Feilong
Xiao‡
^{a},
Danyan
Ji‡
^{b},
Hao
Li
^{a},
Jialiang
Tang
^{a},
Yaping
Feng
^{b},
Liping
Ding
^{c},
Liuxuan
Cao
*^{a},
Ning
Li
^{a},
Lei
Jiang
^{b} and
Wei
Guo
*^{b}
^{a}College of Energy, Xiamen University, Xiamen, Fujian 361005, P. R. China. E-mail: caoliuxuan@xmu.edu.cn
^{b}CAS Key Laboratory of Bio-inspired Materials and Interfacial Science, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, P. R. China. E-mail: wguo@iccas.ac.cn
^{c}Center for Physiochemical Analysis and Measurement, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, P. R. China

Received
19th January 2018
, Accepted 17th February 2018

First published on 19th February 2018

As a type of clean energy resource, salinity gradient power between seawater and river water is important to satisfy the ever-growing energy demand on earth. In the recent years, the use of reverse electrodialysis in biomimetic nanofluidic systems has become a promising way for large-scale and high-efficiency harvesting of the salinity gradient power and surpasses the conventional polymeric ion-exchange membrane-based process. With regard to practical applications, significant efforts have been made towards the design and fabrication of high-performance and economically viable materials and devices. However, while extrapolating from single nanopores to multi-pore membrane materials, the commonly used linear amplification method causes severe deviation from the actual experimental value obtained on nanoporous membranes, particularly at a high pore density. An appropriate simulation method is therefore highly demanded and a great challenge. Herein, we present a general strategy for multi-pore nanofluidic systems by taking the influence of neighbouring nanopores into consideration. We have found that the fourth nearest-neighbor approximation is sufficiently precise for simulation in nanoporous systems. The simulation data are in good agreement with the experimental results. The simulation method provides insights for understanding the pore–pore interaction in porous nanofluidic systems and for the design of high-performance devices.

With regard to practical applications, significant efforts have been made towards extrapolating single-pore nanofluidic devices to multi-pore membrane materials.^{22} In literature, a linear amplification method is widely used to estimate the performance of multi-pore membrane materials based on the experimental measurements on single-pore devices.^{23,24} This assumption may cause severe deviation from the actual value obtained on nanoporous membranes, particularly at a high pore density. For example, we have experimentally observed that once the pore density in the polymer membrane increases to above 1 × 10^{8} pores per cm^{2}, both the diffusion current and the membrane potential obviously deviate from the trend of single-pore linear growth or the value measured on a single-pore device (Fig. 1a and b). At a high pore density, the pore–pore interaction becomes non-negligible.^{25} Thus, the single-pore-based linear approximation may eventually lead to severe overestimation in the overall performance.

From the theoretical aspects, the continuum model based on the coupled Poisson and Nernst–Planck (PNP) equations has been widely proven to be the most effective and convenient method to investigate the ion-transport phenomena through ion-selective nanopores.^{26–30} However, these existing PNP models deal with single nanopores. An appropriate simulation method for the multi-pore nanofluidic systems becomes highly demanded and challenging.

Herein, we presented a general strategy to simulate the osmotic energy conversion in multi-pore nanofluidic systems. Besides the investigated nanopore (the central pore, Fig. 2a), we considered diffusive ion transport through its n-th nearest neighbouring nanopores. The calculated region therefore includes 2n + 1 parallel nanopores (Fig. 2a). In this way, the influence of the adjacent nanopores can be involved in the model as environmental conditions for the central pore. We have found that the fourth nearest-neighbor approximation is sufficiently precise to meet the experimental results. The simulation method provides an insight for understanding the performance degradation due to pore–pore interactions and guidance for the design of high-performance nanofluidic energy devices.

In the model, the size of the reservoirs is set as a_{0} = 5 μm and b = 2 μm. 2a_{1} is the distance between the axes of adjacent pores, corresponding to the pore density in a 2D planar model (Fig. 3a). By changing the value of a_{1}, the pore density can be adjusted from 1 × 10^{6} to 1 × 10^{9} pores per cm^{2} (Table S1, ESI†). The pore length (L), the pore diameter (D), and the surface charge density (σ) were set to 1 μm, 10 nm, and −0.06 C m^{−2}, respectively. The concentration difference across the model nanopore was set as 100|1 mM. Details of the numerical simulation method are presented in ESI† and references therein.

Fig. 3 Diffusive ion transport under the 4th nearest-neighbor approximation. (a) Schematic of the model (ESI†). (b) The local ion concentration at the high-concentration end of the central, the first nearest neighboring, and the fourth nearest neighboring nanopores. (c) Diffusion current through individual nanopores was calculated separately. The applied concentration difference was 100|1 mM. a_{0} = 5 μm, b = 2 μm, and a_{1} = 300 nm, corresponding to the numeric pore density of 3 × 10^{8} pores per cm^{2}. The pore length (L) was 1 μm. The pore diameter (D) was 10 nm, and the surface charge density (σ) was −0.06 C m^{−2}. |

As shown in Fig. 2b, if the calculated region contains only one pore (the single-pore condition), the resulting diffusion current approaches 4.9 pA. With an increase in the number of considered nearest neighbors, the diffusion current gradually drops down to 3.2 pA and eventually reaches a plateau when the 4th nearest-neighbor approximation is considered (totally nine nanopores have been considered in the model). The decrement in the diffusion current through the central pore results from degradation in the effective concentration difference when the neighboring nanopores are taken into consideration (Fig. S3, ESI†). To balance the computation scale of high-order approximation, we proposed that the 4th nearest-neighbor approximation was sufficiently precise to reach convergent results in multi-pore nanofluidic systems.

Furthermore, we separately calculated the ion concentration distribution and the resulting diffusive ionic current through the nine nanopores under the 4th nearest-neighbor approximation (Fig. 3a). At the high-concentration (HC) end, the ion concentration distribution near the central pore orifice is very similar to that of its first nearest neighboring (1st NN) pores, but it is in sharp contrast to that of the fourth nearest neighboring (4th NN) pores near the boundary (Fig. 3b). This trend can also be observed in the diffusion current from individual nanopores (Fig. 3c). The inner pores, which are far away from the boundary, are less influenced by the boundary conditions, but are more sensitive to the presence of neighbouring nanopores. However, the pores close to the reservoir boundary are more affected by the boundary conditions. In porous materials, the number of inner pores (proportional to the membrane area) is far more than that of the boundary pores (proportional to the lateral size). Therefore, the influence of the neighboring pores becomes important in the simulation. In addition, the pore–pore interactions induce more severe ion concentration polarization (ICP) at the inner pores (Fig. S4, ESI†),^{37,38} leading to the degradation of the effective concentration difference and eventually resulting in a lower diffusion current.

From the governing equations (ESI†), the ion transport behavior through the nanopores is co-determined by the local ion concentration (c) and the local electric potential (Φ). Under the 4th nearest-neighbor approximation, we compared the local ion concentration and electric field distribution near the two ends of the central pore and its neighboring nanopores (Fig. 4a). At both the high- and low-concentration ends, the ion concentration and electric potential distribution near the central pore and its first nearest-neighbor (1st NN) pores are almost identical (Fig. 4b–d). However, their divergences become remarkable when the pore is located closer to the boundary such as the cases of the third and fourth nearest-neighbor pores (3rd NN and 4th NN). These evidences verify that higher orders of nearest-neighbor approximation would not further improve the accuracy of the simulation result. In this regard, we think that the 4th nearest-neighbor approximation is therefore sufficient to obtain convergent results.

Finally, we used the 4th nearest-neighbor approximation to simulate the osmotic energy conversion with high-density nanoporous membranes. The pore density is controlled by adjusting the inter-pore distance in a 2D configuration (ESI†). The pore density varies from 1 × 10^{6} to 1 × 10^{9} pores per cm^{2}, corresponding to the experimental parameters. The relationship between the pore density and the inter-pore distance is listed in Table S1 (ESI†). The length and diameter of the individual model nanopore were 1 μm and 10 nm, respectively. The surface charge density was fixed to −0.06 C m^{−2}, which was consistent with the value used in literature.^{39}

As shown in Fig. 5a, the diffusion current density grows non-linearly with pore density, particularly at a high pore density. In the low pore density region (<10^{7} pores per cm^{2}), the increment of diffusion current generally obeys the predicted trend of single-pore linear growth. However, when the pore density further increases to above 1 × 10^{7} pores per cm^{2}, the inter-pore distance shortens down to less than 3000 nm. There is no sufficient room outside the nanopores for ion migration from the bulk to the near-pore region. Thus, the pore–pore interactions become important in this crowded region, and the diffusion current density begins to deviate from the linear growth. This trend is generally consistent with the experimental data shown in Fig. 1a.

Fig. 5 Simulation results of diffusion current (a) and membrane potential (b) with respect to the pore density under the fourth nearest-neighbor approximation. (a) In the low pore density (<10^{7} pores per cm^{2}) range, the diffusion current density generally obeys the single-pore linear growth. But remarkable divergence emerges in the high pore density range. (b) Taking the influence of adjacent pores into consideration once, the degradation in membrane potential can also be found in the high pore density (>10^{7} pores per cm^{2}) range. These simulation results coincide with the experimental data shown in Fig. 1. |

Similarly, by considering the influence of adjacent nanopores, the membrane potential in porous model systems also falls down with the increasing pore density (Fig. 5b). This trend is in accordance with the experimental results shown in Fig. 1b. Considering the symmetry of multiple parallel cylindrical nanopores, we used a 2D planar model in the simulation, not the 2D model with axial symmetry.^{11} Therefore, the simulation results might be somewhat lower than the experimental value. The ICP effect in nanopores can also be influenced by the pore size, surface charge distribution, pore length etc. However, in the present study, we focused mainly on the pore-density dependence. This modelling strategy qualitatively simulates performance degradation when the nanopores become over-crowded in the membrane. Thus, it can be further used to study ion transport in multi-pore nanofluidic systems.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8qm00031j |

‡ These authors contribute equally. |

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