Diffusiophoresis of a Charged Sphere in a Necked Nanopore

Article pubs.acs.org/JPCC

Diffusiophoresis of a Charged Sphere in a Necked Nanopore Ting-Wen Lo,† Chien Hsu,† Kuan-Liang Liu,† Jyh-Ping Hsu,*,† and Shiojenn Tseng*,‡ †

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan

‡

ABSTRACT: In an attempt to improve the performance of single-molecule nanopore based biosensors, we study theoretically the diffusiophoresis of a particle along the axis of a charged, necked nanopore, and the mechanisms involved. Numerical simulation is conducted by varying the geometry of the nanopore and its charged status, the particle position, and the background salt concentration to examine the particle behavior under various conditions and to gather necessary information for future device design. We show that the strength of the local concentration gradient, the charge density of the nanopore, and the presence of the nanopore wall yield complicated and interesting diffusiophoretic behaviors, which are informative to potential applications.

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biosensor devices,26,27 for example, the available space for particle movement is very limited, implying that previous theoretical analyses for an unbounded medium28−34 need to be modified. Several types of boundary effects were investigated, including, for instance, a sphere in a spherical cavity,32,35 normal to a plane,36,37 and along the axis of a nanopore.26,27,32 In, general, the diffusiophoretic behavior of a particle for the case where a boundary is present is much more complicated than that for the case where it is absent. Hsu et al.,32 for example, found that a boundary is capable of influencing appreciably the degree of double-layer polarization (DLP). In addition, the accumulation of co-ions near the high concentration side of a pore results in extra electric force acting on a particle, thereby reducing its mobility.32 Recently, microchannels with a nonuniform cross section were proposed in lab-on-a-chip devices.38,39 Qian et al.,40 for instance, investigated theoretically the electrophoresis of a spherical particle along the axis of a charged converging− diverging nanotube. Xuan et al.41 investigated the wall effect on the electrophoresis of spherical polystyrene particles in a rectangular poly(dimethylsiloxane) microchannel of varying

INTRODUCTION In the past decade, solid-state nanopores have drawn much attention for their potential application in single-molecule biosensors for sensing and characterizing unlabeled entities such as proteins,1−4 DNA, 5−10 and DNA−protein10−12 complexes. In practice, the capture rate of these entities by a nanopore is usually too slow7,13 and their translocation velocity in the nanopore too fast,14−16 resulting in unsatisfactory performance. Several attempts were made to alleviate these problems so that the efficiency can be improved. Yeh et al.,17 for example, proposed using a polyelectrolyte-modified nanopore. Similar ideas were also realized experimentally through coating functional materials or molecules on synthetic nanopores.18−21 Another possibility to improve the performance of a solid-state nanopore is to adopt an auxiliary mechanism. Applying a salt concentration gradient (i.e., diffusiophoresis), for example, is one of the candidates because the direction of particle movement can be controlled by regulating parameters such as type of salt and its concentration.7,22 Attributed to recent advances in nanotechnology, diffusiophoresis is also proposed in various modern applications such as catalytic swimmers (nanomotors).23−25 In practice, diffusiophoresis is usually conducted under conditions where the presence of a boundary can be significant. In the previously mentioned application of diffusiophoresis in © 2013 American Chemical Society

Received: May 21, 2013 Revised: July 15, 2013 Published: September 4, 2013 19226

dx.doi.org/10.1021/jp404975h | J. Phys. Chem. C 2013, 117, 19226−19233

The Journal of Physical Chemistry C

Article

cross section. Nanopore-based DNA sequencing devices, comprising a nanopore and two relatively large reservoirs connected to its two ends,42,43 is another typical example where the cross section of a device is nonuniform. Note that in this case the associated fields are also nonuniform, yielding complicated phenomena.17,44−47 In addition, the particle velocity depends upon its position along the direction of its translocation. This characteristic is seldom discussed in relevant analyses. In an attempt to improve the performance of singlemolecule nanopore based biosensors, we study theoretically the diffusiophoresis of a particle along the axis of a charged, necked nanopore and the mechanisms involved. Parameters including the geometry of the nanopore and its charged status, the particle position, and the background salt concentration are examined to see their influence on the particle behavior through numerical simulation. The results gathered provide necessary information for future device design.

is the particle surface; the nanopore surface comprises Ωb, Ωd, and Ωn; Ωi and Ωo are the nanopore cross sections sufficiently far from the particle. The present problem can be described by the following set of equations:

THEORY Referring to Figure 1, let us consider the diffusiophoresis of a rigid spherical particle of radius a along the axis of a cylindrical

Here, ϕ, v, and p are the electrical potential, the fluid velocity, and the pressure, respectively. ∇ and ∇2 are the gradient operator and the Laplace operator, respectively. Cj, Jj, and Dj are the number concentration, the flux, and the diffusivity of ionic species j, respectively. η, ρ, e, kB, and T are the liquid viscosity, the space charge density, the elementary charge, Boltzmann constant, and the absolute temperature, respectively. We assume the following. (a) Both the surface of the particle and that of the nanopore are nonconductive, impermeable to ionic species, and nonslip and remain at a constant surface charge density. (b) The ionic concentration on the ends of the low (entrance) and high (exit) salt concentration side is maintained at the bulk value. (c) No other external fields are applied. Let Clow (Chigh) be the salt concentration on Ωi (Ωo) coming from ∇n0, Cavg = (Chigh + Clow)/2, and λ = Chigh/Clow, yielding Chigh = 2λCavg/(1 + λ) and Clow = 2Cavg/(1 + λ). The above assumptions lead to the following boundary conditions:

N z en ρ j j = −∑ ε ε j=1

(1)

⎛ ⎞ zje Jj = −Dj⎜∇nj + nj∇ϕ⎟ + nj v kBT ⎝ ⎠

(2)

∇·Jj = 0

(3)

∇·v = 0

(4)

−∇p + η∇2 v − ρ∇ϕ = 0

(5)

∇2 ϕ = −

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Figure 1. Diffusiophoresis of a rigid spherical particle of radius a along the axis of a cylindrical nanopore of radius b along the axis of a nanopore, which is divided into four regions: a throat region of length l, a converging region and a diverging region both having width w and height h, and a straight region; d is the distance between the particle center and the nanopore center. Ωp is the particle surface; the nanopore surface comprises Ωb, Ωd, and Ωn. Ωi and Ωo are the nanopore cross section sufficiently far from the particle.

n ·ε∇ϕ = −σp

on

Ωp

(6)

n ·ε∇ϕ = −σn

on

Ωn

(7)

n · ε∇ ϕ = − σ b

on

Ωb

(8)

n ·ε∇ϕ = −σd

on

Ωd

(9)

ϕ=0

Ωo

on

C1 = C high

C1 = C low

Ωi

Ωo

on

C2 = C high /α

nanopore of radius b subject to an applied uniform concentration gradient ∇n0. For convenience, the nanopore is divided into four regions: a throat region of length l, a converging region, and a diverging region both having width w and height h, and a straight region. The nanopore is filled with an incompressible, aqueous Newtonian fluid containing z2−z1 background salt, where z2 and z1 are the valences of cations and anions, respectively, with α = z2/z1. Let d be the distance between the particle center and the nanopore center. We adopt the cylindrical coordinates (r, θ, z) with the origin at the particle center. ∇n0 is in the z direction and has the salt as that of the background solution. Because the present problem is axial symmetric, only the (r,z) domain needs be considered. Ωp

and

(11)

Ωo

on Ωi

on

on

n ·Jj = 0

Ω b, Ωd , and Ω n

v=0

on

Ωi

Ωp

(14) (15) (16)

v = −Upez

on

p=0

Ω b and Ωd

on

(12) (13)

C2 = C low /α

on

(10)

Ω b, Ωd , and Ω n

(17) (18)

σp, σn, σb, and σd are the surface charge density on Ωp, Ωn, Ωb, and Ωd, respectively; n and ez are the unit outer normal vector 19227



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m−1, a = 10 nm, w = 20 nm, h = 20 nm, l = 30 nm, ρ = 10−3 kg m−3, η = 10−3 Pa s, F = 96500 C mol−1, T = 300 K, and D1(K+) = D2(Cl−) = 2 × 10−9 m2 s−1. Strength of Local Concentration Gradient. After a salt concentration gradient is applied, a charged particle is driven from the entrance (low concentration side) of a nanopore to its exit (high concentration side) due to the nonuniform ionic distribution surrounding it. The concentration of counterions on the high concentration side is higher than that on the low concentration side, inducing a local electric field, known as type I double-layer polarization.32,36 The diffusiophoretic velocity of the particle can be expressed as UP = μ∇ ln Ce with μ and Ce being the diffusiophoretic mobility and the bulk salt concentration, respectively, implying that the driving force is ∇ln Ce = Ce−1∇Ce. For a nanopore having uniform cross region ∇Ce is constant. In our case, both ∇Ce and Ce−1 vary with d, so does ∇ ln Ce. Therefore, we define a scaled local concentration gradient Γ = aCe−1∇Ce of strength Γ. Figure 3a shows the variation of Γ (r = 0) with d at various values of (b/a) for the case where the particle is absent. As can be seen in this figure, due to the application of ∇n0, Ce increases with increasing d, and Γ decreases accordingly, so does the particle mobility. However, as the particle enters the throat region of the nanopore, Γ suddenly increases, and the smaller the b/a the more appreciable that increase is. According to eq 2, the ionic flux comprises a convective term, a diffusion term, and a migration term. For the case where the particle is absent and the nanopore is free of charge, this equation reduces to Jj = −Dj∇Cej, that is, the ionic flux depends only on the ionic diffusivity and the its concentration gradient. Since the diffusivity is constant, the concentration gradient plays the main role, implying that the particle velocity is influenced significantly by Γ. Note that although both Cej and the cross sectional area of the nanopore A are position dependent; JA= −Dj|∇Cej|A remains constant due to mass conservation, implying that ∇Cej varies with A. This is why Γ shows a local maximum in the throat region of the nanopore. Note that Figure 3a itself is unable to verify whether the uniform diameter sections upstream and downstream from the throat are long enough to eliminate errors imposed by the boundary conditions applied at the entrance and exit because Γ(r = 0) is the scaled concentration gradient along the nanopore axis. To verify that, the axial variation of the bulk cation concentration, Ce1(r = 0), is plotted in Figure 3b of the revised manuscript. This figure reveals that the assumed uniform diameter sections upstream and downstream from the throat are long enough. Influence of σn. Figure 4 shows the variation of the particle velocity Up with d for various levels of the fixed charge density σn. This figure reveals that the behavior of Up depends largely upon σn. If σn = 0, Up decreases monotonically with increasing d and has a local maximum in the throat region. As mentioned in the discussion of Figure 3, this arises from the variation of Γ with the cross sectional area of the nanopore. If the throat region is positively charged (σn > 0), Up has a local minimum in the entrance region, has a local maximum in the throat region, and decreases monotonically with increasing d in the exit region. If the throat region is negatively charged (σn < 0), Up decreases monotonically with increasing d in the entrance region, has a local minimum in the throat region, and has a local maximum in the exit region. In the entrance and exit regions of the nanopore (i.e., |d| exceeds ca. 7), if its throat region is negatively (positively) charged, Up is larger (smaller)

and the unit vector in the z direction, respectively; Up is the particle velocity. Up can be determined from the fact that the total force acting on the particle in the z direction vanishes at steady state. That is, FE + FH = 0, with FE and FH being the z components of the electrical and the hydrodynamic forces acting on the particle, respectively. FE and FH can be evaluated by17,48 FE = FH =

∬Ω ∬Ω

(σ E·n) ·ez dΩ p p

(σ H·n) ·ez dΩ p p

(19)

(20)

σE and σH are the Maxwell stress tensor and the hydrodynamic stress tensor, respectively.

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RESULT AND DISCUSSION Code Verification. COMSOL MultiPhysics (version 4.3a, www.comsil.com) operated in a high performance cluster is adopted to solve the present problem. Mesh independence is checked to ensure that the results obtained are sufficiently accurate. Nonuniform elements are employed with a larger number of elements assigned locally as necessary. The total number of the mesh used depends upon domain size and the conditions assumed. Typically, complete mesh consisting of around 170 000 and 300 000 elements is sufficient for b/a = 6 and 10, respectively. Its applicability is first verified by solving the diffusiophoresis of an isolated, rigid, negatively charged sphere in an infinite aqueous solution, which was solved analytically for the case of low surface potential.31 Let U* = U/ U0 be the scaled particle mobility with U0 = εγ(kBT/e)2/(aη) being a reference velocity and σ*p = σpa/[ε(kBT/e)] be the scaled particle surface charge density. Figure 2 shows the

Figure 2. Variation of the scaled mobility U* with the scaled particle surface density σp at κa = 1; curve 1, analytical result of Keh and Wei;31 curve 2, present result.

variation of the scaled particle mobility U* with the scaled particle surface charge density σp. This figure reveals that the result based on the software adopted is in good agreement with that based on the other two methods. The deviation of the result of Keh and Wei31 at large σp is expected because their result is valid only for low σp. Numerical Simulation. To investigate the particle behavior under various conditions, a thorough numerical simulation is conducted. For illustration, we assume that the nanopore is filled with an aqueous KCl solution with the following values: σp = −0.0089 C m−2, σb = σd = 0 C m−2, ε = 6.93 × 10−10 F 19228



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Figure 3. Variations of the strength of the scaled local concentration gradient Γ(r = 0) (a) and Ce1(r = 0) (b) with z/a for various values of b/a at Cavg = 10−2 M and σn = 0 C m−2 for the case where the particle is absent; curve 1, b/a = 10; curve 2, b/a = 8; curve 3, b/a = 6.

and positive for σn < 0; these trends are reversed near the nanopore exit. This is because the applied concentration gradient drives both cations and anions toward the low concentration side, and the local electric field induced by the charged nanopore attracts (repels) counterions (co-ions). Therefore, the counterion concentration near the nanopore entrance (exit) is higher (lower) than that of co-ions, known as ion concentration polarization (ICP). If the sign of the particle surface charge is the same as that of the nanopore throat region, ICP has the effect of attracting the particle toward the entrance of the nanopore and reducing the particle mobility near the nanopore exit. On the other hand, if the sign of the particle surface charge is different from that of the nanopore throat region, ICP has the effect of repelling the particle toward the nanopore entrance and raising the particle mobility near its exit. In our case, the nanoparticle is negatively charged. If the throat region is also negatively charged, the particle velocity in that region is slower than that for the case where it is uncharged, as seen in Figure 4. The decrease in the particle velocity results from the enhancement of the ICP. As the particle enters the throat region, the overlapping of the double layer of the former and that of the latter makes the diffusion of coins through the nanopore from the high concentration side to the low concentration side difficult. As a result, co-ions accumulate near the high concentration side (or exit) of the nanopore, which repels the nanoparticle and slows down its velocity. On the other hand, if the throat region is positively charged, the resulting ICP tends to attract the nanoparticle to the high concentration side of the nanopore, thereby raising the nanoparticle velocity. The influence of ICP on the nanoparticle behavior can also be illustrated by the flow field. Figure 6 shows the contours of the velocity field on the plane θ = 0 at two different values of σn. In Figure 6a, where both the nanoparticle and the nanopore are negatively charged, the liquid flows near their surfaces are driven by the induced electric field coming from type I DLP; both of these flows are toward the low-concentration side. On the other hand, the flow of the liquid sufficiently far from both the nanoparticle and the nanopore is toward the high concentration side. This counterion-enriched flow, induced by the attraction (repulsion) of co-ions (couterions) on the high (low) concentration side, suggests the presence of ICP. The flow field shown in Figure 6b, where the nanoparticle surface is negatively charged and the nanopore surface positively charged, is more complicated than that seen in Figure 6a. The flow pattern in Figure 6b results

Figure 4. Variation of the particle velocity Up with d/a for various levels of σn at a = 10 nm, Cavg = 10−2 M, and b/a = 10; curve 1, σn = 0.0071 C m−2; curve 2, σn = 0.0036 C m−2; curve 3, σn = 0 C m−2; curve 4, σn = −0.0036 C m−2; curve 5, σn = −0.0071 C m−2.

than that for the case where it is uncharged. These can be explained by Figure 5, which summarizes the variation in the

Figure 5. Variation of the ionic concentration difference at r = 0 (Cn1 − Cn2) with d/a for various levels of σn at Cavg = 10−2 M and b/a = 10; curve 1, σn = 0.0071 C m−2; curve 2, σn = 0.0036 C m−2; curve 3, σn= −0.0036 C m−2; curve 4, σn = −0.0071 C m−2.

difference of the bulk ionic concentration at r = 0 (Cn1 − Cn2) with d for various levels of σn, where Cn1 and Cn2 are the concentrations of cations (K+) and anions (Cl−), respectively, for the case where the particle is absent. Figure 5 reveals that near the nanopore entrance, Cn1 − Cn2 is negative for σn > 0, 19229



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Figure 6. Contours of the velocity field on the plane θ = 0 at a = 10 nm, b/a = 6, and Cavg = 10−2 M for σn = −0.0107 C m−2 (a) and σn = 0.0107 C m−2 (b).

Figure 8 suggests that in the entrance and the exit regions (throat regions), the Up in a charged nanopore is always larger (small) than that in an uncharged nanopore. In addition, the difference between the Up of a negatively charged nanopore and that of an uncharged nanopore increases with increasing |σn|. This is because a closer boundary or a larger |σn| yields a more significant ICP. Therefore, the smaller the b/a the larger the difference between the Up of a negatively charged nanopore and that of an uncharged nanopore is, as can be seen from Figure 8a,b. Figure 9 summarizes the variation in the difference of the bulk ionic concentration at r = 0 (Cn1 − Cn2) with d at various values of b/a. This figure indicates that the smaller the b/a (more important the boundary effect), the more significant the ICP effect is. In DNA sequencing, raising the DNA capture rate by the nanpore of a device, and reducing its translocation velocity inside the nanopore are crucial challenging problems.15,18,19,48 We show that these problems can at least be partially solved by adopting our design. Due to the ICP effect, a negatively (positively) charged particle can be attracted by a negatively (positively) charged nanopore toward its opening, thereby facilitating its capture rate, and its translocation velocity in the nanopore is reduced due to its electrical repulsion with the polarized counterions near the nanopore outlet. The influence of the averaged salt concentration, Cavg, on the particle velocity, Up, is shown in Figure 10. Because the higher the Cavg the thinner the double layer, if Cavg is sufficiently high (curve 5), the interaction between the double layers of the nanoparticle and the nanopore is unimportant. In this case, the presence of the nanopore influences only hydrodynamically on the nanoparticle behavior by raising the hydrodynamic drag. In this case, the behavior of Up is similar to that when the nanopore is uncharged (e.g., curve 1 of Figure 8).

from the competition of the effect of ICP of the nanoparticle and that of the nanopore. As seen, the flow of the liquid sufficiently far from both the nanoparticle and the nanopore is toward the high concentration side, implying that the ICP effect of the nanopore is more significant than that of the nanoparticle. Note that no matter how low the charge density of the nanopore surface is, the associated ICP always drives the negatively charged particle to the high concentration side. This explains the trend of Up in the throat region of Figure 4, that Up(σn > 0) > Up(σn = 0) > Up(σn < 0). Influence of Boundary. The boundary effect can be measured by the ratio b/a: the smaller the b/a the more significant that effect is. The influence of this effect on the particle velocity Up is illustrated in Figure 7 for the case of σn =

Figure 7. Variation of the particle velocity Up with d/a for various values of b/a at a = 10 nm, σn = 0, and Cavg = 10−2 M; curve 1, b/a = 10 curve 2, b/a = 8; curve 3, b/a = 6.

0. Outside the throat region (|d| exceeds ca. 7), the smaller the b/a, the smaller is the Up. However, this is not the case in the nanopore throat region. These behaviors of Up result from the competition between the hydrodynamic force acting on the particle and the local concentration gradient it experiences. Outside the throat region, the hydrodynamic force acting on the particle increases with decreasing b/a due to a more significant boundary effect. On the other hand, the increase of Γ due to the shrinkage in the cross sectional area of the nanopore dominates in the throat region. The variations of the particle velocity Up with d at various combinations of σn and b/a are illustrated in Figure 8 for the case where σn ≤ 0. Recall that the particle is negatively charged.

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CONCLUSIONS We study theoretically the diffusiophoresis of a particle along the axis of a charged, necked nanopore and the mechanisms involved. Taking an aqueous potassium chloride solution as an example, we conclude the following. (1) Before entering the throat region of a nanopore, the closer the particle to the throat the slower its mobility is. However, as it enters that region, the closer the particle to the throat center the faster the mobility, 19230



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Figure 8. Variations of the particle velocity Up with d/a for various combinations of σn and b/a at a = 10 nm and Cavg = 10−2 M; curve 1, σn = 0 C m−2; curve 2, σn = −0.0036 C m−2; curve 3, σn = −0.0071 C m−2; curve 4, σn = −0.0107 C m−2; (a) b/a = 6; (b) b/a = 8.

In the entrance and exit regions of the nanopore, if the nanopore is negatively (positively) charged, the mobility is larger (smaller) than that if it is uncharged. These arise from the effect of ion concentration polarization (ICP). (4) For a positively (negatively) charged nanopore, the mobility in its throat region is larger (smaller) than that for an uncharged nanopore. The decrease (increase) in the mobility for the case where the sign of particle surface charge is the same as (different from) that of the nanopore arises from the electrical repulsion (attraction) between the two. (5) Outside the throat region of an uncharged nanopore, the narrower the nanopore the smaller the mobility, but this is not the case in its throat region. This is because the hydrodynamic drag coming from the nanopore wall dominates outside the throat region, but the local concentration gradient dominates in the throat region. (6) For a negatively charged particle, the magnitude of its mobility in the entrance and exit regions (throat region) of a negatively charged nanopore is always larger (smaller) than that of an uncharged nanopore, and the higher the charged density the larger the difference between the Up of a negatively charged nanopore and that of an uncharged nanopore. This is because the higher the charge density the more significant the ICP. (7) The narrower the nanopore (more important the boundary effect) the more significant the ICP effect is. (8) In nanopore based DNA sequencing, raising the DNA capture rate by the nanpore and reducing its translocation velocity inside the nanopore are crucial challenging problems. These problems can at least be partially solved by adopting our design. (9) The higher the averaged salt concentration (thinner double layer), the less significant the ICP effect is.

Figure 9. Variation of the ionic concentration difference at r = 0 (Cn1 − Cn2) with z/a for various levels of b/a at Cavg = 10−2 M and σn = −0.0036; curve 1, b/a = 10; curve 2, b/a = 8; curve 3, b/a = 6.

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Figure 10. Variations of the particle velocity Up with d/a for various values of Cavg at a = 10 nm, σp = −0.0089 C m−2, σn = −0.0107 C m−2, and b/a = 6; curve 1, Cavg = 10 mM; curve 2, Cavg = 20 mM; curve 3, Cavg = 30 mM; curve 4, Cavg = 40 mM; curve 5, Cavg = 50 mM.

AUTHOR INFORMATION

Corresponding Author

*Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail: jphsu@ ntu.edu.tw; [email protected]. Notes

and the narrower the nanopore the more appreciable this effect. These behaviors arise from the variation in the strength of the local concentration gradient. (2) The behavior of the mobility depends highly on the charge density of the nanopore. If it is uncharged, the closer the particle to the throat region the slower the mobility is, which can be explained by the variation in the local concentration gradient. (3) For a negatively charged nanopore, the mobility decreases monotonically as it approaches the throat region, shows a local minimum in its throat region, and shows a local maximum in the outlet region.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Science Council of the Republic of China.

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REFERENCES

(1) Han, A.; Creus, M.; Schurmann, G.; Linder, V.; Ward, T. R.; de Rooij, N. F.; Staufer, U. Label-Free Detection of Single Protein

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