Ion transport and fluid flow through nanofluidic devices play important role in a broad spectrum of technological applications and biological processes,^[1] including supercapacitor,^[2] energy harvesting and conversion,^{[3, 4]} chemical separation, biosensors,^[5–7] and drug delivery.^[8] Due to the recent development of nanotechnology, nanofluidic devices have attained considerable attention. Therefore, a deep mechanistic understanding of ion transport and fluid flow through nanofluidic devices is not only helpful for us to reveal some biological activities, but also useful for the design of novel nanofluidic devices.

On one hand, nanofluidic system structures can be categorized into two groups, nanochannels and nanopores, according to the structural characteristics of nanofluidic system. More specifically, for nanochannels, the length is much larger than the radius, while for nanopores, the length and the radius are comparable to each other. Across a broad range of applications, such as molecular separation, detection, nanosensing, medicine release and seawater desalination (purification)^[9–12] for nanofluidic materials, carbon nanotubes (CNTs) have regarded as one of the most promising candidates for nanofluidic devices.^[13] Owing to the development of nanotechnology, CNTs can be easily synthesized to mimic nature ion channels, desalination, and energy harvesting. For example, the diameter and length of CNTs can be tuned, which makes the geometry of CNTs controllable. Aside from the regular structures with controllable geometry, CNTs also have smoothness and hydrophobic interior surfaces because of their graphitic walls. In addition, functional groups and charges on the surfaces of CNTs have significant influences on ion transport.^[14–16] Particularly, the exceptional mechanical, optical, electrical, and thermal characteristics make CNT to be a promising material as an ion transporter and being extensively investigated.^[17–23]

On the other hand, three major interactions, i.e., steric interactions, van der Waals interactions and electrostatic interactions, play a nontrivial role in nanofluidic systems with their varying depending on characteristic length scale. There are two treatments to nanofluidic systems concerning their theoretical investigations, depending on their characteristic length scale. When characteristic length is less than 5 nm, as the flow features in such nanofluidic systems strongly rely on the interactions among individual ions, thus the transport behaviors of ions are usually analyzed by using molecular dynamics driven by steric interactions, the dominant interaction player. While the characteristic length scales increase to range 5 nm–100 nm, electrostatic interactions have a profound effect on ion transport. Thus, in this case, continuum dynamics are often employed to elucidate the transport features. Typically, Debye length , ranging from 1 nm to 100 nm, is one of the most important characteristic length scales, where ϵ is the dielectric constant of the solvent, ϵ₀ is the permittivity of vacuum, is the Boltzmann constant, T is the absolute temperature, n_i and q_i are the bulk ion concentration and the charge of ion species i, respectively. In addition, as for van der Waals interactions, the characteristic length scales typically range from 1 nm to 50 nm.

Extensive phenomena in nanofluidics have been explored experimentally, including the current rectification, giant electroosmotic currents and cation-induced stochastic pore blocking, double-layer overlaps, ion permittivity, diffusion, surface charge effect and entropic forces.^{[1, 24–30]} In order to simulate nanofluidics in CNT, the ions are usually modelled to transport through a CNT by coupling water,^[31] figure 1 is a cross-section view of ions and water molecules passing through a CNT. To better understand the underlying mechanisms of the considerable interesting phenomena of ion transport in CNTs, the rest part of this review paper is organized as follows. In Section 2, we will focus on the introduction of Poisson–Nernst–Plank (PNP) equations which are generally used to quantitatively illustrate the transport phenomena of ions in nanochannels or nanopores. In Section 3, we will pay attention to the application of molecular dynamics (MD) simulation in CNTs. Specifically, by using MD simulation, the effects of nanotubes size, functionalized nanotubes, ionic conditions, and unusual parameters on the transport properties are discussed. In Section 4, a summary is presented.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Cross-section view of ions and water molecules (green spots) passing through a CNT.

Based on the three major interactions in nanofluidic systems (nanochannels and nanopores), when the size of system is larger than 5 nm, continuum dynamics is generally used to model the transport phenomena of ions. In this situation, what govern ion transport in nanofluidic systems are the well-known Poisson–Nernst–Plank (PNP) equations, which combine Poisson equation and Nernst–Plank equation. One of the prevailing method to predict the transport behaviors of ions inside such nanofluidic systhems is to numerically solve the PNP equations. Normally, nanofluidic systems do not include any chemical reactions between components, and therefore the flux of ions is constant in time, which can be described by a diffusion equation

Here the electrostatic potential obeys Poisson equation 2 where is the total charge density, ϵ is the dielectric constant of the medium, and ϵ₀ is the permittivity of vacuum.

The ion flux as given by Eq. (1) is governed by Nernst–Planck equation 3

In practice, the current rectification is one of the most important phenomena in nanochannels and nanopores. In order to quantitatively understand the underlying physical mechanism, numerical solutions to PNP equations have been used extensively to shed light on some related interesting phenomena. In a pioneering work reported by Cervera and coauthors,^[32] a theoretical model based on PNP equations was presented to give a quantitative description of the ionic transport across synthetic conical nanopores (Fig. 1 in Ref. [32]). Subsequently, by using spherical coordinates numerically solve the corresponding PNP equations, and eventually concluded a total electric current equation. This model was compared with corresponding experimental results with a relatively good agreement. To get more insights into the asymmetric properties of ion transport in charged conical nanopores, some tactics were also utilized to solve PNP equations, such as provide good initial values, decompose the equations, and multiply a coefficient in the nonlinear term proposed by Liu et al.^[27] The fact that theoretical results accord well with experimental observations, indicates that PNP equations are a useful tool to quantitatively understand the transport properties of ions in nanofluidic systems. In order to study the ion current rectification in bipolar nanofluidic diode, a model based on PNP equations has been proposed by Constantin and Siwy to predict the ionic concentrations and electrical-potential profiles within the bipolar nanofluidic diode, as well as the current-voltage curves.^[33] On the basis of PNP equations and Navier–Stokes equations, the characteristics of ion transport in a fluidic unipolar nanopore have been studied by Singh, and it is found that the length and location of surface charge on the walls of fluidic unipolar nanopore have a significant influence on ion transport and ion current rectification.^[34] Through the combination of PNP and Navier–Stokes equations, the ionic transport and the liquid flow in conical nanopores were modeled.^[35] Furthermore, the influence of the power-law index, surface charge of the nanopore, applied voltage, and the ratio of the tip radius to the Debye length on ionic current rectification and electroosmotic flow was investigated.^[35] Recently, concerning the non-electrostatic interactions between the solution and channel, and between solvent molecules, a generalized PNP theory was derived by Park et al.^[36]

In the past decade, CNTs have been widely investigated in theory, experiment and simulation^{[31, 37–43]} due to their exceptional mechanical, optical, electrical, and thermal characteristics,^[21–23] which makes CNT a promising material as an ion transporter in ion, water and electrolyte solution transportation.^[44–50] For example, CNTs has been increasingly used to upgrade water purification technologies due to their mechanical and chemical characteristics, high aspect ratio and small influence on the environment.^[51]

In summary, we have reviewed the numerical simulations and theoretical modeling in ionic transport and fluidic flow in nanofluidic systems. PNP model is widely utilized as a theoretical tool to model the transport behaviors. Furthermore, by employing MD simulations, it is shown that nanotubes sizes, functionalized nanotubes, ionic conditions, and unusual parameters influence the transport properties. According to the conclusions obtained from the literatures, it is found that (i) CNTs with large radius are more liable to conduct ions, while CNTs with small radius tend to hinder ions from passing through CNT. (ii) The ionic current, recognition capability and selectivity can be significantly increased for the functionalized CNTs. (iii) The ability of drive water flux for different ionic species are different, and the variation tendency of ions flux are different for different ionic concentration. (iv) Some unusual parameters such as cation-π interactions can profoundly affect the ionic transport behaviors. However, despite a number of deep insights, the understanding of ionic transport properties under nanoscale confinement is far from complete. Precise fabrication of biomimetic nanofluidic materials so as to mimic the exact biological process is still a big challenge. Developing new theoretical methods to model various ion, electrolyte and molecular transport properties are still in need, as the transport behaviors are rather sensitive to chemical and physical structural features, such as chemical groups, chemical reactions and polarization. Yet theoretical models rarely take the above factors into account. The combination of experiment, theory and simulation is crucial to uncover the fundamental mechanistic origin of ionic transport in confined one-dimensional environment. Such a new horizon of understanding can also provide useful guidelines to the design of novel nanofluidic devices.