, that is, the number of pores connected to each node in the pore network, Gd is the conductance of the pore with pore size of d, G(d) = K(d)A(d)/L, and dmin and dmax are the minimum and maximum pore sizes. EMA has been used extensively to describe transport phenomena in heterogeneous media.51 Numerous studies were devoted to addressing its reliability and limitations, primarily for pore networks with low conductance, in particular near the percolation threshold of micropores.52 Because this work is focused on ion transport in electrode materials with high porosity, the network connectivity of micropores is expected to be remote from the percolation limit. In this case, previous studies indicate that EMA is quantitatively accurate in comparison with numerical simulations.53 Under conditions such that the permeability is limited by the low coordination number, we may estimate the effective conductivity of the porous electrode by using a linear relationship:54

keff

l NGeff o o o o o o A total o o =m o o o ij Z − o o jj o o o 4 o nk

=

Geff Wdm

Z>6

2 yzijj Geff yzz zzjj zz Z ⩽ 6 {jk Wdm z{

(14)

AUTHOR INFORMATION

(10)

∫0 ∑ Zieρi (z)ui(z)dz

ij (lnd − lndm)2 yz Vt zz expjjjj− zz 2 2π dσ 2 σ k {

where dm is the mean pore diameter, σ is the standard deviation of the log-transformed pore diameter, and Vt is the total pore volume of the porous electrode materials. To avoid undersized or oversized pores, the pore size is limited within a reasonable range [dmin, dmax], where the values of the cumulative distribution function of the log-normal distribution are between 0.001 and 0.999.

where υi represents the ionic mobility. Throughout this work, we assume υ = υ+ = υ− = 4.8 × 1011 mol s kg−1, which corresponds to the mean value of Na+ and Cl− mobilities in an aqueous solution. The diffusivity of each ion, Di, is related to its mobility, υi, by the Einstein relation Di = kBTυi.49 The electrical current is related to the ionic distributions and the velocity profiles inside the pore calculated from the classical DFT and the NS equation, respectively:50

I=W

Article

(13)

The linear interpolation yields the permeability of the cubic lattice when Zcor = 6 and vanishes at Zcor = 2, the percolation limit. To illustrate the polydisperse effect, we consider a hierarchical porous material with the pore size described by the log-normal distribution:36 F

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano (10) O’Brien, R. Electroosmosis in Porous Materials. J. Colloid Interface Sci. 1986, 110, 477−487. (11) Lagadec, M. F.; Zahn, R.; Muller, S.; Wood, V. Topological and Network Analysis of Lithium Ion Battery Components: The Importance of Pore Space Connectivity for Cell Operation. Energy Environ. Sci. 2018, 11, 3194−3200. (12) Melnikov, D. V.; Hulings, Z. K.; Gracheva, M. E. ElectroOsmotic Flow through Nanopores in Thin and Ultrathin Membranes. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2017, 95, No. 063105. (13) Thompson, W. H. Perspective: Dynamics of Confined Liquids. J. Chem. Phys. 2018, 149, 170901. (14) Breitsprecher, K.; Holm, C.; Kondrat, S. Charge Me Slowly I Am in a Hurry: Optimizing Charge−Discharge Cycles in Nanoporous Supercapacitors. ACS Nano 2018, 12, 9733−9741. (15) Peters, P. B.; van Roij, R.; Bazant, M. Z.; Biesheuvel, P. M. Analysis of Electrolyte Transport through Charged Nanopores. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2016, 93, No. 053108. (16) Conway, E.; Farrell, T.; Psaltis, S. Mathematical Modeling of Ion Transport through Nanopores. J. Phys. Chem. C 2018, 122, 23728−23738. (17) Gao, J.; Liu, X. L.; Jiang, Y. A.; Ding, L. P.; Jiang, L.; Guo, W. Understanding the Giant Gap between Single-Pore- and MembraneBased Nanofluidic Osmotic Power Generators. Small 2019, 15, 1804279. (18) Wu, R.; Liao, Q.; Zhu, X.; Wang, H. Pore Network Modeling of Cathode Catalyst Layer of Proton Exchange Membrane Fuel Cell. Int. J. Hydrogen Energy 2012, 37, 11255−11267. (19) He, Y.; Huang, J.; Sumpter, B. G.; Kornyshev, A. A.; Qiao, R. Dynamic Charge Storage in Ionic Liquids-Filled Nanopores: Insight from a Computational Cyclic Voltammetry Study. J. Phys. Chem. Lett. 2015, 6, 22−30. (20) Bonthuis, D. J.; Netz, R. Unraveling the Combined Effects of Dielectric and Viscosity Profiles on Surface Capacitance, ElectroOsmotic Mobility, and Electric Surface Conductivity. Langmuir 2012, 28, 16049−16059. (21) Ma, Y.; Yeh, L. H.; Lin, C. Y.; Mei, L.; Qian, S. PH-Regulated Ionic Conductance in a Nanochannel with Overlapped Electric Double Layers. Anal. Chem. 2015, 87, 4508−4514. (22) Lian, C.; Liu, H.; Li, C.; Wu, J. Hunting Ionic Liquids with Large Electrochemical Potential Windows. AIChE J. 2019, 65, 804− 810. (23) Kondrat, S.; Wu, P.; Qiao, R.; Kornyshev, A. A. Accelerating Charging Dynamics in Subnanometre Pores. Nat. Mater. 2014, 13, 387−393. (24) Lian, C.; Zhao, S.; Liu, H.; Wu, J. Time-Dependent Density Functional Theory for the Charging Kinetics of Electric Double Layer Containing Room-Temperature Ionic Liquids. J. Chem. Phys. 2016, 145, 204707. (25) Xin, Y.; Zheng, Y. X.; Yu, Y. X. Density Functional Theory Study on Ion Adsorption and Electroosmotic Flow in a Membrane with Charged Cylindrical Pores. Mol. Phys. 2016, 114, 2328−2336. (26) Wu, P.; Qiao, R. Physical Origins of Apparently Enhanced Viscosity of Interfacial Fluids in Electrokinetic Transport. Phys. Fluids 2011, 23, No. 072005. (27) Qiao, R. Effects of Molecular Level Surface Roughness on Electroosmotic Flow. Microfluid. Nanofluid. 2006, 3, 33−38. (28) Ai, Y.; Zhang, M.; Joo, S. W.; Cheney, M. A.; Qian, S. Effects of Electroosmotic Flow on Ionic Current Rectification in Conical Nanopores. J. Phys. Chem. C 2010, 114, 3883−3890. (29) Gillespie, D. High Energy Conversion Efficiency in Nanofluidic Channels. Nano Lett. 2012, 12, 1410−1416. (30) Wu, J. Z. Density Functional Theory for Chemical Engineering: From Capillarity to Soft Materials. AIChE J. 2006, 52, 1169−1193. (31) Lian, C.; Liu, K.; Van Aken, K. L.; Gogotsi, Y.; Wesolowski, D. J.; Liu, H. L.; Jiang, D. E.; Wu, J. Z. Enhancing the Capacitive Performance of Electric Double-Layer Capacitors with Ionic Liquid Mixtures. ACS Energy Lett. 2016, 1, 21−26.

(32) Hoffmann, J.; Gillespie, D. Ion Correlations in Nanofluidic Channels: Effects of Ion Size, Valence, and Concentration on Voltageand Pressure-Driven Currents. Langmuir 2013, 29, 1303−1317. (33) Kong, X.; Jiang, J.; Lu, D.; Liu, Z.; Wu, J. Molecular Theory for Electrokinetic Transport in pH-Regulated Nanochannels. J. Phys. Chem. Lett. 2014, 5, 3015−3020. (34) Lian, C.; Kong, X.; Liu, H.; Wu, J. Flow Effects on Silicate Dissolution and Ion Transport at an Aqueous Interface. Phys. Chem. Chem. Phys. 2019, 21, 6970−6975. (35) Lian, C.; Gallegos, A.; Liu, H.; Wu, J. Non-Scaling Behavior of Electroosmotic Flow in Voltage−Gated Nanopores. Phys. Chem. Chem. Phys. 2017, 19, 450−457. (36) Kirkpatrick, S. Percolation and Conduction. Rev. Mod. Phys. 1973, 45, 574−588. (37) Do, D. D.; Do, H. D. Pore Characterization of Carbonaceous Materials by Dft and Gcmc Simulations: A Review. Adsorpt. Sci. Technol. 2003, 21, 389−423. (38) Marini Bettolo Marconi, U.; Melchionna, S. J. L. Charge Transport in Nanochannels: A Molecular Theory. Langmuir 2012, 28, 13727−13740. (39) Sahimi, M. Effective-Medium Approximation for Density of States and the Spectral Dimension of Percolation Networks. J. Phys. C: Solid State Phys. 1984, 17, 3957−3966. (40) Wang, X.; Mehandzhiyski, A. Y.; Arstad, B.; Van Aken, K. L.; Mathis, T. S.; Gallegos, A.; Tian, Z. Q.; Ren, D. D.; Sheridan, E.; Grimes, B. A.; Jiang, D. E.; Wu, J. Z.; Gogotsi, Y.; Chen, D. Selective Charging Behavior in an Ionic Mixture Electrolyte-Supercapacitor System for Higher Energy and Power. J. Am. Chem. Soc. 2017, 139, 18681−18687. (41) Gogotsi, Y.; Simon, P. True Performance Metrics in Electrochemical Energy Storage. Science 2011, 334, 917−918. (42) Siria, A.; Poncharal, P.; Biance, A. L.; Fulcrand, R.; Blase, X.; Purcell, S. T.; Bocquet, L. Giant Osmotic Energy Conversion Measured in a Single Transmembrane Boron Nitride Nanotube. Nature 2013, 494, 455. (43) Cheng, C.; Jiang, G.; Garvey, C. J.; Wang, Y.; Simon, G. P.; Liu, J. Z.; Li, D. Ion Transport in Complex Layered Graphene-Based Membranes with Tuneable Interlayer Spacing. Sci. Adv. 2016, 2, No. e1501272. (44) Lian, C.; Liu, H.; Henderson, D.; Wu, J. Can Ionophobic Nanopores Enhance the Energy Storage Capacity of Electric-DoubleLayer Capacitors Containing Nonaqueous Electrolytes? J. Phys.: Condens. Matter 2016, 28, 414005. (45) Yu, Y. X.; Wu, J. Z. Structures of Hard−Sphere Fluids from a Modified Fundamental−Measure Theory. J. Chem. Phys. 2002, 117, 10156−10164. (46) Bonthuis, D. J.; Gekle, S.; Netz, R. R. Dielectric Profile of Interfacial Water and Its Effect on Double-Layer Capacitance. Phys. Rev. Lett. 2011, 107, 166102. (47) Bonthuis, D. J.; Netz, R. Beyond the Continuum: How Molecular Solvent Structure Affects Electrostatics and Hydrodynamics at Solid-Electrolyte Interfaces. J. Phys. Chem. B 2013, 117, 11397−11413. (48) Bharadwaj, R.; Santiago, J. G. Dynamics of Field-Amplified Sample Stacking. J. Fluid Mech. 2005, 543, 57−92. (49) Monroe, C. W. Ionic Mobility and Diffusivity. In Encyclopedia of Applied Electrochemistry; Kreysa, G.; Ota, K. I.; Savinell, R. F., Eds.; Springer: New York, 2014; pp 1125−1130. (50) Blum, L. Mean Spherical Model for Asymmetric Electrolytes 0.1. Method of Solution. Mol. Phys. 1975, 30, 1529−1535. (51) Sahimi, M. Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches; John Wiley & Sons: Weinheim, Germany, 2011. (52) David, C.; Gueguen, Y.; Pampoukis, G. Effective Medium Theory and Network Theory Applied to the Transport Properties of Rock. J. Geophys. Res. 1990, 95, 6993−7005. (53) Koplik, J. On the Effective Medium Theory of Random Linear Networks. J. Phys. C: Solid State Phys. 1981, 14, 4821−4837. G

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano (54) Jurgawczynski, M. Predicting Absolute and Relative Permeabilities of Carbonate Rocks Using Image Analysis and Effective Medium Theory. Ph.D. Thesis, University of London, 2007.

H

DOI: 10.1021/acsnano.9b03303 ACS Nano XXXX, XXX, XXX−XXX