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C: Physical Processes in Nanomaterials and Nanostructures
Effect of Intra-Channel Ion Transport on Transient Characteristics of Nanochannels Xingye Zhang, Xin Zhu, Zhen Cao, Chaoming Gu, and Yang Liu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04714 • Publication Date (Web): 01 Aug 2018 Downloaded from http://pubs.acs.org on August 2, 2018
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The Journal of Physical Chemistry
Effect of Intra-Channel Ion Transport on Transient Characteristics of Nanochannels
Xingye Zhang, Xin Zhu, Zhen Cao, Chaoming Gu, Yang Liu*
College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, P. R. China. *E-mail:
[email protected] ABSTRACT: This study numerically and theoretically investigates the transient responses of nanofluidic channels under step-like voltage biases. Time-dependent simulations based on Poisson-Nernst-Planck equations demonstrate that, in the Ohmic regime of electrical conductance, the transient characteristics are distinctively different from those in the limiting regime. While the latter has been previously identified as due to the dynamics of ion depletion in the reservoirs, the present study shows that the former is mainly due to the propagation of ion accumulation inside the channels. Our further analysis derives expressions for the characteristic transient times in the Ohmic regime based on the ambipolar nature of the ion transport. The derived expressions are correlated with numerical simulations for examining the dependence of the characteristic transient times on important device parameters as well as the effect of electro-osmotic flow.
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I. INTRODUCTION Nanofluidic systems based on nanochannels have been making progress toward potential applications including lab-on-a-chip sample processing and sensing1-5. Extensive studies have been carried out on nanochannels’ electrical properties and the associated transport processes, including the nonlinear current-voltage relations6,7, concentration polarization8-10, vortex formation10-12, and charge inversion13,14. In understanding the transport of ions and fluids in nanochannel devices, numerical simulations have played an important role. Particularly, the dc conductance15-20 and instability properties21,22 have been numerically and theoretically studied in depth. It is now well received that, as the voltage bias increases, the dc conductance of nanochannels is commonly identified into three regimes: Ohmic, limiting, and over-limiting.
The transient response under a step-like voltage or current bias is another important electrical property of nanochannels; it provides information both for understanding the intrinsic transport physics and for actual application designs. Experimentally, the dynamic response of a permselective membrane under step-wise voltage biases has been directly measured23. In that work, the observed current transients in the limiting and over-limiting regimes have been respectively attributed to the electro-diffusional and electro-convective development of ion concentrations in the reservoirs. Theoretical and numerical studies of the transient characteristics have also been carried out for permselective membranes, mostly based on one-dimensional models24-26. Recently, a two-dimensional, three-layer model has been analytically solved for the case of ideal permselectivity, and the voltage transient response to a step-like current bias is also obtained27.
The aforementioned transient modeling works have been mainly focused on the dynamics of ion concentration polarization, particularly the ion depletion process that occurs inside the reservoirs as the bias increases. The porous membranes have been modeled as effective permselective medium, while the transient effect due to the ACS Paragon Plus Environment
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intra-channel ion transport is usually not considered. In lab-on-a-chip systems, nevertheless, it is commonly a single nanochannel or small arrays of them instead of porous membranes that connect reservoirs. The details of ion transport inside the nanochannels can play an important role, as shown in recent steady-state modeling works14,16,20. Particularly at low biases in the Ohmic regime, the intra-channel transport is critical when it becomes the bottleneck of the overall resistance. Experimentally, it has already been observed that the current transient in the lower Ohmic regime is drastically different from that in the limiting and over-limiting regimes: the former rises to the steady state, while the latter decays to the steady state23. However, the difference experimentally observed has not been quantitatively explained.
In this work, we provide a detailed analysis on the dynamics of ion transport inside the channels and how it affects the current transients in response to voltage steps. Two-dimensional, time-dependent numerical simulations are carried out in the framework of Poisson-Nernst-Planck equations. The difference in the transient characteristics between the Ohmic and limiting regimes is investigated. The models of characteristic transient time in the Ohmic regime are mathematically derived and, together with numerical simulations, used to evaluate the dependence on important device parameters. Furthermore, the role of convective ion transport and the effect of electroosmotic flow (EOF) are examined.
II. METHODS We use both numerical and analytical methods in this study. For the simulations, the structure of nanofluidic channels under study has an axisymmetric geometry as schematically shown in Fig. 1(a). The cylindrical channel has a negative surface charge density −σ and connects two micro-reservoirs filled with a KCl solution of
bulk ionic concentration . An electric bias, , is applied across the two reservoirs
and generates an ion current, . We adopt the conventional continuum modeling approach16,28,29, in which the coupled electrostatics and electro-diffusional transport of ACS Paragon Plus Environment
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ions are modeled by the Poisson-Nernst-Planck (PNP) equations. If the fluid transport (i.e. EOF) is also to be accounted for, the coupled PNP-Stokes (PNP-S) equations need to be solved. Here, we consider channels with hydrophilic channels surfaces, so that the no-slip boundary condition applies. It has been previously shown that, in such channels, the effect of EOF on ion fluxes is usually negligible for the case of strongly overlapped electrical double layers (EDLs)30. We mostly focus on this case, for which the numerically more tractable PNP model is appropriate. In the last part of this paper, we extend the study to the case of weakly overlapped EDLs, for which the PNP-S model is simulated to examine the effect of EOF.
Self-consistent transient simulations of the PNP equations are conducted using the finite element method (COMSOL MultiPhysics). The default geometric and physical parameters are given in the Supporting Information. Among them, the channel radius is 10 nm, and the surface charge density – is -0.01 q/nm2, where q is the
elementary charge. The bulk ionic concentration, , has a default value of 1 mM.
The corresponding Debye length Λ is ~10 nm. Under these conditions, the EDLs are strongly overlapped.
The analytical method is based on a simplified model that considers homogenized 1-D PNP equations along a uniform, long channel14,16:
&
+ − + Σ = 0,
+ −
∙ + =
∙ − =
&
∂% #$ ,
∂% #$ ,
(Eqn.1) (Eqn. 2)
(Eqn. 3)
where is the relative permittivity of water, is the vacuum permittivity, is
the elementary charge, ± represents the cation/anion diffusion coefficient, and ±
refers to the cation/anion electrophoretic mobility. The basic variables are the
potential φz, t, cation concentrations ,, $, and anion concentration ,, $, ACS Paragon Plus Environment
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which are all functions of the longitudinal position (,) and time ($). For brevity of notations, we do not explicitly write the z- and t-dependence in the equations.
Σ ≡ −2 %/ is the effective volume density of the surface charges for a cylindrical
geometry. Here, / is the Faraday’s constant. III. RESULTS AND DISCUSSION
1. Transient Responses in the Ohmic and Limiting RegimesWe firstly simulate a
relatively short channel (0 = 0.5 μm). The simulated steady-state I-V curve is plotted
in Fig. 1(b). The typical dc characteristics are observed: the I-V is almost linear at low voltage biases (Ohmic regime); as the voltage bias increases, it gradually becomes sub-linear, showing its transition to the limiting regime. For the simulated bias range, the over-limiting regime is not reached. The transient behavior of the total ionic current at the anode is then simulated in response to a step-like voltage bias from 0V to at $ = 0 5. Transients of two values, 1 V and 20 V, are
examined in Fig. 1(c), representing the Ohmic and limiting regimes, respectively. At
the initial stage, there are very fast (sub-µs) transients observed in both curves. They are the displacement current component, coming from the rapidly increasing electric field along the channel. In this study, we are mainly interested in the subsequent, relatively slower transients. They reflect the developing of ion concentrations and are therefore non-trivial. It is observed that the main characteristics of the two transients are distinctively different. The 1 V transient rises to approach its final steady state, while the 20 V transient decays to do so. Such difference is consistent with experimental observation: the measured transient shows rising behavior at low biases in the Ohmic regime and decay behavior as the bias transits to the limiting regime23. In that study, the former has been attributed to the de-passivation of the anode without further analysis.
To better understand the cause of the different transient behaviors in the two regimes, we examine the time evolution of cation concentration profile along the symmetry
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axis. For the 20 V case as shown in Fig. 2(a), there is initially no external bias, and the ion distribution is in the equilibrium condition. From the onset of 20 V at $ = 0 5, the development of ion concentration polarization becomes evident. On
one hand, the ion accumulation propagates inside the channel from the cathode end to the anode end. This propagation of ion accumulation is relatively fast, reaching its final steady state around 3 µs. On the other hand, significant ion depletion develops at the channel-reservoir junction of the anodic side at a much slower rate. As particularly shown in Fig. 2(b), the extending of the ion depletion from the junction toward the anode has a settling time longer than 200 µs. It is in the same order of magnitude of the diffusion time constant
06 % , where 0 is the reservoir length, and = 1.97 ×
10; ?
To quantify the effect of intra-channel ion transport, we consider a relatively long
nanochannel with 0 = 10 ?
The dependence of the characteristic transient times (KL ) on various device parameters
is examined. The values extracted from PNP simulations and those calculated using
Eqns. 11&12 are compared. The simulated transient responses of @ are shown for
biases ranging from 10 V to 50 V in Fig. 4(a) and for channel lengths ranging from 2 µm to 10 µm in Fig. 4(b), respectively. It is clear that, while fixing other parameters, increasing the voltage bias or the channel length results in longer KL .
The KL values extracted and plotted in Figs. 4(c)&(d). Figure 4(c) shows that KL is
inversely proportional to the electric bias. This is distinctively different from the
dynamic response of diffusion-governed ion depletion at junctions, whose time scale is set by the thermal voltage (~0.026 V) and therefore about three orders of magnitude slower and independent of the applied electric bias27,32. Figure 4(d) shows that KL
has a quadratic dependence on the channel length. A fitting of the data gives the ratio 06% a value of ~1.02 × 10_
