Ultrahigh Water Flow Enhancement by Optimizing Nanopore

Jun 4, 2019 - Ultrahigh Water Flow Enhancement by Optimizing Nanopore ... for the world today that is facing growing water scarcity and energy demands...
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Article Cite This: Langmuir 2019, 35, 8867−8873

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Ultrahigh Water Flow Enhancement by Optimizing Nanopore Chemistry and Geometry Keliu Wu,*,†,‡ Zhangxin Chen,*,†,‡ Jing Li,†,‡ Jinze Xu,‡ Kun Wang,‡ Ran Li,‡ Shuhua Wang,‡ and Xiaohu Dong† †

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, China Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada

‡

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S Supporting Information *

ABSTRACT: The high permeability of nanoporous membranes is crucial for separation processes and energy conversions, especially for the world today that is facing growing water scarcity and energy demands. Unfortunately, further improving permeability, without sacrificing the required selectivity for specific applications, is still extremely challenging. Here, we shed light on the mechanisms of extremely high water permeability of artificial nanopores with the aquaporin-inspired pore geometry and propose a simple yet practical optimization strategy by using computational research to relate nanopore chemistry and geometry to permeability performance. We demonstrated that an ultrahigh water flow enhancement, up to 7 orders of magnitude, can be achieved by optimizing the combination of chemical and geometrical parameters of bioinspired artificial nanopores. Moreover, we addressed an existing debate over the water flow enhancement spanning over 10−1 to 105, attributed to the huge differences in chemical and geometrical properties. Our work provides a guideline to the design and optimization of nanofluidic devices with excellent performance.

1. INTRODUCTION Scarcity of freshwater is a critical underlying reason for many serious global problems that impact public health, energy and food production, environmental concerns, and industrial outputs.1 Water desalination and purification technologies, assuming sufficiently low cost yet high efficiency, are potentially promising to mitigate these issues.2 The emergence of membranes consisting of nanopores [e.g.,, carbon nanotubes (CNTs)3,4 and graphene nanopores5,6] has tremendous potential to promote the progress of these technologies; however, these artificial membranes still suffer a ubiquitous and detrimental trade-off between permeability and selectivity, failing to achieve a high performance.7 Breaking this trade-off, for example, achieving higher permeability yet still retaining the desired selectivity, has attracted great interest thanks to enabling higher energy efficiency.8,9 Aquaporins10 exhibit an excellent performance by possessing both extremely high selectivity and permeabilitywater molecules transport at astonishingly high rates, up to 3 × 109 molecules/s,7 whereas neutral solutes and protons are almost fully blocked11, which is unattainable for current artificial membranes. Thus, tremendous efforts have been made to focus on directly incorporating such biological structures into synthetic membranes.12 For example, aquaporin Z-based biomimetic membranes can potentially achieve a huge enhancement in water permeability, up to ∼800 times that of commercially available membranes.13 Unfortunately, most of © 2019 American Chemical Society

the existing biomimetic membranes suffer from three major issues: low ion rejection, difficult scale-up, and insufficient stability for industrial applications. 14 Therefore, many researchers have shifted to developing bioinspired artificial membranes,15 which are further accelerated by the growing availability of nanomaterials that are readily controlled in terms of pore size, size distribution, geometrical features, chemical functionalization, and scale-up to large areas.16 In the last decade, even though the water flow enhancement in nanopores, including CNTs, graphene nanopores, and biological water channels, has been demonstrated by solid evidence from experiments and molecular dynamics (MD) simulations,17 the underlying physical mechanisms and how to control this phenomenon are still under debate.18−20 The most critical breakthroughs going forward would lead to successful fabrications of bioinspired artificial membranes for specific applications.21,22 We need to obtain a deep and systematic understanding of optimal structures of aquaporins that yield an excellent permeability−selectivity combination. To address these issues, the following features must be investigated: (i) water transport behavior at the nanoscale, differing from that of its bulk counterpart;23,24 (ii) effects of nanopore chemistry25,26 (e.g., wall wettability and chemical functionalization) and Received: April 22, 2019 Revised: June 4, 2019 Published: June 4, 2019 8867

DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873

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Langmuir geometry27,28 (e.g., diameter, length, and shape) on the water transport; (iii) fundamental modeling at scales varying from atomistic to a continuum that handles the first two features,29,30 advances nanopore creation techniques, and guides the design of bioinspired artificial membranes for different applications. These features, by the full use of nanomaterials with favorable chemical and geometrical properties, can be expected to improve the performance of bioinspired artificial membranes, even matching and surpassing that of biological proteins.31 In this work, we first investigate the water flow behavior in a nanopore with an hourglass shape mimicking aquaporin-1,32 GlpF,33 and aquaporin Z34 (Figure 1), indicating that the total

hydrophobic CNT); (2) behavior, such as that from a single-file chain, to stacked hexagonal rings, to disordered bulk-like water with the increasing diameter of the nanopore.37 We use two key parameters, true slip (Huang/Bocquet slip38) length, ls,t, and effective viscosity, μ, to characterize the effects of the above unique physical phenomena on the flow behavior and to modify the Hagen−Poiseuille equation. A pressure drop in the central nanopore is expressed as30

Δp4 =

8μ(θ , r )L T ×Q π(r 4 + 4r 3ls,t(θ))

(1)

where θ is the contact angle for water at the nanopore walls, r is the effective hydrodynamic radius of the central nanopore, ls,t, is the Huang/Bocquet slip length as a function of θ, μ is the effective viscosity depending on θ and r, LT is the central nanopore length, Q is the water flux, and Δp4 is the pressure drop in the central nanopore. The Huang/Bocquet slip describes the tangential velocity jump of water molecules at the nanopore walls.30,39 The weak interaction between water molecules and a hydrophobic nanopore with a smooth wall can induce a huge Huang/Bocquet slip length (e.g., 30−80 nm for water confined in graphene nanopores40,41). The Huang/Bocquet slip length can be readily obtained by ls,t = 0.41/(cos θ + 1)2.30,38 Note that the equation cannot consider the effect of roughness on the Huang/Bocquet slip length, and this will give a significant deviation for some cases, such as water flow through CNTs with atomic roughness.42,43 Similarly, the viscosity of water in the vicinity of nanopore walls, named the interfacial region influenced by the wall force, also depends on the wall chemistry.30 Compared with bulk water viscosity, several distinguishing features present: it decreases due to a reduction of hydrogen bonds in a hydrophobic nanopore;44 it increases because of a strong adhesion force in a hydrophilic nanopore;45 it sharply varies along the nanopore radial direction by several orders of magnitude within ∼1 Å variation in the interfacial region;46 and it approaches the bulk water viscosity in the central part of a nanopore,47 named the bulk-like region. Thus, the effective viscosity can be obtained by a volume-weighted sum of the interfacial region viscosity and bulk-like region viscosity30,40 (Supporting Information, A). Note that in eq 1, an effective hydrodynamic radius accessible to water moleculessubtracting one molecular size near the wall (∼0.25 nm) from the nanopore radiusshould be used for accurately calculating the water flux. 2.2. Hydrodynamic Resistance. In addition to the internal resistance, water entering/exiting the central nanopore produces an extra hydrodynamic resistance induced by the water streamlines bending outward from the central nanopore. This is named the hydrodynamic entrance (or end) effects.48 Experiments and simulations have widely confirmed that hydrodynamic end effects dominate the capacity of water flow through a nanopore with a large slip and/or one with a small length as compared with its radius.49 In other words, this physical phenomenon occurs when the internal resistance is relatively small, for example, water flow in CNTs/ graphene nanopores;50 in these cases, permeability is mainly controlled by the hydrodynamic resistance at the entrance (or end). Biological nanopores, such as aquaporins with both extremely high selectivity and permeability, have an entrance (or end) with an hourglass shape,10 which may contribute to their high permeation. Therefore, to optimize the performance of bioinspired artificial water nanopores, the hydrodynamic resistance for water flowing through a nanopore with the hourglass shape needs to be modeled. We divide the entrance (or end) into three parts: the first entrance (the interface region between the reservoir and the cone), the cone, and the second entrance (the interface region between the cone and the nanopore), as shown in Figure 1. The hydrodynamic resistance in the first entrance (① in Figure 1) is expressed by a modified Sampson equation51 as

Figure 1. Schematic of the bioinspired artificial nanopore discussed in this work. The central nanopore, with the diameter Dint and the length LT, (the partial slip assumed for the water flow is shown by the red line), connects with two cones (entrance and end) with the length LE and the opening angle α. Green lines with arrows are water streamlines, where their density represents the magnitude of the water flow rate; namely, higher the density, faster the water flow (or vice versa). Symbols ①, ②, ③, and ④ represent the first entrance, the second entrance, the entrance cone, and the central nanopore, respectively.

resistance mainly consists of the internal resistance in a central nanopore and the hydrodynamic resistance at the entrance (or end). Second, we develop an approximate analytical model, within the framework of continuum hydrodynamics, to model the water flow enhancement for nanopores with various chemistries and geometries. Finally, we propose a simple, yet practical, optimization strategy to significantly enhance the permeability without sacrificing the required selectivity for specific applications. Our work enables a better understanding of structure−property−performance relationships and provides the guidelines for the design of the future bioinspired artificial membranes. In addition, we address the prolonged controversy over water flow enhancement reported in the literature,35 and demonstrate that the modified continuum hydrodynamic model, reflecting the interactions of water with nanopore walls, can describe the water flow through nanopores, which significantly saves time and cost compared with experiments and MD simulations.

2. METHODS 2.1. Internal Resistance. To model the internal resistance, namely the flow behavior, of water confined in the central nanopore (④ in Figure 1), it is necessary to consider the unique water structure and dynamics at a nanoscale,36 including (i) the pattern and average number of hydrogen bonds, which significantly deviate from those in bulk,31 caused by the additional force exerted from the nanopore wall as a function of the nanopore chemistry and size36 (e.g., water molecules form a well-ordered hydrogen bond network in a

Δp1 =

Cμ∞ r′3

×Q

(2) 48

where C is a prefactor depending on the water vorticity, mainly controlled by the particular geometrical configuration and the wall 8868

DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873

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Langmuir wettability, r′ is the local effective hydrodynamic radius of the cone close to the reservoir, calculated by r′ = r + LE tan α (where LE and α are the cone length and angle, respectively), μ∞ is the viscosity of water in the reservoir, which is equal to bulk water viscosity, and Δp1 is the pressure drop at the first entrance. Note that: (i) we do not consider the conical shape effect, and we directly simulate the pressure drop in the first entrance by treating a generalized Sampson geometry, namely the geometry for water flowing into a plain straight nanopore,51 which is approximately valid for a cone with a small angle; (ii) the bending of water streamlines in the reservoir mainly contributes to the hydrodynamic resistance in the first entrance,52 which is a viscosity-dependent resistance.48 Thus, the viscosity of water in the corresponding reservoir is adopted in eq 2; (iii) the prefactor C = 3 in the Sampson equation is obtained by the assumption of a no-slip boundary condition.51 It increases to 3.75 in the case of the perfect slip boundary condition, which means that a transition between no slip and the perfect slip in a cone induces an additional hydrodynamic resistance by 25%.27,52 Based on this simply motivated argument, a practical equation is proposed to approximately model the effect of different boundary conditions

C = 3 + (3.75 − 3) sin(θ /2)

down because the entrance and end regions overlap and interfere with each other. To conveniently validate eq 6 and to evaluate the permeability performance of the membranes consisting of nanopores shown below, we define water flow enhancement as the measured water flux divided by that calculated by the Hagen−Poiseuille equation: ΔpHP = 8μ∞LTQHP/(πrint4), where ΔpHP and QHP are the pressure drop and water flow rate in the Hagen−Poiseuille equation, respectively, and rint is a nanopore radius. The measured water flow can be expressed by combining and rearranging eqs 1−6. Thus, the water flow enhancement, ε, is derived as ÅÄ 4 4 1 1 ÅÅÅÅ Cπ rint Cπ sin α rint μ(θ , r ) ÅÅ = + 3 3 Å ε L T ÅÅÅ 4 r′ 4 μ∞ r Ç ÑÉÑ LE ÑÑ r 4 μ(θ , r(z)) ÑÑ + int × 2 dz Ñ 4 3 0 μ∞ (r(z) + 4r(z) ls,t(θ )) ÑÑÑÑÖ rint 4 μ(θ , r ) + 4 3 (r + 4r ls,t(θ)) μ∞ (7)

∫

(3)

Equation 7 accounts for the hydrodynamic entrance (or end) effects (the first item on the right-hand side) and the water flow behavior (the second item on the right-hand side) at the nanoscale, providing the important information that the flow enhancement is a chemistry- and geometry-dependent quantity. This highlights why experimental and simulation results reported35 in the literature spanned over 10−1 to 105 and, in some cases, even contradicted each other due to the huge differences in chemistry and geometry. When the nanopore length, LT, is short, the hydrodynamic entrance (or end) effects are relatively strong, resulting in a small flow enhancement. When the nanopore length increases to ∞, the flow enhancement will asymptotically approach a theoretical maximum value, merely depending on the intrinsic properties without relevance to the entrance (end) of the nanopore.54 Also, the accurate water viscosity and Huang/Bocquet slip length in the nanopore are crucial for obtaining the exact flow enhancement. Both are functions of the nanopore size and its wall wettability.30 Here, we do not consider the effect of nanopore size on the Huang/Bocquet slip length, namely, the curvature dependence of the friction coefficient of water confined in the nanopore41 (Supporting Information, C). We collected 21 experimental data sets and 164 simulation results for nanoporous materials with various chemistries and geometries (Supporting Information, Table S1). Comparison with those obtained from our model (Figure 2), indicates, overall, that the simple model we developed is reliable and can capture the main physical phenomena in the water flow through nanoporous materials. Some deviations exist due to different types of origins, including the uncertainty caused by experimental challenges in quantitative nanofluidic measurements, unrealistic initial and boundary conditions restricted by the limited calculation capacity of MD simulations, as well as some limitations of our model (Supporting Information, C).

Here, C is successfully degraded to 3 and 3.75 for the no-slip and perfect slip boundary conditions with a contact angle of 0° and 180°, respectively (more details are provided in the Supporting Information, B). Similarly, we calculate the pressure drop at the second entrance (② in Figure 1) by

Δp2 = C sin α

μ(θ , r ) ×Q r3

(4)

In this situation, we must consider the conical shape effect by a factor “sin α” due to a large angle difference of “90° − α”, significantly deviating from the generalized Sampson geometry. The conical shape effect naturally vanishes as the cone angle increases to 90° with sin(α) = 1, recovering the generalized Sampson geometry.27 In addition, the viscosity μ(θ,r) inside the nanopore can be used in eq 4 due to a small variance in the local effective hydrodynamic radius near the junction between the cone and the nanopore, where the bending of water streamlines occurs. Beyond hydrodynamic resistance at the two entrances above, the inner viscosity dissipation in the cone (③ in Figure 1) also occurs because of a finite slip at the wall.27 This supplementary resistance can be treated as an independent obstruction to the water permeability and is expressed by a theory of the approximate lubrication due to a small angle α as

Δp3 =

∫0

LE

dz

8μ(θ , r(z)) ×Q π(r(z)4 + 4r(z)3 ls,t(θ ))

(5)

where z is the axial coordinate along the nanopore, r(z) is the local effective hydrodynamic radius of the cone calculated by r + z tan α, μ(θ,r(z)) is the effective viscosity at point z in the cone, and Δp3 is the pressure drop in the cone. Up to this point, we can model the hydrodynamic resistance in the end (or entrance) with the hourglass shape by summing various contributions from eqs 2, 4, and 5 in series. We also infer that the end (or entrance) effects depend on the chemistry (contact angle) and the geometry (cone length and angle) of the cone. 2.3. Total Resistance. The total resistance, composed of the internal resistance in the central nanopore and the hydrodynamic resistance in the end (or entrance), is expressed as

Δp = 2Δp1 + 2Δp2 + 2Δp3 + Δp4

3. RESULTS AND DISCUSSION Improving the performance of a membrane consisting of nanopores, that is, enhancing permeability and selectivity, is very important to reduce the cost of applications of interest. Generally, high selectivity is mainly achieved by optimizing the nanopore critical diameter,55 in which water will pass rapidly but ions or contaminants will be blocked.36 The critical diameter is almost kept as an optimal constant for a specific application.56 A high water flux is mainly governed by the high permeability and the short length of the nanopore due to the flux being inverse of its length.7 The nanopore length should be kept at a specific value, ensuring sufficient mechanical strength to avoid damage in the practical environment. Thus, maximizing a water flux is only achieved by minimizing the internal resistance and the hydrodynamic entrance (or end)

(6)

where Δp is the total pressure drop, and the factor “2” accounts for the entrance and end of the nanopore with the hourglass shape. Here, we assume that the nanopore length and cone length are larger than 2−5 nm, ensuring that all entrance and end effects can be fully decoupled.53 Below 2−5 nm for any one of them, eq 6 will break 8869

DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873

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Langmuir

possessing extremely high selectivity−permeability combinations. We adopt the optimization strategy as follows: first, internal resistance in the central nanopore is reduced by investigating the nanopore chemical parameter, for example, the contact angle; second, minimizing the hydrodynamic resistance is obtained by searching for the optimal combination of chemical and geometrical parameters, for example, the contact angle, cone angle, and cone length, for the entrance (or end) of the nanopore. This optimization strategy is naturally common in biology. For example, the inner surface of an aquaporin contacting with water is mostly hydrophobic, generating a large Huang/Bocquet slip length and a small effective viscosity originating from the decreasing average number of hydrogen bonds of water, resulting in a small internal resistance.30 In addition, the hourglass shape geometry of an aquaporin considerably decreases the hydrodynamic resistance in the entrance (or end), achieving extremely high permeability.32 Using our model, we confirmed that optimizing nanopore chemistry and geometry significantly enhances the permeability capability, while maintaining the required selectivity in specific applications. Moreover, the geometrical optimization of an entrance (or end) always has an enhancement factor which is systematically larger than that for the corresponding plain straight nanopore, as shown in Figure 3. For any given aspect ratio, the enhancement factor increases with a decreasing diameter for cases with geometrical optimization and a contact angle larger than 120°, such as CNTs. For any given diameter, the enhancement factor always decreases with a decreasing aspect ratio for all cases with/without geometrical

Figure 2. Comparisons of the water flow enhancements obtained by the model, experimental, and simulation results published. The data in black is from experiments; the data in red and blue are from simulations. The data in red denotes the case with significant entrance (end) effects, and the solid-filled data is the case for the water flow through boron nitride nanotubes. Each data set and the corresponding error bar for the model are the mean and standard deviation induced by the uncertainty in the contact angle for the water− nanopore systems. The diagonal line is a guide for the eye.

resistance. This is optimized by the available controlling parameters, including the chemical and geometrical parameters of the system, but excluding the optimal constant nanopore diameter and length. Moreover, this can be potentially realized by mimicking an aquaporin with the hourglass shape geometry

Figure 3. Correlations of water flow enhancement with the contact angle for 2, 10, and 100 nm diameter nanopores (a−d). The aspect ratio is 1, 5, 500, 500 000, respectively. Here, Dint is the intrinsic diameter of the nanopore; the aspect ratio is defined as the central nanopore length divided by the nanopore intrinsic diameter; “without optimization” represents the plain straight nanopores; and “geometrical optimization” represents the nanopores with the optimal entrance (or end) geometry. We chose the values of the intrinsic diameter and aspect ratio above according to the typical structures of practical membranes. 8870

DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873

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Langmuir optimization and any contact angle. These results agree well with those of previous simulations27,52 and experiments,3,4 indicating that the enhancement factor is a strong function of the nanopore chemical and geometrical parameters. For water flow through a small and short nanopore with a strong hydrophobic wall (cases of Dint = 2 nm in Figure 3a,b), the internal resistance is very small and can be negligible. Thus, the hydrodynamic resistance in the entrance (or end) dominates the permeability of the nanopore and should be preferentially minimized by optimizing the entrance (or end) geometry to improve the permeability. For water flow through a large and long nanopore, regardless of wall wettability (cases of Dint = 100 nm in Figure 3c,d), however, the internal resistance becomes dominant and should be preferentially reduced by increasing the hydrophobic property using the chemical functionalization of the wall. For water flow through a medium dimension nanopore with a hydrophobic wall (cases of Dint = 10 nm in Figure 3b,c), the internal and hydrodynamic resistances both control the permeability, which can be further improved by simultaneously optimizing the chemistry and geometry of the nanopore. It is worthy to note that with geometrical optimization, the water flow enhancement factor is even up to 7 orders of magnitude for a 2 nm diameter nanopore with the aspect ratio of 500 000 and contact angle larger than 178° (Figure 3d). These factors to achieve the theoretical high value provide the direction in optimization design. Generally, the enhancement factor (hydrodynamic resistance) increases (decreases) with the increasing contact angle; moreover, the enhancement factor increases with the decreasing central nanopore diameter at a great contact angle, whereas it decreases with the decreasing central nanopore diameter at a small contact angle; it always increases with the increasing central nanopore length;30 its variation with the geometry parameters of the entrance (or end) is complex and will increase with an even distribution of the hydrodynamic resistances at the entrance (or end). To design a biomimetic nanopore, we search the optimal geometry of the entrance (or end) by varying the combination of the cone angle and length to obtain the highest permeability, noting these cases for the given diameter and length of the central nanopore (Supporting Information, A). According to the basic principle of even distribution of the hydrodynamic resistance at the entrance (or end), different optimization strategies for three types of central nanopores with a diameter of 2 nm, classified by wettability, should be adopted, as shown in Figure 4, and as follows: (i) for nanopores with the contact angle increasing from 0° to 99°, the hydrodynamic resistances in the second entrance and cone should be shifted to that in the first entrance by simultaneously reducing the cone angle and length; (ii) for nanopores with the contact angle varying from 100° to 119°, the shifted change of each hydrodynamic resistance is similar to that for the first type of nanopores, which is achieved by considerably decreasing the cone angle and keeping almost the same cone length; (iii) for nanopores with the contact angle rising from 120° to 180°, the hydrodynamic resistance should be shifted from the second entrance to the first entrance and cone by first, rapidly decreasing the cone angle and then, considerably increasing the cone length. It is worth noting that the optimization strategy will change with the diameter of the central nanopore but be independent of its length (Figure 4, Supporting Information, Figures S1−S4). The central nanopore diameter influences, in the reverse trend, the hydrodynamic resistance at the first and

Figure 4. The optimal geometrical parameters varying with the contact angle for a 2 nm diameter central nanopore with the aspect ratio of 5. Here, α* and LE* are the optimal cone angle and length, respectively. The arrows represent the increasing direction of the contact angle.

second entrances and that in the cone, whereas its length almost does not influence any resistance at the two entrances and in the cone. In addition, the optimal cone angles, falling in the range of 1°−23° for the cases with the central nanopore diameter of 2 nm (Supporting Information, Figure S5), are similar to the geometrical parameters from a variety of aquaporins.10 With an increasing contact angle, the optimal cone angle for all cases always decreases. The optimal cone length becomes larger for cases with a contact angle more than 120° (Supporting Information, Figures S5−S7), agreeing well with the previous studies from MD simulations.57 These outcomes are attributed to the hydrodynamic resistance in the central nanopore and cone, significantly reduced by the rising Huang/Bocquet slip length induced by the increasing contact angle, and the hydrodynamic resistances in the first and second entrances (or ends), decreased by only increasing the optimal cone length and decreasing the optimal cone angle, respectively. The permeability can be further enhanced by continuously optimizing the nanopore geometry (e.g., changing the cone shape to the superellipse shape) and chemistry [e.g., specifically and chemically functionalizing and charging the entrance (or end)]. The locally concentrated hydrodynamic resistance, near the two angular points at reservoir−cone and cone−nanopore connections, gradually weakens, creating a more even distribution along the entire entrance (or end). In addition, inspired by aquaporins, a nanopore with 0.3 nm diameter can possess the dipolar orientation and ordering of the water-wires, which is obtained by the inner chiral surface interactions. This nanopore will have a high permeability for water molecules, which is attributed to a driving force exerted from the dielectric polarization that is induced by the dipolar orientation of the water-wires in nanopores. Moreover, this nanopore will have simple size exclusion for all ions.58 It is also worthy to note that for a nanopore with the charged wall, as the diameter increases, fast water flux is produced, whereas the high ion rejection is still maintained by electrostatic interactions between the ions and wall charges.59 These mechanisms to improve the performance of extremely high selectivity−permeability combinations should be further studied in future. 8871

DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873

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4. CONCLUSIONS In summary, we developed a simple yet effective analytical model, within the framework of continuum hydrodynamics, to evaluate the water flow enhancement for nanopores with various chemistries and geometries. Maximizing a water flux through an artificial nanopore can be achieved by minimizing the internal resistance and the hydrodynamic resistance, and this can be realized by mimicking an aquaporin with an hourglass shape. The internal resistance is reduced by increasing the contact angle; the hydrodynamic resistance can be minimized by searching the optimal combination of contact angle and geometrical parameters, including cone angle and cone length for the entrance (or end). The water flow enhancement factor is up to 7 orders of magnitude for 2 nm diameter nanopore with the aspect ratio of 500 000, the contact angle larger than 178°, and the geometrical optimization of the entrance (or end). The optimal cone angle, ranging from 1° to 23°, is like that from a variety of aquaporins, and it decreases with the increasing contact angle. This may deduce that the hourglass shape and hydrophobic inner walls of aquaporins may originate from natural evolution. Our work not only enriches the existing knowledge on water flow behavior at the nanoscale, but also guides the design of efficient nanofluidic devices for application, such as desalination, purification, energy harvesting, nanofluidic circuits, fuel cells and drug delivery.

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ACKNOWLEDGMENTS

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REFERENCES

This work was supported by Science Foundation of China University of Petroleum, Beijing (no. 2462018YJRC033), Beijing Natural Science Foundation of China (no. 2184120), National Natural Science Foundation of China (no. 50974128), NSERC/Energi Simulation and Alberta Innovates Chairs, the Energi Simulation/Frank and Sarah Meyer Collaboration Centre, and the University of CalgaryBeijing Research Site.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.9b01179. Methods; the dependence of prefactor C on the boundary condition; challenges in model validation; summary of various parameters used to reproduce Fig. 2; optimal geometrical parameters varying with the contact angle for 2 nm, 10 nm, and 100 nm diameter central nanopore with varying aspect ratio; and relationships of the optimal cone angle and cone dimensionless length with the contact angle for 2 nm, 10 nm, and 100 nm diameter central nanopore with varying aspect ratio (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: (403) 966-3673 (K.W.). *E-mail: [email protected]. Phone: (403) 220-7825 (Z.C.). ORCID

Keliu Wu: 0000-0002-0021-5007 Kun Wang: 0000-0001-9442-5349 Shuhua Wang: 0000-0003-0915-0024 Author Contributions

K.W. and Z.C. conceived and directed the project. J.L. and J.X. performed data analysis. R.L. and S.W. provided theoretical support. K.W. and Z.C. proposed the optimization strategy. K.W., X.D., and J.X. optimized the nanopore chemistry and geometry. K.W. and Z.C. prepared the manuscript. All authors provided comments on the manuscript. Notes

The authors declare no competing financial interest. 8872

DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873

Article

Langmuir

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DOI: 10.1021/acs.langmuir.9b01179 Langmuir 2019, 35, 8867−8873