Impact of Water-Depletion Layer on Transport in ... - ACS Publications

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Impact of Water-Depletion Layer on Transport in Hydrophobic Nanochannels Yuhui He,†,‡ Makusu Tsutsui,*,‡ Xiang Shui Miao,† and Masateru Taniguchi*,‡ †

School of Optical and Electronic Information, Huazhong University of Science and Technology, LuoYu Road, Wuhan 430074, China The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

‡

S Supporting Information *

ABSTRACT: Recent experiments showed that by fabricating nanofluidic channels with hydrophobic materials, the measured amplitudes of both electroosmotic flow (EOF) and ionic current deviated significantly from the conventional electrokinetic modeling indication. Among these unexpected observations, the complicated dependence of EOF on the surface charge concentration of the channel wall remains most confusing. In this work we give a complete and unified picture for the phenomena by outlining the competing two mechanisms in the water-depletion layer around the channel wall: the decreasing trend of fluidic flow due to the redistribution of net charges, and the increasing trend because of the reduced solution viscosity there. Our quantitative evaluation illustrates that the alternate dominating by the two mechanisms leads to the observed transport behaviors. Furthermore, by considering the decreasing of ionic mobility in the depletion layer, our calculations show quantitative agreement with the latest experiments using BN nanotube channels. where Q is the measured flow rate, uz is the channel-axial component of the fluid field u⃗, and R is the radius of the nanochannel. Based on the conventional electrokinetic modeling, the relation between σw and uz̅ is derived as a function u̅z(σw) as shown in Figure 1. Then by substituting the measured u̅z into the inverse function u̅−1 z (σ), the electrokinetic surface charge density σek is obtained. The other approach is to deduce σw from the change of electrical current caused by those counterions in the solution, σc = I−1(σ), as seen in Figure 1. Since they all denote the channel-wall surface charge density, we make some discrimination between the symbols: σw is the real value; σek is that derived from EOF means, named as the electrokinetic surface charge density; σc is from the electrical conductance method, named as the conductive surface charge density σc; σ0 is from the titration method. In this work we assume that σ0 = σw, and we focus on the difference between σek and σc. For SiO2 nanochannnels, recent experimental and theoretical studies found that the EOF and ion transport could be satisfactorily described by the Poisson−Plank−Navier equations, and the amounts of surface charge density σw on the channel wall determined by various groups showed nice agreement.14−16 The dependence of σw on the surface chemistry and the tuning by solution temperature have also been studied.17,18 Besides, the regulation of σw by the pH value of the solution was also reported, and it was attributed to the pH-regulated chemical reaction at the channel wall surface:

E

lectrokinetic transport in nanofluidics has attracted quite a bit of research interest.1−3 As shown in Figure 1a the electrical double layers (EDLs), which are induced by the channel wall surface charges, now get overlapped due to the nanoscale shape restriction. In this manner, a significant proportion of the liquid body gets charged, and under the applied voltage there would be kinetic motion of the fluid due to the electrical body force within it. This novel approach of manipulating fluid motion electrically may find potential applications in diverse fields. One of the most intensively studied topics is to utilize nanofluidic channels as platforms for biosensors.4 By electrically regulating fluidics in nanochannel/ nanopores, the motion of DNA polymers has been demonstrated to be controllable.5−7 The demonstrated strategy may address those crucial demands of biosensors operating in a nanofluidic environment, such as the nanopore-based genome sequencing proposal.8 Other promising applications of electrically regulating nanofluidic transport include but are not limited to fluidic transistors,9−11 energy conversion,12,13 and so on. The amount of surface charges on the channel wall is one crucial parameter that determines the amplitude of the induced electroosmotic flow (EOF). Thus, an accurate approach to measure the surface charge density σw is essential for evaluating the electrokinetic properties of the target nanochannels. Nowadays, apart from the basic charge titration method, there have been two strategies shown in Figure 1a to evaluate σw. The first one is to derive it from the measured EOF. Here we define the averaged fluidic velocity uz̅ as 2

uz̅ = Q /πR = 2

∫0

R

Received: June 22, 2015 Accepted: November 9, 2015

2

uzrdr /R

© XXXX American Chemical Society

(1) A

DOI: 10.1021/acs.analchem.5b03061 Anal. Chem. XXXX, XXX, XXX−XXX

Article

Analytical Chemistry

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σ0↑ ⇒ ΔI = (I |expr − I |theo )↑

(3)

THEORETICAL CHALLENGES Apart from the above experimental studies, previous investigations on fluidic transport through hydrophobic surfaces suggested the existence of an atomic-thick water-depletion layer at the interface as shown in Figure 2a and b.22−26 By discussing

Figure 1. (a) Schematic view of ion and fluidic transport generated by a voltage through the nanochannel. Net charges are induced by the surface charges on the channel wall: σw → ρe. Consequently, there emerges both electroosmotic flow (EOF) and the tuning of electrical current under the voltage. From the measured amplitudes of EOF, the electrokinetic surface charge density is derived by σek = u̅−1 z (σ), while from the obtained values of electrical current the conductive surface charge density is estimated through σc = I−1(σ). (b) and (c): The trends of σek(σ0) and σc(σek) demonstrated by experiments. Here σ0 is the surface charge density determined by titration. The dashed lines are where σek = σ0 and σc = σek. The circled numbers ① ∼ ④ denote four significant phenomena which are stated in the context.

Figure 2. (a) and (b): Schematic view of ions and water molecules in hydrophobic nanotubes. There exists a water-depletion layer close to the wall surface. Many more potassium ions (purple balls) are present in the depletion layer to screen the negative surface charges on the nanotube surface. Here, for demonstration purposes, the thickness of the layer and the sizes of ions are exaggerated a bit. (c) and (d): The radial distribution of the relative permittivity ε and the relative change of water viscosity η/η0. Both of them decrease sharply in the waterdepletion layer (R − δ < r < R).

the water transport in thin hydrophobic capillaries, de Gennes first proposed that a gaseous film may be present at the fluid/ wall interface which causes the slippage phenomenon.27 Then Poynor et al. claimed through their synchrotron X-ray data that a depletion layer was unambiguously observed when water met a hydrophobic surface.28 The existence of the depletion layer was further supported by molecular dynamics simulation from several groups,24,26,29 and the layer thickness was estimated to be about 0.5 or 0.7 nm. Within this layer, the density of water is assumed to be significantly reduced.26 By assuming a substantial drop of the solution viscosity η in the depletion layer:

MOH ⇌ MO− + H+.15,16,19 However, in nanochannels fabricated by hydrophobic materials, it was found that there existed obvious disagreements between the values of surface charge concentration measured by titration σ0 and those by the above means.20 It has been summarized that there are four types of transport behaviors for the hydrophobic nanochannel as seen in Figure 1b and c:21 (1) Given a small concentration of surface charges on the channel wall, σw, the EOF is larger than the theoretical estimation: u̅z|expr > uz̅ |theo. Therefore, the electrokinetic surface charge density σek derived from u̅−1 z (σ) is larger than the titration report σw. (2) However, once σw is larger than a critical value, the measured EOF gets smaller than the theoretical estimation: u̅z|expr < u̅z|theo. Consequently, σek < σw. (3) Given sufficiently large σw, the higher the imposed salt concentration C0, the larger the electrokinetic surface charge density σek, as derived through u̅−1 z (σ).

C0↑ ⇒ σek↑

⎧ (r ≤ R − δ ) ⎪ ηb η=⎨ ⎪ ⎩ ηd (R − δ < r < R )

(4)

calculation of nanofluidics based on the Navier equation showed nice agreement with the fluid transport experiments using carbon nanotubes.23 Recently, Bonthuis and Netz proposed a modified electrokinetic model by considering that another important parameter, the relative permittivity εr, is also decreased remarkably in the depletion layer:21

(2)

(4) As the density of surface charges on the channel wall grows larger, the conductive surface charge density σc surpasses the electrokinetic one σek in quite a rapid manner. From the formalism shown in Figure 1 we are aware that the gap between the experimentally measured electrical current and the theoretical expectation increases with the channel-wall surface charge concentration:

⎧ (r ≤ R − δ ) ⎪ εb εr = ⎨ ⎪ ⎩ εd (R − δ < r < R )

(5)

The variations of η and εr are shown in Figure 2c and d. Moreover, they assumed that the mobilities of ions within the depletion layer are enhanced by a factor of ηb/ηd.21 Here we B


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Figure 3. Radial distribution of net charge density ρe/e = (nK − nCl) in a R = 29 nm nanotube when the surface charge density (a) σw = −1 mC/m2 and (b) − 50 mC/m2. Insets plot the electrical body force near the channel wall. The corresponding distribution of fluidic velocity uz is plotted in (c) and (d). Device parameters: the thickness of water-depletion layer δ = 0.5 nm, the length of the nanotube L = 900 nm, and the applied longitudinal voltage V = 0.1 V. Here for demonstration purposes, different scales are used for the bulk and depletion regions. From this figure to Figure 5, the dash lines characterize the globally homogeneous system with εd = 80.1 and ηd = η0 (model I); the dotted lines mark the bulk-depletion system where only the change of solution permittivity is considered εd = 1 and ηd = η0 (model II); the real lines denote the bulk-depletion system where within the depletion region εd = 1 and ηd = η0/15 (model III). The dotted lines overlap exactly with the real ones in (a) and (b), and the numbers shown in the figures mark the percentages of the amount of the net charges in the bulk and depletion regions, respectively.

By resorting to the physical pictures of both permittivity ε and viscosity η decrease in the depletion layer (Figure 2c and d), we can quickly point out that the first factor is the ηdecrease. Yet whether the ε-decrease is the second factor, and why there is a transition of the dominating factors require careful discussion. On one hand, the ε-decrease in the depletion layer causes more σw-induced counterions to shift to this layer. This can be clearly found by comparing the dashed and real lines in Figure 3a or b. Such redistribution of charges then results in enhanced electrical body force fe in the depletion layer since fe ∝ ρ. Consequently, it seems that the EOF should be promoted due to a η-decrease in that layer. Yet, we note that the presence of more net charges in the depletion layer is at the expense of having less of them in the bulk region, since the total amount of charges is fixed by σw. This leads to decreased fe in the bulk region, and therefore, EOF should be attenuated there. Hence, that the overall EOF is enhanced or reduced now remains unclear. In other words, whether the ε-decrease is the decreasing factor of EOF calls for more elucidation. In light of the above discussion, further exploring of the physical mechanisms which are capable of providing a complete and unified understanding for electrokinetics in hydrophobic nanochannels is necessitated.

recall that, according to the above definitions, the motility of both ions and water is increased in the close-to-wall depletion layer. This differs fundamentally from the traditional concept of the Stern layer, where ions are mobile and water is stagnant. Based on the above image, they evaluated the associated changes of ion distribution and mobility near the hydrophobic surface. They indicated that the low permittivity within the depletion layer would shift those counterions induced by channel wall surface charges. This redistribution of excess charges would, on one hand, lead to the saturation of electroosmotic mobility at large surface charge density, and on the other hand, result in excess surface conductivity due to the low viscosity there. Their numerical results reproduced those well-established experimental observations as listed above. Therefore, the model established by Bonthuis and Netz has illustrated an elegant physical picture, and laid a framework for understanding transport in hydrophobic nanochannels. However, several profound issues concerning the physical mechanisms remain mysterious. First of all, observations ① and ② suggest that there probably exist two competing factors which dominate the EOF alternately with increasing σw. To be more specific, one mechanism causes the enhancement of EOF so that the derived σek is larger than σ0, while the other reduces the amplitude of EOF and thus the deduced σek becomes smaller than σ0. Then, for σw smaller than a critical value, the first mechanism overwhelms the second while for larger σw the situation reverses.

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METHODS In order to address the proposed question, we consider three different conditions in the water-depletion region (R − δ < r < R) as below C


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Analytical Chemistry I: εd = εb; ηd = ηb; (dash lines in Figure 3) II: εd = 1; ηd = ηb; (dotted lines in Figure 3) III: εd = 1; ηd = ηb/15;21 (real lines in Figure 3) Here we have assumed step-like variations of permittivity and viscosity at the bulk/depletion interface. Such abrupt changes are widely used21,23 and are based on the steep decrease of solvent concentration there as suggested by molecular dynamics simulation.26 The first case, I, in fact corresponds to the global homogeneous situation, while the second, II, considers only the change of permittivity in the depletion layer, and the third, III, takes into account both changes of solution permittivity and viscosity. The above strategy can also be viewed as starting from the conventional model in which the changes in the waterdepletion layer are absolutely excluded; then we decrease the solution permittivity in the target region to see what would happen; in the following we further lower the viscosity in that area to evaluate the hydrodynamic effect. The advantage of the above approach is that the electrostatic and fluidic effects now become separable: by comparing I and II, the effect of the permittivity decrease is singled out; by comparing II and III, the impact of viscosity decreasing on EOF is then demonstrated. As we are going to illustrate, it is crucial to discriminate these two effects, since they contribute conversely to EOF and the mixing of them leads to those complicated observations.

solving of the Poisson−Boltzmann equation along the channel radial direction ⎛ eV ⎞ 1 ∂ ⎛⎜ ∂V ⎞⎟ = −ρe = 2C0e sinh⎜ rεr ⎟ r ∂r ⎝ ∂r ⎠ ⎝ kBT ⎠

with boundary conditions at the two ends of the channel radius ⎧ ∂V ⎪ ⎪ ∂r ⎨ ⎪ ∂V ⎪ εr ∂r ⎩

EFFECT OF PERMITTIVITY-DECREASE IN THE DEPLETION LAYER We first explore the influence of a regional permittivitydecrease on EOF. Here we introduce one central concept for describing the system: given fixed channel-wall surface charge density, the total amount of electrical driving force over the cross-section of the nanochannel is invariant; the introduction of a low-permittivity layer around the channel wall merely shifts the distribution of the force toward that region. The first half of the above statements stems from the fact that the full amount of σw-induced net charges is determined by the charge neutrality requirement:

∫0

2π

∫0

fe rdr = 2πEz

∫0

∂uz ∂r

(6)

(7)

εr ε0kBT 2C0e 2

(12) (13)

In the above, we have assumed that the nanochannel is cylindrically shaped and sufficiently long so that axial symmetry exists and translational invariance is applied along the longitudinal direction. In our calculation, Ez is estimated by assuming the tube length L = 900 nm and the applied longitudinal voltage Vz = 0.1 V. The calculated liquid velocities along the channel radial direction, uz(r), are plotted in Figures 3c and d for different surface charge concentrations. By comparing the results of situation I and those of II, we derive the first important conclusion of this work: when considering only the factor of decreased permittivity in the water-depletion layer, EOF would decrease. Yet the physical mechanisms are more complicated than thought. As indicated by the characters in Figure 3c, f b and fd denote the electrical body force in the bulk and depletion regions, respectively. From previous analysis, we are aware that the sum of them is fixed while the decrease of permittivity in the depletion region would enhance fd on one side and reduce f b on the other side. Therefore, the velocity gain in the bulk

In the above, fe is the density of the electrical body force, which points along the direction of the applied longitudinal electrical field Ez. Ez is supposed to be translationally invariant along the channel axis. The second half of the judgment can be derived from the expression of the Debye length, which characterizes the length of the screening charge distribution: λD =

(11)

=0 r=0

uz|r = R = 0

R

ρe rdr = −2πEzRσw

(10)

and the boundary conditions at the two ends of the channel radius:

where ρe is the charge concentration within the solution caused by those excessive counterions. Hence, the overall electrical body force on the liquid remains invariant as long as σw is fixed: R

= σw r=R

1 ∂ ⎛ ∂uz ⎞ ⎜rη ⎟ = fe = Ezρe r ∂r ⎝ ∂r ⎠

R

ρe rdr = −2πRσw

=0 r=0

In the above, we have assumed the commonly used KCl as the electrolyte with imposed concentration C0. We point out that in the numerical calculation we do not linearize the right-handside of eq 9, since that treatment would introduce nonnegligible error with larger values of σw. Figures 3a and b plot the calculated charge distribution along the channel radial direction, ρe(r), in a R = 29 nm nanotube while under different amounts of channel wall surface charges σw = −1 mC/m2 and −50 mC/m2. The thickness of the depletion region is assumed to be δ = 0.5 nm, and the permittivity there is εd = 1.23,26 In the figures, the dashed, dotted, and solid lines represent situations I, II, and III, respectively. In Figures 3a and b, the dotted lines overlap with solid lines exactly. Furthermore, those numbers near the lines show the percentages of net charges in the corresponding regions according to the calculation. By comparing those values, we find that the decreasing of permittivity in the depletion layer does relocate more charges into this layer. Then, the corresponding EOF is calculated via the Navier equation:

■

2π

(9)

(8)

In the above C0 is the concentration of imposed monovalent salt. It indicates that smaller permittivity in the depletion region results in stronger capability of screening those wall surface charges, which means denser charge distribution in the screening layer. A quantitative investigation relies on the D


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we find several times enhancement of the velocity gain in the depletion region:

region decreases, as seen by comparing the magnitudes of Δub for dashed lines and for dotted lines: (Δub)II < (Δub)I

(Δud)III > (Δud)II

(14)

However, the figure demonstrates that the velocity gain in the depletion region, Δud, also decreases, (Δud)II < (Δud)I

while on the other hand the velocity gain in the bulk region remains unchanged: (Δub)III = (Δub)II

(15)

∂uz ∂r

= fi

⎧ 1 ∂ ⎛ ∂uz ⎞ ⎜r ⎟ = fe ⎪ ηb ⎪ r ∂r ⎝ ∂r ⎠ ⎪ ⎨ ∂uz =0 ⎪ ∂r r = 0 ⎪ ⎪ u| ⎩ z r = R − δ = ui

(16)

r=R−δ

Here f i is the dragging force at the bulk/depletion interface (in SI unit of N/m2), as characterized in Figure 3c. According to the requirement of force balance for the bulk region, we are aware that f i should satisfy: 2π (R − δ)fi = 2πEz

∫0

(17)

Now it becomes clear that the overall forward driving force on the water within the depletion layer remains actually invariant: R

Fd = 2π (R − δ)fi + 2πEz

∫R−δ ρe rdr = −2πEzRσw

(21)

In the above, ui is the fluid velocity at the bulk/depletion interface, which is obtained from solving eq 11, eq 13, and eq 16 for the depletion region. Remember that from case II to III, electrostatics no longer changes. Thereby in the above equation the electrical driving force fe stays unvaried, while the only difference between the two cases is ui. From Figures 3c and d we are aware that ui acts as the initial pining point while the enhancement, Δub, does not change due to the same profiles of electrical body force. In light of the above discussion, we conclude that the decrease of solution viscosity in the depletion layer would cause substantial enhancement of fluid speed in that layer; such increasing would promote the flow rate in the bulk region with a uniform amount, while there is no relative change of flow speed profile inside the bulk.

R−δ

ρe rdr

(20)

The great improvement of velocity in the depletion region is just as expected, while the invariance of Δub can be understood from the motion equation for the bulk region:

Should not (Δud)II > (Δud)I since now ( fd)II > ( fd)I? In order to answer the above nontrivial question, we have to resort to the kinetic description of the depletion layer region. The Navier equation for motion, eq 11, still applies. Nonetheless, the boundary condition at the bulk/depletion interface becomes as follows

−η

(19)

(18)

That is to say, the decreasing of permittivity in the depletion layer does not change the overall driving force on the layer; instead, it tunes only the proportion of the force at the boundary and that on the body. Therefore, the physical picture demonstrated by eq 15 is that, by putting a larger ratio of the overall driving force on the body while a smaller ratio on the interface, the efficiency of fluid boosting decreases. Here a visual display of this important conclusion can be found by comparing the dashed and solid lines in the insets of Figures 3a or b: by reducing the solution permittivity in the waterdepletion layer, the locating of electrical driving force moves much closer to the wall where the friction is strongest; consequently, the fluid motion would be weakened. Then by combining the results shown in eq 14 and eq 15, we find that both of the velocity gain in the bulk region and that in the depletion region are reduced when the permittivity of the close-to-wall region decreases. As a result, we arrive at the conclusion that the factor of permittivity-decreasing in the water-depletion region would attenuate EOF. We recall that there were similar transport phenomena. By imposing a higher concentration of KCl into SiO2 nanopore systems, a smaller amplitude of cationic EOF was generated, and accordingly, larger translocation speed of anionic DNA was reported.5,30 The increasing of salt concentration in this system caused the profile shifting of net charges closer to the wall and thus the tuning of the electrical driving force, resulting in smaller EOF.6

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DEPENDENCE ON WALL SURFACE CHARGE DENSITY Now we have demonstrated that there are two competing mechanisms manipulating EOF in hydrophobic nanochannels: one is the permittivity-decrease induced reduction of fluid speed, and the other is the viscosity-decrease caused enhancement of that speed. Whether EOF is reinforced or attenuated depends on the comparative strengths of the two factors. In experiments, one tunable parameter is the channel-wall surface charge density σw. Observations ① and ② suggest that at smaller σw the enhancing factor governs the transport properties, while at higher σw the situation reverses. Such a change is clearly illustrated in Figure 4a: the difference between the averaged flow rates calculated based on situations I and II, u̅I − u̅II, is plotted as green line, and that based on situations III and II, u̅III − uI̅ I, is plotted as a gray line. The former characterizes the amplitude of the flow rate reduction by the permittivitydecrease factor, while the latter represents the magnitude of fluid speed enhancement by the viscosity-decrease factor. There exists a turning point before which the former prevails yet after which the latter does. So what is the physical mechanism that takes responsibility for this transition? Here we point out the role of quite different proportions of charge distribution in bulk/depletion regions under different surface charge concentrations. As demonstrated in Figures 3a and b, at small surface charge concentration σw = −1 mC/m2, the ratio of charges within the bulk region decreases from 84%

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EFFECT OF VISCOSITY-REDUCTION IN THE DEPLETION LAYER Now we turn to the second stage, which is the evaluation of the viscosity-decreasing effect. By comparing situations II and III, represented by dotted lines and solid ones in Figures 3c and d, E


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The variation trend of σek(σw) shown in Figure 4b agrees with experimental reports ① and ② shown in Figure 1b.

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DEPENDENCE ON SALT CONCENTRATION Previous work suggested the relation C0↑ ⇒ u̅z↑ ⇒ σek↑ concerning the dependence of EOF on salt concentration.21 Here we indicate a more complicated relation:

The calculation results for EOF at different concentrations C0 = 1 mM and 50 mM are plotted in Figure 5a with red and blue lines, respectively. On one hand, by comparing the conventional global homogeneous model (dash lines) and the bulk/ depletion one (real lines) for either concentration, the experimental observations ① and ② shown in Figure 1b are once again reproduced. On the other hand, we find that smaller salt concentration would cause a larger amplitude of EOF, given the high density of channel-wall surface charges: C0↓ ⇒ uz̅ ↑

The physical mechanism is as follows: from discussion in the previous sections we are aware that the shifting of net charge distribution toward the water-depletion layer would lead to lower efficiency of stimulating EOF. This is exactly what the larger concentration of imposed salt would achieve because greater salt concentration leads to denser screening charges at the place close to the channel wall. The above mechanism is further verified by our numerical calculations, as shown in Figures 5b and d. The numbers in Figure 5b indicate that when the salt concentration C0 increases from 1 mM to 50 mM, the proportion of the induced charges dwelling in the depletion region would promote from 96% to 99%, while on the other hand, that within the bulk region would fall from 4% to 1%. Hence, the electrical body force within the depletion region remains slightly strengthened, while that in the bulk region is weakened substantially by 75%. From the previous discussion we are aware that not only will there be velocity loss in the bulk region due to the electrical driving force attenuating, but the velocity gain in the depletion region will also be decreased due to the attenuating of the bulk/depletion interface force. The discussed mechanisms are clearly demonstrated in Figure 5d if we compare the fluid velocity gain Δu in the same region while under different salt concentrations. Therefore, it is the relation shown in eq 23 that is valid. Then, the mechanism for the right half of eq 22 can be found in Figure 5a. Although a smaller concentration of salt fuels a larger amplitude of EOF, the gap between the bulk/depletion model and the global homogeneous one becomes even larger for the case of smaller salt concentration. That is to say, when translated into the σek(σw) language, which demonstrates the difference between experiments (simulated by bulk/depletion model) and conventional theory (the global homogeneous model), a smaller concentration would show more a obvious gap from the traditional expectation. Finally we plot in Figure 5c the calculated σek(σw) relations for various salt concentrations (the result of C0 = 10 mM has been substituted for that of C0 = 30 mM, since the former has already been shown in Figure 4b). Due to the above elucidated

Figure 4. (a) Difference between the averaged fluidic velocities u¯z calculated by different models I, II, and III as a function of the surface charge concentration σw. (b) Calculated σek as a function of the channel-wall surface charge density σw. Here the imposed salt concentration C0 = 10 mM.

to 36% when the permittivity-decrease effect is taken into account; at large concentration σw = −50 mC/m2, however, that ratio decreases from 56% to 2%. Thus, the relative change inside the bulk region is much smaller at small σw than at large 84 % 56 % σw: 36 % < 2 % . The above difference is physically reasonable since stronger surface charge density would attract a larger proportion of screening charges into the close-to-wall layer, leaving relatively huge changes within the bulk region. Yet a direct consequence is that the fluid speed reduction in the bulk region now becomes much more substantial at larger σw than at smaller ones, due to the significant weakening of the electrical driving force. This can be found by comparing (Δub)II/(Δub)I in Figure 3d and in Figure 3c. The relatively huge decreasing of the flow rate in the bulk region now could not be compensated by the speed enhancement in the depletion region. The above mechanism explains why at larger surface charge density EOF becomes reduced. Finally we evaluate the σek(σw) relation under a given salt concentration C0 = 10 mM and plot it in Figure 4b. The process is as follows: we first obtain the numerical results of u̅z(σw) based on a conventional model without the waterdepletion effect (situation I); we then fit the results with a

(

logarithmic function uz̅ = α1 ln 1 −

σw α2

(23)

); in the following, the

inverse function is obtained as σ = α2[1 − exp(uz̅ /α1)]; next, the numerical results of u̅z(σw) based on the bulk/depletion model (situation III) are obtained; the electrokinetic surface charge density σek is then calculated by putting the newly obtained u̅z into the previous σ(u̅z) function. In this way, the relation σek(σw) is deserved for the bulk/depletion model III.

F


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Figure 5. (a) Calculated averaged fluidic velocity uz̅ as a function of the nanochannel wall surface charge density σw under different salt concentrations: C0 = 1 mM (red lines) and C0 = 50 mM (blue lines). The arrows show the difference between the bulk/depletion model III and conventional homogeneous model I for either salt concentration. (b) and (d): Radial distribution of net charge density ρe/e = (nK − nCl) and fluidic velocity uz in a R = 29 nm nanochannel where σw = −50 mC/m2 under different salt concentrations: C0 = 1 mM (red lines), C0 = 10 mM (black lines), and C0 = 50 mM (blue lines). (c) Calculated σeK as a function of the channel-wall surface charge density σw. The red, dark-yellow, and blue lines denote cases with imposed salt concentration C0 = 1 mM, 30 mM, and 50 mM, respectively.

increasing amount of σw, an even larger ΔG would be obtained. Therefore, we expect that the calculated conductive surface charge concentration, σc, would be larger than σ0, and the gap between them would grow even larger for increasing σ0. The above analysis of varying trends shows consistency with the experimental observation listed as ④.

mechanisms, the variation trend showed in the figure agrees with experimental observation ③.

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DEPLETION EFFECT ON ELECTRICAL CONDUCTANCE Compared to the complexity concerning EOF, the effect of the depletion layer on the electrical conductance is much more straightforward. As demonstrated previously, the decreasing of permittivity in the depletion layer leads to moving of more net charges from the bulk region into the near-wall depletion layer. Meanwhile, these counterions obtain much higher electrophoretic mobility there than in the bulk, since solution viscosity is substantially reduced in the space R − δ < r < R. Mathematically we estimate the electrical conductance based on the following expression: G ≈ 2πR2C0μb /L + 2πR |σw|[(1 − β)μb + βμd ]/L

■

(24)

In the above, β denotes the percentage of net charges which dwell within the depletion layer. Comparing to the expression by conventional homogeneous model: G0 ≈ 2πR2C0μb /L + 2πR |σw|μb /L

(25)

the change is about ΔG = β(μd − μb )

TRANSPORT IN BORON NITRIDE NANOTUBES

It has long been found that nanotubes, owing to their atomically flat surfaces, are capable of conducting fast mass transport when utilized as nanochannels.31−33 A recent experimental work on ionic transport through boron nitride (BN) nanotubes reported a series of intriguing conducting behaviors, including large amplitudes of conductivity under small salt concentration, small streaming conductance upon mechanical pressure propelling, and giant conductance stimulated by a sanity gradient.34 The findings highlight the prospect of using BN nanotubes as osmotic power harvesting devices. Nonetheless, the experiments have posed several profound difficulties for the theoretical understanding based on conventional electrokinetic modeling. The first challenge is the unusual relation between the measured electrical conductivity κ and the nanotube radius R. From eq 25 we are aware that when the imposed salt concentration is very small (C0 ≤ 1 mM) the conductivity should be saturated and depend inversely on R. Yet in the BN nanotube experiments κ was enhanced by 11 times when the tube radius decreased less than 3-fold from 40 to 15 nm. The second challenge is the inconsistency between

(26)

since μd ≫ μb and 0 < β < 1, the electrical conductance would be greatly enhanced. Besides, β would increase with larger σw, as demonstrated in Figure 3a and b. This indicates that, with G


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Figures 3a and b the depletion region is as thin as δ = 0.5 or 0.7 nm, as suggested by previous studies.23,26 Obviously, inside such a one-atom layer those sharply arising curves derived from the continuum Poisson−Boltzmann model are inappropriate to accurately characterize the counterion concentration there. Therefore, in this section we apply eq 9 only for the bulk region (0 ≤ r < R − δ), while we introduce the 2-dimensional surface charge density σd to quantify the charge distribution within the depletion layer (R − δ ≤ r ≤ R). The variable σd is more appropriate to describe the nearly one-atom thin layer of charges than the 3-dimensional quantity ρe, which shows unrealistically large variation in that space. The electrostatic law then demands

the nanotube surface charge amount suggested by the electrical current measurement I(V) and that by the streaming conductance assessment I(Δp).34 This has been clearly demonstrated in the experimental part, where a R = 29 nm BN nanotube was used with pH = 7 and the imposed salt concentration C0 = 10 mM. On one hand, a surface charge density as large as σw = −200 mC/m2 was estimated according to the conventional electrokinetic model; on the other hand, such a significant amount of σw should have led to at least a streaming conductance Sstr = 38 pA/bar according to conventional electrokinetic modeling, while the real observation was about 3.8 pA/bar.34 The above two issues suggest there exist some undiscovered physical mechanisms beyond the conventional framework of the electrokinetic description. Theoretically, several recently developed models may be capable of providing novel insight into the above challenges. It was demonstrated through DFT calculation that as the channel radius varies the constructive or destructive interference of EDL superposition would lead to oscillating of channel capacitance.35 Therefore, the variation of effective surface charge concentration under different channel radii as stated above may be interpreted through this capacitance oscillation effect. Another potential explanation emphasizes the tuning of channel-wall surface charge amounts by surface chemistry.36 The joint DFT/Poisson calculation suggested that the effective σw would be substantially tuned by solution pH and the estimated ionic current and EOF showed nice agreement with the experiments. Here we apply the bulk/depletion model demonstrated in previous sections to this BN nanochannel system. By outlining the role of the counterions within the water-depletion layer around the BN nanotube wall, we are capable of interpreting the experimental results in a qualitative way. First, since those counterions induced by nanotube surface charges possess much higher electrophoretic mobility within the depletion layer, with fairly small amount, the induced K+ ions in the depletion layer would cause a large magnitude of electrical current I(V). That is, the unusual enhancing of conductivity with decreasing channel radius now can be attributed to the presence of the water-depletion layer. Second, those highly mobile ions, on the other hand, do not make an extra contribution to the mechanical pressure-driven streaming current I(Δp), where only the convectional motion of ions contributes. Therefore, the high mobility and small proportion of the counterions inside the water-depletion layer are responsible for the large electrical conductivity and meanwhile small streaming conductance reported in the experiments. Below we are going to demonstrate our quantitative modeling and calculation. Further modified Poisson−Boltzmann equation for electrostatics. We start from the Poisson−Boltzmann description of the net charge distribution as shown in eq 9. Here we have assumed translational invariance along the tube axial direction, since in the experiments the tube lengths were about several micrometers. The effects of ion concentration polarization at the two ends of channels are usually neglected in such high aspect-ratio nanochannels.30 Yet we recall that in the previous sections the purpose of our simulation was to qualitatively reproduce the trends of experimental observations. Here in this section, we hope to quantitatively rebuild the experimental results and hence acquire those important parameters for the electrokinetics in BN nanotubes. From this viewpoint some more delicate modifications of the proposed bulk/depletion model are necessary. As shown in

2π (R − δ)Dr (r )|R − δ + 2π (R − δ /2)σd = −2πRσw

(27)

where Dr is the tube radial component of electrical displacement field D⃗ . To facilitate the understanding of the physical picture, we define the effective surface charge density on the virtual wall of the interface as σb = −Dr (r )|R − δ

(28)

From the viewpoint of the bulk region, the net charges within the space (0 ≤ r < R − δ) are induced by these virtual surface charges with density σb. In other words, as sketched by those dash charges at the bulk/depletion interface (r = R − δ) in Figure 6a, σb has the same physical meaning to the induced charges in the bulk region as σw to those in the whole region. Therefore, the second boundary condition of eq 9 for the bulk ∂V

region becomes εb ∂r

r = (R − δ)−

= σb , where εb is the permittivity

of water in the bulk region. Figure 6a plots the calculated concentration distribution of K+ and Cl− along the channel radial direction. The parameters are set as in the experiments where a R = 29 nm and L = 900 nm BN nanotube with the imposed salt concentration C0 = 10 mM was used. The real lines with blue and red color in the range 0 ≤ r < 28.5 nm denote the cation and anion concentrations in the bulk region, while the blue real line in 28.5 ≤ nm r ≤ 29 nm shows the equivalent 3-dimensional net charge density ρd = σd/δ in the water-depletion region. The surface charge densities {σb, σd} were set as {−1, 2.76} mC/m2 with which the calculated conductivity was found to match best with the experiments (the red line shown in Figure 2b of ref 34). As a comparison, simulation results by assuming overall homogeneous water density were plotted with the dash-dot lines. We find higher location of the blue-real line than the dash-dot one in the water-depletion region, indicating that ρd is larger than that of the nondepletion model. This shows good agreement with our qualitative anticipation: within the gaseous water-depletion layer, the permittivity ε is much smaller than that of the bulk region; consequently, the screening charge density becomes substantially larger, as indicated by the smaller Debye length λD ∼ √(ε). Therefore, the existence of the waterdepletion layer contributes to the enhancing of electrical conductance not only by the greatly improved ion mobility there, but also by the larger charge density within compared to the nondepletion model. Mathematically, this indicates that quite small amount of σw is required to induce the large amplitude of I(V) reported in the BN nanotubes. We show in the Supporting Information that this is the origin of the very small streaming conductance observed under pressure driving. Electrokinetics. We then employ eqs 11, 12, and 13 to evaluate the fluidics under voltage/pressure driving. We find H


Article

Analytical Chemistry Table 1. Fitted Surface Charge Densities for the Experimentsa R

σb

σd

σw

15 22 29 40

−0.1 −0.9 −1.0 −3.0

3.96 2.84 2.76 0.698

3.99 3.69 3.71 3.66

a We fit the curves of conductivity versus imposed sanity concentration κ(C0) as shown in Figure 2b of ref 34, where R is in units of nm and σ is in units of mC/m2.

measurements. This is shown in Figure 6b, where our numerical results are plotted by the real lines while the experimental ones are plotted by the symbols (See the Ionic conductance from experiments section in the Supporting Information). The corresponding variation trends of σb and σd with the nanotube radius are illustrated in the inset of the figure. We stress that by using our model σw stays roughly invariant as −3.7 mC/m2 under various nanotube radii. For such poorly charged channel-walls, the surface chemistry induced variation of surface charge density σw becomes negligible.36 Then, while the planar density of the overall screening charges in the solution remains the same as |σw|, the distribution of these charge densities in the bulk and depletion regions, |σb| and σd, is shown to be tube radius-dependent. We attribute this behavior to the tuning of radial distribution of electrical potential V(r) in differently sized nanotubes. Assuming that σb remains the same under various R, it would result in larger density of the screening charges ρ̅e in smallerradius nanotubes according to the charge-neutrality requirement ρ̅eπR2 = −σb2πR. The larger ρ̅e has to be achieved by larger magnitude of V following the Boltzmann distribution law. The overall consequence is that for smaller-radius nanotubes the total density of the screening charges would exceed that on the tube surface (−σb + σd > |σw|), which obviously breaks the electrostatic law. Therefore, from the above reduction to absurdum we derive that |σd| has to decrease in smaller-radius nanotubes. Hence, the fitting values shown in Table 1 are physically reasonable not only by the fact that the surface charge density on the tube wall remains constant for differently sized BN nanotubes, but also by the fact that the variation trends of the screening charge density inside the solution are rational. Finally we point out that there exists significant disparity between the value of the BN surface charge density σw suggested by us and that by the authors of the original experiments.34 The two models gave fundamentally different physical pictures to the huge electrical conductance observed in BN nanotubes. We attribute the phenomena to the substantially increased mobility of ions in the gaseous layer adjacent to the BN wall, while the authors of the original experiments ascribed it to the large amount of counterions induced by large σw. Therefore, the quantities of σw estimated by us are small and remain almost invariant in BN tubes with different radii. As a comparison, those by the authors of the experiments are huge and vary significantly for BN tubes with different sizes. Physically, those surface charges on the BN tube wall were assumed to result from the chemical reaction BN3 + H2O ⇌ BN3−OH− + H+.34 The interaction effects, such as hydrogen bonds between water molecules and Coulomb repulsion between the OH− groups, may play crucial roles in determining the magnitude of σw. Quantitatively, whether σw as

Figure 6. (a) Radial distribution of K+ (blue lines) and Cl− (red lines) concentrations. The real lines are by our modified electrokinetic model while the dash-dot lines are by assuming the overall homogeneity of water density in the nanotube. The nanotube radius R = 29 nm and the imposed salt concentration C0 = 10 mM. The negative charges, indicated by dash circles, represent the equivalent surface charge density σb on the virtual wall at the bulk/depletion interface. σd is the surface charge density in the depletion layer, and σd/δ characterizes the equivalent 3-dimensional charge density there. Note that, for demonstration purposes here, different scales for the bulk and waterdepletion regions are used. (b) Simulated electrical conductance of BN nanotubes as a function of the imposed salt concentration under various radii (real lines) compared with the experimental observations (symbols). The inset plots the net charge density in the depletion layer σd and the opposite value of the effective surface charge density for the bulk region σb as functions of the nanotube radius.

that by assuming ηd = 0.02ηb23 the results matched best with the experiments. The ionic current I is estimated as the sum of electrophoretic and convectional motion: I = 2πe

∫0

R

[(μK nK + μCl nCl)Ez + (nK − nCl)uz]rdr (29)

The electrical conductivity κ and streaming conductance Sstr are then estimated through κ = (I/V)·(L/πR2) and Sstr = I/Δp. In the calculation, the mobility of ions in the depletion region is assumed to be μd = μb·ηb/ηd, where μb is the ion mobility in the bulk region.21 By using the above assumption, we no longer rely on the concept of the Stern layer. Conventionally it was regarded that ions were mobile while water molecules were fixed in the Stern layer. As indicated by Bonthuis and Netz, such a concept was especially awkward for a hydrophobic surface.21 Hence the bulk/depletion model developed here may provide a different and reasonable understanding of transport in BN nanochannels. By setting {σb, σd} as in Table 1, we find good agreement between our theoretical calculation of κ and the experimental I



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large as 1 C/m2 can be achieved by this reaction calls for future ab initio or quantum mechanical/molecule mechanics studies. On the other hand, the dependence of σw on the tube radii R assumed by the authors of the experiments (Figure 2b in ref 34) may be interpreted through the field effect.14,15

CONCLUSION In summary, we have presented a unified and complete physical picture for understanding the electrokinetic transport in hydrophobic nanochannels. By separating the effects of permittivity-reduction and viscosity-decrease in the close-towall water-depletion layer, we are able to identify the opposite roles of these two effects in manipulating the fluidic motion. By further comparing the varying trends of the two factors with device parameters, we have shown which one would dominate the fluid kinetics under various densities of channel-wall surface charges and salt concentrations. The demonstrated picture has reproduced those complicated experimental observations. Moreover, we have applied the above bulk/depletion model to the very recent experiments on electrokinetics in BN nanotubes. We have found that the counterions within the water-depletion layer near the tube wall play a crucial role for the observed large electrical conductivity and small streaming conductance. The significant increase of both mobility and concentration of the counterions in the depletion layer enables a fairly small amount of the charges to generate a large electrical current under voltage driving, while such a small concentration leads to low-level streaming conductance. Our modeling and simulation are physically convincing not only by the quantitative agreement with experimental reports, but also by the fact that the derived density of surface charges on the tube wall is invariant under various tube radii. ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.5b03061. Ionic conductance from experiments; Why the conventional Poisson−Boltzmann equation is inappropriate in BN nanotubes; Failure of the slip-length model for BN nanotubes; Why σb decreases with R; Streaming conductance in BN nanotubes (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is partially supported by KAKENHI Grant No. 26220603. A part of this work was supported by ImPACT Program of Council for Science, Technology, and Innovation (Cabinet Office, Government of Japan), “Nanotechnology Platform Project (Nanotechnology Open Facilities in Osaka University)” of Ministry of Education, Culture, Sports, Science and Technology, Japan [No: F-12-OS-0016], and by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 24681032. J


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Analytical Chemistry (36) Kong, X.; Jiang, J.; Lu, D.; Liu, Z.; Wu, J. J. Phys. Chem. Lett. 2014, 5, 3015−3020.

K


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