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Viscoelectric Effects in Nanochannel Electrokinetics Wei-Lun Hsu, Dalton J. E. Harvie, Malcolm R. Davidson, David Edwin Dunstan, Junho Hwang, and Hirofumi Daiguji J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06798 • Publication Date (Web): 29 Aug 2017 Downloaded from http://pubs.acs.org on September 8, 2017

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The Journal of Physical Chemistry

Viscoelectric Effects in Nanochannel Electrokinetics Wei-Lun Hsu,*,† Dalton J. E. Harvie,‡ Malcolm R. Davidson,‡ David E. Dunstan,‡ Junho Hwang,† and Hirofumi Daiguji† †Department of Mechanical Engineering, University of Tokyo, Tokyo 113-8656, Japan ‡Department of Chemical and Biomolecular Engineering, University of Melbourne, Victoria 3010, Australia E-mail: [email protected]

1 ACS Paragon Plus Environment


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Abstract Electrokinetic transport behavior in nanochannels is different to that in larger sized channels. Molecular dynamics (MD) simulations in nanochannels have demonstrated two poorly understood phenomena which are not observed in microchannels, being : the decreases of (i) average electroosmotic mobility at high surface charge density, and (ii) channel conductance at high salt concentrations, as the surface charge is increased. However, current electric double layer models do not capture these results. Here we provide evidence that this inconsistency primarily arises from the neglect of viscoelectric (VE) effects in conventional electrokinetic continuum models. We propose a modified continuum model that includes the concepts of a VE immobile layer and a viscoelectric layer (VEL), and that reproduces the above mentioned nanochannel-specific phenomena. It is shown that the theoretical VE coefficient, estimated based on previous MD results (2.3 × 10−16 m2 /V2 ), is close to previous experimental estimates (5-10 × 10−16 m2 /V2 ).

Introduction Electrokinetic transport of aqueous electrolytes in nanofluidic channels is of essential importance to plenty of cutting-edge technologies such as capacitive deionizaion, 1 nanofluidic batteries 2 and bio-nanosensing. 3 At these tiny scales where the surface/volume ratios are much higher than that in microchannels, the surface conduction contributes a considerable proportion to the overall channel conductance. As such, one would expect that an increase in surface charge leads to higher channel conductance. In experiments, due to the technical difficulties to directly probe the surface charge of nanochannels, a reverse procedure is often adopted that surface charge levels are estimated based on channel conductance measurements. However, the logic of this analysis relies on a critical presumption that − ionic mobilities are equivalent to their bulk values even in nanospace. Although extensively used in current literature, 4,5 this assumption lacks scientific evidence. Indeed, it is against by 2 ACS Paragon Plus Environment

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a number of studies that the mobilities of ions at charged solid/liquid interfaces can be profoundly influenced by solution conditions. 6–11 In some cases, they are close to the bulk values 12,13 while in the others the ionic mobilities are significantly suppressed. 7–11 As these studies reported ionic mobilities in the vicinity of an open surface, direct measurements of ionic mobilities in nano-confinements are not acquirable to date. Fortunately, according to the Stokes-Einstein equation that describes the dependence of ionic mobilities on viscosity down to the molecule level. It provides an alternative to evaluate ionic mobilities in nanospace via viscosity measurements. Stimulated by Zhu and Granick’s 14 pioneering work on the measurements of the shear viscosity of aqueous electrolytes in a confined nanospace, there has been growing interest in understanding transport behavior and viscous property in nano-confinements. 15–19 Evidence has shown that the solvent viscosity in nanospace can vary up to several orders of magnitude higher than its bulk value (implying a significant reduction in ionic mobilities). However, while experimental techniques can detect the average viscous behavior within a nano-confinement, the resolution of the techniques is still not sufficient to accurately probe local behavior. Consequently, despite considerable experimental evidence indicating that the shear viscosity of an aqueous solution does increase near an interface, the details of the physical mechanism(s) responsible remain opaque. At these length scales, atomic simulation can be used to investigate transport processes. This computational method has been shown to closely predict experimental measurements, while also providing details of local physical quantities. 20 For example, Freund’s Monte Carlo simulation 21 indicated that the viscosity within a nanochannel varied non-uniformly, increasing significantly near charged interfaces, while remaining similar to the macro-scale value elsewhere. Qiao and Aluru’s molecular dynamics (MD) study 22 further investigated electrokinetic transport behavior in nanochannels and found that the MD results were not reproduced by classical electrokinetic continuum theories (that neglect any viscosity and ionic mobility variations). Other atomic simulation studies also converged toward a similar conclusion that the transport behavior (of both the solvent and solute) in nanospace is



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different to that at larger scales. 23–27 Clearly a continuum model that predicts electrokinetic transport behavior in nanochannels would be a useful tool across a range of application areas, from porous flow (as used in leach mining operations) through to membranes (used for reverse osmosis of sea water). However, despite the abundant evidence from both experiments and atomic simulation indicating the classical electrokinetic theory is not sufficient to describe physical phenomena at the nanoscale, the missing mechanism has not yet been identified. This may be owing to the complication that, after over a hundred years of development since Gouy-Chapman’s diffuse model of the electric double layer (EDL), there have been a plethora of modifications to the Poisson-Boltzmann and Navier-Stokes equations that could possibly affect the transport behavior. These include steric effects of ions, 28 electrostatic correlations, 29 the Stern layer capacitance, 30 dielectric saturation, 31 viscoelectric effects, 32 etc. Among these effects, the viscoelectric (VE) effect, which has attracted relatively little attention in the past few decades, has recently reemerged as a critical topic in modern electrokinetics due to its capability of consistently accounting for a number of experimentally observed phenomena in microfluidic systems that can neither achieve by the Gouy-Chapman nor Stern capacitance theories. 32 While our previous study 32 on VE effects is grounded on experimental results at the extended-nano (101 -103 nm) or larger scales, here we lay the focus onto the sub-10 nm scale to evaluate the applicability of the VE theory to nanochannels. Specifically, we aim to clarify whether VE effects are responsible for the unique transport phenomena in nanochannels reported in previous MD simulation studies, being (i) the decrease of average electroosmotic mobility at high surface charge density, and (ii) the decrease of channel conductance at high salt concentrations, as the surface charge is increased. We modify the classical GouyChapman model by taking the VE effect and Stokes-Einstein equation into account. The predicted electrokinetic behavior from the modified continuum VE model is then compared with previous MD results. 22


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Figure 1: Schematic of the transport behavior in a positively charged slit nanospace. Within the viscoelectric layers (VELs), the viscosity is increased and ionic diffusivities and dielectric permittivity are decreased, attributed to the orientation of water molecules. The solution is effectively immobile in the vicinity of the surface (𝑖.𝑒. within the viscoelectric (VE) immobile layers) due to the high viscosity. 𝐻 denotes the channel height.

Theory The channel geometry and solution conditions considered follow the previous MD study by Qiao and Aluru 22 that a potassium chloride (KCl) aqueous solution was confined within a long slit channel of height 𝐻 (= 3.49 nm), as illustrated in Figure 1. Both surfaces possessed the same amount of positive charge uniformly distributed along each wall. An external electric field 𝐸ext (|𝐸ext |= 0.2 V/nm) was applied parallel to the surfaces, simultaneously yielding an electroosmotic flow in the opposite direction of 𝐸ext and an electric current in the same direction of 𝐸ext , whereby the local nanochannel conductance in the 𝑦-direction per unit length along the channel 𝐺L (𝑦) can be obtained as : (︂ 𝐺L (𝑦) = 𝑒

𝑒𝒟K+ 𝑣(𝑦) + |𝐸ext | 𝑘B 𝑇

)︂

(︂ 𝑛K+ (𝑦) − 𝑒

𝑣(𝑦) 𝑒𝒟Cl− − |𝐸ext | 𝑘B 𝑇

)︂ 𝑛Cl− (𝑦)

(1)

in which 𝑦 is the direction normal to the surfaces with 𝑦 = 0 on the channel centerline, 𝑒 the element charge, 𝑣(𝑦) the electroosmotic velocity, 𝑘B the Boltzmann constant, 𝑇 the temperature (= 300 K), 𝒟K+ the ionic diffusivity of K+ , 𝒟Cl− the ionic diffusivity of Cl− , 𝑛K+ (𝑦) the K+ concentration and 𝑛Cl− (𝑦) the Cl− concentration.



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We employ the electric Poisson equation and a modified Navier-Stokes equation (considering an electric body force from 𝐸ext ) to calculate 𝑛K+ (𝑦), 𝑛Cl− (𝑦) and 𝑣(𝑦) : 𝑧𝑒 𝑑2 𝜑(𝑦) 𝜌𝑒 = − (𝑛K+ (𝑦) − 𝑛Cl− (𝑦)) = − 𝑑𝑦 2 𝜖r 𝜖0 𝜖r 𝜖0

(2)

𝑑 𝑑𝑣(𝑦) (𝜂 ) + 𝜌𝑒 |𝐸ext | = 0 𝑑𝑦 𝑑𝑦

(3)

In these expressions, 𝜑(𝑦) is the electric potential, 𝜌𝑒 the space charge density, 𝜖r the relative permittivity of the solution, 𝜖0 the permittivity of vacuum, 𝜂 the viscosity and 𝑧 the ionic valence of binary electrolytes (=1 for the KCl solution). To highlight the significance of the VE effect in the nanofluidic transport behavior, we compare it with other two most discussed physical effects in modified electrokinetic theories, being the ionic steric (S) and dielectric (DE) effects. That is, we consider that the presence of ions influences the transport behavior in three ways : (i) S effects − Since the classical Boltzmann equation assumes the ions are point-like (which becomes invalid when the channel size is just several nanometers), a modified Boltzmann distribution considering S effects of ions is employed and 𝜌𝑒 is expressed as :

𝜌𝑒 =

2𝑧𝑒𝑛0 sinh( 𝑘𝑧𝑒𝜑 ) B𝑇 1 + 2𝜆 sinh2 ( 2𝑘𝑧𝑒𝜑 ) B𝑇

28

(4)

where 𝑛0 is the bulk KCl concentration and the bulk volume fraction of ions 𝜆 is given by : 𝜆 = 2𝑛0 𝑎3

(5)

where 𝑎 is the size of hydrated ions and given by 𝑎 = 6.6 ˚ A. 22 (ii) DE effects − Due to the high electric field near the surfaces, the water molecules are electrically saturated and thus the dielectric permittivity at the interface is lower than the bulk value. 33 A permittivity modification was proposed by Booth 34 based on the Onsager 35


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and Kirkwood 36 theories of polar dielectrics and re-interpreted by Hunter 37 as :

𝜖r = 𝜖r,0 (1 − 𝑏|𝐸EDL |2 )

(6)

where 𝜖r,0 and 𝐸EDL are the relative permittivity of the solution in the absence of an electric field (= 81) and local electric field within the EDL, respectively. The coefficient 𝑏 is estimated to be 4×10−18 m2 /V2 for water. 37,38 (iii) VE effects − Attributed to the interactions between the water molecules and charged surfaces, the vibration frequency of orientated water molecules in the vicinity of the walls is reduced. Consequently, the tangential (to the surfaces) motion of water molecules is largely inhibited, giving rise to a higher apparent viscosity. As shown in Freund’s MD simulations 21 that water molecules within a 10 ˚ A distance to the walls are significantly orientated, which is consistent with recent experimental detection. 39 A formula was proposed by Andrade and Dodd 40 based on experimental observations at low electric field magnitude and, later theoretically verified by Lyklema and Overbeek 41 for water :

𝜂 = 𝜂0 (1 + 𝑓 |𝐸EDL |2 )

(7)

where 𝜂0 and 𝑓 are the viscosity of the solution in the absence of an electric field (= 0.743 mPa·s) and VE coefficient, respectively. For arbitrary electric field magnitude, eq 7 can be revised based on the theory of polarization :

41

∞ ∑︁ ∆𝐸a 𝛼𝑚2 𝐸i2 (𝑓 |𝐸EDL |2 )𝑛 2 𝜂 = 𝜂0 exp( ) = 𝜂0 exp( ) = 𝜂0 exp(𝑓 |𝐸EDL | ) = 𝜂0 𝑘B 𝑇 𝑘B 𝑇 𝑛! 𝑛=0

(8)

where ∆𝐸a is the increased activation energy due to the presence of 𝐸EDL , constant 𝛼 a structural coefficient, 𝑚 the dipole moment, and 𝐸i the internal electric field magnitude which is proportional to |𝐸EDL |. If 𝑓 |𝐸EDL |2 ≪ 1, eq 8 converges to eq 7, however there is no theoretical basis for neglecting the higher order terms of eq 8 (generally for EDLs), and



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indeed, in this paper we find that their effect is significant (a typical value of surface |𝐸EDL | = 1.18 ×108 V/m, when the surface charge density equals 80 mC/m2 , making 𝑓 |𝐸EDL |2 > 1). In this study, where we employ eq 8, it is found that 𝑓 = 2.3 ×10−16 m2 /V2 achieves the closest fit to MD simulation results, which is close to previous experimental estimates of 𝑓 for water by Hunter and Leyendekkers 42 based on eq 7 (=5-10 × 10−16 m2 /V2 ). The difference may be due to the fact that 𝜂 was several times higher than 𝜂0 near the interface in Hunter and Leyendekkers’s experiments, resulting in non-negligible higher order field terms (which were neglected in their analysis) and a consequential overestimation of 𝑓 . Other studies have similarly concluded that 𝑓 should be lower than 1 × 10−15 m2 /V2 . 37,43 Based on the Stokes-Einstein equation, the ionic diffusivity 𝒟𝑖 (in which 𝑖 denotes K+ or Cl− ) concurrently decreases due to the increase of hydrodynamic drag force on ions when migrating in a viscous solution :

32

𝒟𝑖 =

𝒟𝑖,0 𝜂0 𝜂

(9)

where 𝒟𝑖,0 is the ionic diffusivity in the absence of an electric field (= 1.96 × 10−9 m2 /s for K+ and 2.03 × 10−9 m2 /s for Cl− ) and 𝜂 varies as per eq 8. In a previsous study by Kaji et al., 44 a decrease in diffusion coefficients of spherical nanoparticles within a quartz (negatively charged) nanopillar chip was experimentally observed supporting eq 9. On the boundaries, it is assumed that the nanochannel walls are non-conductive and that the surface charge is entirely balanced by the net charge within the solution, and non-slip. 22 We derive the following boundary conditions at 𝑦 = ±𝐻/2 : 𝑑𝜑 𝜎 =∓ 𝑑𝑦 𝜖r 𝜖0

(10)

𝑣=0

(11)

in which 𝜎 denotes the surface charge density. For comparison, we employ the following continuum models : a classical Gouy-Chapman (GC) model that ignores S, DE and VE effects (𝑎 = 𝑏 = 𝑓 = 0), a S model (𝑎 = 6.6 ˚ A 8 ACS Paragon Plus Environment

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and 𝑏 = 𝑓 = 0), a DE model (𝑏 = 4×10−18 m2 /V2 and 𝑎 = 𝑓 = 0), and a VE model (𝑓 = 2.3 ×10−16 m2 /V2 and 𝑎 = 𝑏 = 0). In addition, we add Qiao and Aluru’s continuum model for reference. 22 This model is basically the same as the GC model except the boundaries of eqs 10 and 11 are located on the center of the first layer of water molecules and ions adjacent to walls, respectively. Specifically, the boundaries for 𝑣 and 𝜑 are 𝑦 = (± 𝐻/2 ∓ 1.6) ˚ A and 𝑦 = (± 𝐻/2 ∓ 3.3) ˚ A, respectively. On the other hand, in Qiao and Aluru’s MD simulation, the water molecules were modeled using the Extended Simple Point Charge (SPC/E) model, and the ions were modeled as charged Lennard-Jones (LJ) atoms that details are described elsewhere 22 (all the MD results are provided by Qiao extracted from ref 22).

Results and Discussion Figure 2 shows profiles of 𝑛K+ , 𝑛Cl− , 𝜑, 𝑣 and 𝜂 along the 𝑦-direction for five continuum models and the MD results. Despite the fact that it provides close estimates to the MD results in 𝑛K+ , 𝑛Cl− and 𝜑, Qiao and Aluru’s continuum model greatly overestimates 𝑣. The effect of the boundary shifts on these results can be clarified by comparing the results of Qiao and Aluru’s model and the GC model. Because the gradient of 𝜑 with respect to 𝑦 for the GC model is smaller than that for Qiao and Aluru’s model, 𝑣 for the GC model is smaller than that for Qiao and Aluru’s model (still much larger than the MD results, nevertheless). Both S and DE effects do not greatly change these profiles but slightly increase 𝑣 especially around the channel center in comparison with the GC results. In contrast, when the VE effect is considered, 𝑣 is significantly reduced and becomes close to the MD simulation results, implying the interfacial water orientation is important to the electroosmotic behavior. Notably, the 𝑣 profile obtained from the MD simulation is greatly suppressed near the surfaces. The increased 𝜂 near the surfaces is essential to this 𝑣 profile and a higher average 𝜂 cannot reproduce it. The average electroosmotic mobility 𝜇 (over the nanochannel cross sectional area) and



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Figure 2: (a) Ion concentrations 𝑛Cl− (solid curves) and 𝑛K+ (dashed curves), (b) electric potential 𝜑 (the MD results of 𝜑 are derived based on the MD results of 𝑛Cl− in Figure 2a using the classical Boltzmann equation), (c) electroosmotic velocity 𝑣 and (d) viscosity 𝜂 profiles along the 𝑦-direction at 𝜎 = 80 mC/m2 . The error noise of the MD simulation results is comparable to the size of the circular yellow symbols used. The gray regions indicate the boundaries of Qiao and Aluru’s continuum model. 22 local electroosmotic velocity at different 𝜎 are shown in Figure 3, where 𝜇 is obtained as : 1 𝜇= 𝐻

∫︁

𝐻 2

−𝐻 2

𝑣 𝑑𝑦 |𝐸ext |

22

(12)

As seen in Figure 3a, the VE model predicted electroosmotic behavior closely agrees with the MD results over the whole range of 𝜎. Importantly, 𝜇 becomes much less sensitive to 𝜎 at high levels of 𝜎. A slight decrease of 𝜇 while increasing 𝜎 even occurs when 𝜎 > 90 mC/m2 .


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Figure 3: (a) Variation of average electroosmotic mobility 𝜇 (along with an inset plot showing variation of relative permittivity 𝜖r at the wall interfaces as a function of 𝜎 under the DE model) and (b) electroosmotic velocity 𝑣 distributions along the 𝑦-direction at different levels of surface charge 𝜎 at KCl concentration 𝑛0 = 0.6 M. Note that, despite the hydrated ion size (𝑎 = 6.6 ˚ A) being roughly 19 % of 𝐻 and 𝜖r near the walls dropping over 10% from its bulk value (as shown in the inset of Figure 3a), it is shown that once VE effects are included, S and DE effects are significantly suppressed as evidenced by the small difference between the results from the VE model and a continuum model (S/DE/VE) that simultaneously considers S, DE and VE effects. Figure 3b shows 𝑣 profiles across the nanochannel at different 𝜎 based on the VE model. In the low 𝜎 regime (𝑖.𝑒. 30 and 60 mC/m2 , under which VE effects are relatively weak), the centerline electroosmotic velocity 𝑣c increases with the increase of 𝜎 as does the averaged 𝜇. At higher 𝜎 levels (≥ 90 mC/m2 ), VE immobile layers (in which the solution becomes immobile) are gradually formed in the vicinity of the walls due to the high local viscosity 11 ACS Paragon Plus Environment


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(highlighted in gray). It is found that this insensitiveness of 𝑣 upon 𝜎 arises from the presence of the VE immobile layers. By double integrating eq 3 from the centerline (𝑑𝑣/𝑑𝑦 =0 and 𝑣 = 𝑣c ) to the location at which the solution begins becoming immobile (𝑖.𝑒. the boundary of the VE immobile layers, where 𝑣 ≈ 0), we derive a modified Smoluchowski equation : 𝜖0 (𝜖r,0 𝜑c − 𝜖r,IL 𝜑IL ) 𝑣c = |𝐸ext | 𝜂0

(13)

where 𝜑c , 𝜖r,IL and 𝜑IL are the electric potential at the centerline, relative permittivity at the boundary of the VE immobile layers and electric potential at the boundary of the VE immobile layers, respectively. This equation indicates that when the VE immobile layers exist, 𝑣c is no longer determined by 𝜑 on the channel walls (which is a function of 𝜎 by eq 10). Instead, it depends upon a 𝜎-independent parameter 𝜖r,IL 𝜑IL , rendering almost constant 𝑣c when 𝜎 ≥ 90 mC/m2 . In contrast, the local 𝑣 (in between the centerline and the VE immobile layer boundary) decreases in response to the thickening of the VE immobile layers. As a consequence, the derived 𝜇 decreases at larger 𝜎. Given that this average mobility decrease is a result of the slower electroosmotic velocities within the EDLs, it occurs under conditions where the VE immobile layer is significant at small 𝐻 and low 𝑛0 . We herein investigate the channel size effect on the average electroosmotic mobility. We consider the channel size varying from 2 to 10 nm in which the KCl concentration 𝑛0 is fixed at 0.6 M. A characteristic length that estimates the EDL thickness, √︁ being the Debye length 𝜅−1 = 𝜖𝜖2𝑒0 2𝑘𝑛B0𝑇 , at this concentration is 0.4 nm. As we can see in Figure 4a, the decrease in 𝜇 appears in the small channels (𝐻 = 2 and 3.49 nm) when 𝐻/2 is comparable to 𝜅−1 (considering both sides of the channel). At larger channels (𝐻 = 5 and 10 nm), the decrease becomes marginal which almost vanishes for 𝐻 = 10 nm. On the other hand, for the same channel the decrease is apparent at lower concentrations due to a similar mechanism. As seen in Figure 4b, we vary the KCl concentration between 0.1


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Figure 4: Variation of average electroosmotic mobility 𝜇 as a function of surface charge density 𝜎 (a) at different channel size 𝐻 and KCl concentration 𝑛0 = 0.6 M, and (b) at different 𝑛0 and 𝐻 = 3.49 nm. and 1M at fixed channel height 𝐻 =3.49 nm. The Debye lengths 𝜅−1 for these concentrations (from the low to high) are 0.98, 0.57, 0.40 and 0.31 nm, respectively. As expected, the decrease is most obvious at the lowest concentration (𝑛0 = 0.1 M) while, at 𝑛0 = 1 M, 𝜇 becomes nearly independent of 𝜎 at high levels of 𝜎 (𝑖.𝑒., ≥ 60 mC/m2 ) and hence the decrease is not significant. We investigate ion transport behavior using the nanochannel conductance per unit length along the channel 𝐺 given by : ∫︁

𝐻 2

𝐺=

𝐺L 𝑑𝑦

(14)

−𝐻 2

As seen in Figure 5a, at 𝑛0 = 1 M, 𝐺 decreases as 𝜎 increases for the VE-modified models 13 ACS Paragon Plus Environment


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Figure 5: Variation of nanochannel conductance per unit length 𝐺 as a function of surface charge density 𝜎 at KCl concentration 𝑛0 = (a) 1M and (b) 0.1M, respectively. Insets show local nanochannel conductance per unit length 𝐺L at different surface charge density 𝜎. (c) Variation of 𝐺 as a function of 𝑛0 at different 𝜎 levels. At low 𝑛0 (EDL > VEL) and low 𝜎, 𝐺, which increases with 𝜎, is dominated by charging effects. At high 𝑛0 (EDL = VEL), VE effects dominate and 𝐺 decreases at larger 𝜎. In inset figures, the VELs are highlighted by the red shadows.


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(𝑖.𝑒., the VE and S/DE/VE models). Conversely, in a lower concentration solution with 𝑛0 = 0.1 M (Figure 5b), this relationship reverses at low 𝜎, namely 𝐺 increases with the increase of 𝜎, before plateauing at 𝜎 ≥ 60 mC/m2 . These continuum-based results are consistent with previous MD simulations. 22 Note that, at high 𝑛0 (= 1 M in Figure 5a), S effects, which amplify VE effects due to ion jamming, 45 become non-negligible, although the qualitative conductive behavior in response to the 𝜎 increase remains similar. We herein define a viscoelectric layer (VEL), as illustrated in Figure 1, in which VE effects are significant. At high concentrations, the size of the region covered by the VELs is equivalent to the EDL region. On the other hand, at low concentrations, when the EDLs become overlapped, the VELs only occupy the wall adjacent region. As a result, a ‘VE free zone’ remains around the centerline. At high concentrations (𝑒.𝑔., 𝑛0 = 1 M), as seen in the inset of Figure 5a, 𝐺L around the centerline area is constant, implying that the EDLs are not overlapped. Within the EDLs (equivalent to VELs), in which 𝐺L is suppressed by 𝜎, 𝐺L decreases with the increase of 𝜎, due to higher 𝜂 and thus lower 𝒟𝑖 (based on eq 9). In consequence, 𝐺 decreases gradually with the increase of 𝜎. At low concentrations (𝑒.𝑔., 𝑛0 = 0.1 M), as seen in the inset of Figure 5b, 𝐺L is altered with 𝜎 across the whole of the nanochannel. When 𝜎 increases, 𝐺L is altered by two competing effects : (i) VE effects and (ii) charging effects. The former suppresses 𝐺L and the latter, which refers to the increase of net charge within the solution, enhances 𝐺L . At low 𝜎, charging effects dominate over the VE effects, while at high 𝜎, two factors compete and offset each other. These effects are summarized in Figure 5c showing the calculated nanochannel conductance versus KCl concentration (𝐺 − 𝑛0 ) curves for different 𝜎 based on the VE model. At high 𝑛0 , 𝐺 decreases with the increase of 𝜎 in the same way as the previous MD simulation results. 22



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Conclusion We proposed a modified continuum model of nanochannel electrokinetics that includes the concepts of a viscoelectric immobile layer and a viscoelectric layer, and successfully reproduced two poorly understood nanochannel-specific phenomena, being: (i) the decrease of average electroosmotic mobility at high surface charge density, and (ii) the decrease of channel conductance at high salt concentrations, as the surface charge is increased. In terms of the average electroosmotic mobility, at low surface charge density (< 90 mC/m2 ), the centerline electroosmotic velocity increases with the increase of surface charge density as does the average electroosmotic mobility. Whereas, at high surface charge density (≥ 90 mC/m2 ), the viscoelectric immobile layers are gradually formed in the vicinity of the walls due to the high local viscosity resulting in a decreased average electroosmotic velocity as the surface charge increases. In addition, the surface transport properties characterized by viscoelectric effects significantly screen the variations of surface electric properties due to ion steric and dielectric effects in nanochannel electroosmotic behavior. In terms of the channel conductance, the channel conductance increases as the surface charge is increased when charging effects are dominant at low salt concentrations (< 0.2 M). Whereas, at high salt concentrations (≥ 0.2 M), the channel conductance decreases with an increase in surface charge since the ionic mobilities are significantly suppressed by viscoelectric effects. These results provide a novel perspective on comprehending nanochannel electrokinetics that will shed light on various nanofluidic applications.

Acknowledgements The authors acknowledge Rui Qiao of Virginia Tech for providing us with the MD simulation results and some additional information of ref 22. This research was supported by the Grantin-Aid for Young Scientists (B), Japan Society for the Promotion of Science (17K17682).


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