Ionic Diffusoosmosis in Nanochannels Grafted with End-Charged

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B: Fluid Interfaces, Colloids, Polymers, Soft Matter, Surfactants, and Glassy Materials

Ionic Diffusoosmosis in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes Raja Sampath Maheedhara, Harnoor Singh Sachar, Haoyuan Jing, and Siddhartha Das J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b04827 • Publication Date (Web): 03 Jul 2018 Downloaded from http://pubs.acs.org on July 4, 2018

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The Journal of Physical Chemistry

Ionic Diffusoosmosis in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes Raja Sampath Maheedhara, Harnoor Singh Sachar, Haoyuan Jing, and Siddhartha Das∗ Department of Mechanical Engineering, University of Maryland, College Park, MD-20742, USA E-mail: [email protected] Abstract Abstract: In this paper, we develop a theory to study the imposed axial saltconcentration-gradient-driven ionic diffusioosmosis (IDO) in soft nanochannels or nanochannels grafted with end-charged polyelectrolyte (PE) brushes. Our analysis first quantifies the diffusioosmotically induced electric field, which is primarily dictated by the imposed concentration gradient (CG) with little contribution of the induced osmosis. This induced electric field triggers an electroosmotic (EOS) transport, while the net diffusioosmotic (DOS) transport results from a combination of this EOS transport and a chemiosmotic (COS) transport arising from the pressure gradient induced by the applied CG. Our results unravel that the DOS transport is massively enhanced in nanochannels grafted with PE brushes with weak grafting density stemming from the significantly enhanced EOS transport caused by the localization of the EOS body force away from the nanochannel walls. This augmentation is even stronger for cases where the COS transport aids the EOS transport. On the other hand, the DOS transport gets severely reduced in nanochannels grafted with dense PE brushes owing to the severity of the brush-induced additional drag force. We anticipate that these findings

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will help to unravel an entirely new understanding of induced electrokinetic transport in soft nanochannels.

Introduction Grafting nanoscale solid-liquid interfaces (e.g., surfaces of metallic nanoparticles or the inner walls of nanochannels) by charged polyelectrolyte (PE) brushes have been extensively used for a myriad of applications ranging from targeted drug delivery,

1–3

oil recovery 4 , water

harvesting, 5 selective sensing of ions and charged biomolecules, 6–9 gating and current rectification, 10–12 and many more. These applications depend on the functionalities rendered to these interfaces by such PE grafting and have been quantified in terms of the responses of these brushes to the substance-specific cues or some specific environmental cues. For such examples, often only the thermodynamics and the electrostatics of these brushes have been theoretically probed. 13–21 On the other hand, relatively less research has been conducted for probing the detailed fluid physics at these PE-brush-grafted nanoscale interfaces. More significantly, the existing studies invariably neglect the role of thermodynamics (e.g., coupled description of brush configuration and brush electrostatics) in the fluid mechanics description. 22–34 This becomes evident from the fact that these papers invariably consider a constant, ion-concentration-independent brush height while modelling fluid flow at the brush-grafted interfaces. Very recently Chen and Das addressed this limitation and provided possibly the first theoretical model where the PE brush thermodynamics has been accounted for in the description of the flow field in nanochannels grafted with such PE brushes. 35 In this paper, electroosmotic (EOS) transport through the nanochannels grafted with end-charged PE brushes (such interfaces designed with end-charged PE brushes have been considered previously as well 36–38 ) was investigated – the fluid mechanics model indeed accounted for the correct thermodynamics and the electrostatic behavior of these end-charged brushes. 35 More importantly, this study established that the presence of significantly tall brushes with

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weak grafting density will actually augment the EOS transport in brush-grafted nanochannels in comparison to the brush-free nanochannels. This was an extremely counter-intuitive finding given the fact that the presence of the brushes has always been believed to retard the nanofluidic transport on account of the additional brush-induced drag. 23 This highly nonintuitive phenomenon was explained by noting that the end-charged brushes localized the EDL (electric double layer) and hence the EDL-induced EOS body force at the non-grafted end of the brushes, which is significantly away from the wall (the location of the maximum drag force). As a consequence, the EOS body force of a given strength will have a much larger effect on the velocity field triggering a much augmented velocity field. For large salt concentrations, the EDLs became thin and consequently, this effect of the localization of the EOS body force away from the wall became even more prominent leading to a larger strength of the EOS transport. Of course, for the PE brushes having a much larger grafting density, the brush-induced drag force outweighed this effect associated with the localization of the EDL charge density and hence one indeed witnessed the classical result where the presence of the brushes severely weakened the flow field. In the present paper, we shall study the ionic diffusioosmosis (IDO) in such nanochannels grafted with end-charged PE brushes. Diffusioosmosis refers to the liquid transport in presence of an applied gradient in concentration in ions or nonionic solute – this gradient triggers a driving force (in form of pressure gradient and/or an electric field) that triggers the fluid motion. 39–43 There has been signifiant effort in fundamentally understanding the DO in presence of both ionic 39,43–54 and non-ionic solute gradients 42,55–62 and applying that knowledge for a host of applications such as driving liquid flows in microchannels, 57 amplifying the strength of interfacial flows, 60 designing self-powered microdevices for sensing purposes, 62 photo-induced manipulation of particles at solid-liquid interfaces, 63 strategizing new methods for phase separations, 64 and many more. In this paper, we study for the first time this new electrokinetic problem for soft nanochannels (i.e., nanochannels grafted with end-charged PE brushes). Our theory accounts for the thermodynamics and electrostatics of the brushes in

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developing the fluid mechanics model for the IDO in brush-grafted nanochannels. Our theory leads us to an integro-differential equation that is solved numerically to unravel the physics of the IDO. Firstly, we quantify the diffusioosmotically induced electric field, establishing the manner in which the imposed concentration gradient (CG) dictates this electric field with insignificant contribution from the induced osmotic effect. Secondly the diffusioosmotic (DOS) flow is described as a combination of the induced electroosmotic (EOS) transport (caused by this induced electric field) and a chemisoosmotic (COS) transport caused by the induced pressure gradient owing to this imposed CG. We compare the results of the PEbrush-grafted nanochannels with that of the brush-free nanochannels. While there is a less difference in the overall induced electric field between these two systems by virtue of the fact that the electric field is primarily dictated by the effect of the imposed CG, the velocity fields vary massively between the brush-free and brush-grafted nanochannels. The influence of charge localization induced enhancement of the EOS velocity field 35 ensures that for sufficiently long but weakly grafted brushes, the induced EOS transport and consequently the resulting DOS transport gets massively enhanced for the nanochannels grafted with the PE brushes. Furthermore, for conditions where the COS transport augments the EOS transport, this enhancement is even larger. Moreover, such dominant role of the EDL (and hence EOS body force) localization effect leads to a concentration-dependent behavior of the DOS flow field in brush-grafted nanochannels that is completely opposite to what occurs in brush-free nanochannels. Finally, we study the case where the grafting density of the brushes is much larger and therefore, on account of the dominant influence of the brush-induced drag force, the DOS transport gets severely reduced as compared to the DOS transport in brush-free nanochannels. In summary, our paper studies for the first time the electrokinetic problem of IDO in brush-grafted soft nanochannels unraveling fluid physics that can be instrumental for novel flow based applications involving functionalized, soft nanochannels.

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∇n∞ > 0

uEOS uCOS

Einduced

Figure 1: Schematic showing DOS transport in nanochannels grafted with end-charged brushes. The DOS flow is a combination of the EOS (caused by the induced electric field) and the COS flows, as illustrated in the schematic. The schematic shown here corresponds to non-negative values of the dimensionless diffusivity difference β.

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Theory We consider a nanochannel of half height h and length L grafted with end-charged PE brushes (see Fig. 1). The nanochannel is connected to bulk microfluidic reservoirs. We have previously studied the electrostatics, thermodynamics, ionic current, and electrokinetic transport in such nanochannels. 35,37,38 The calculation of the electrokinetic transport in these papers 35 is unique in the sense that it accounts for the appropriate coupling of the brush configuration and the resulting EDL electrostatics while computing the electrokinetic fluid flow. In the present paper, we would use this information on the coupled brush configuration and the resulting EDL electrostatics in order to quantify the ionic DOS transport. The IDO is triggered by imposing a constant axial ionic concentration gradient of strength ∇n∞ = dn∞ /dx, such that L∇n∞ /n∞ (x = 0) (here n∞ is the bulk number density of the electrolyte ions or the electrolyte ion concentration in the microfluidic reservoirs). We shall first provide the coupled equilibrium description of the configuration of the end-charged PE brush and the induced EDL electrostatics. This description has already been provided in our previous paper 35 – we repeat it here for the sake of completion. Next, we shall use this equilibrium description to quantify the electric field induced due to the diffusioosmotic effect and the resulting nanofluidic velocity field (occurring due to the combined influence of the induced EOS and the COS velocities).

Equilibrium Thermodynamics and Electrostatics of the End-Charged Brushes The equilibrium behavior of the end-charged brushes is obtained by minimizing the total free energy F of the brushes and the surrounding electrolyte:

F = FB,els + FB,EV + FB,elec + FEDL ,

6


(1)

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where FB,els , FB,EV and FB,elec are the elastic, excluded volume, and electrostatic free energies associated with the brushes and FEDL is the free energy associated with the induced EDL. These individual free energies can be expressed as: 3d2 FB,els = , kB T 2Np a2k

(2)

ωNp2 σ FB,EV = , kB T d

(3)

ψs σch , σ

(4)

FB,elec = FEDL

1 = σ

Z

0

fEDL dy.

(5)

−h

In the above equations, d is the brush height, Np is the polymer size (or the number of monomers in the polymer chain), ak is the Kuhn’s length, ω =

1−2χ 3 ak 2

(where χ is the Flory

exponent), ψs is the electrostatic potential at the non-grafted charged end of the brushes, σch is the constant (pH-independent) charge density of the ends of the PE brushes, σ is the grafting density, and kB T is the thermal energy. Finally, fEDL is the density of the EDL energy and can be expressed as:

fEDL

0 r dψ 2 n+ n− | | + eψ (n+ − n− ) + kB T n+ ln =− − 1 + n− ln − 1 , (6) 2 dy n∞ n∞

where 0 is the permittivity of free space, r is the relative permittivity of water, ψ is the electrostatic potential, n± is the number density of the symmetric monovalent electrolyte ions, and n∞ is the bulk number density of the ions. The final free energy equation obtained by using eqs.(2-6) in eq.(1) is minimized to obtain the equilibrium conditions. The minimization with respect to ψ yields the Poisson equation: e (n+ − n− ) d2 ψ = − . dy 2 0 r

7


(7)


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On the other hand, the minimization of this free energy equation with respect to n± leads to the well-known Boltzmann distribution: eψ . n± = n∞ exp ∓ kB T

(8)

Finally the minimization of the free energy equation with respect to ψs yields the boundary condition at the location of the brush tip (i.e., y = −h + d) for the nanochannel bottom half (see Chen and Das 35 for detailed derivation):

dψ dy

− y=(−h+d)+

dψ dy

=− y=(−h+d)−

σch , 0 r

(9)

where d is the equilibrium brush height. The Poisson-Boltzmann equation, obtained by using eq.(8) to replace n± in terms of ψ in eq.(7), is solved numerically in presence of the boundary condition expressed in eq.(9) as well as the conditions expressed below:

dψ dy

= 0,

y=−h

dψ dy

= 0, (ψ)(y=−h+d)+ = (ψ)(y=−h+d)− .

(10)

y=0

Once the equilibrium ψ and n± distribution have been obtained, we use it in eqs.(5,6) to compute the equilibrium value of FEDL . Of course, this equilibrium FEDL will depend on the brush height d. This equilibrium FEDL is used in the overall free energy, which is then minimized with respect to d numerically to obtain the equilibrium brush height (i.e., equilibrium value of d). This is the same procedure that we have used in our previous papers. 35,37

Diffusioosmotically Induced Electric Field In order to obtain the diffusioosmotically induced electric field, we need to equate the net current in the system to zero, which is identical to making the integral of the difference

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between the cationic and anionic fluxes to zero, i.e., Z

h

(J+ − J− ) dy = 0,

(11)

−h

where the fluxes (J± ) can be expressed as: J± = −D±

e ∇n± ± n± (∇ψ − E) + n± u. kB T

(12)

In the above equation, D± are the diffusivities of cations and anions, u is the induced velocity field (a combination of the EOS and COS velocity fields) and E is the induced electric field. Using eq.(12) in eq.(11), we shall eventually get the dimensionless diffusioosmotically induced electric field (please see the Appendix for detailed derivation): h io −ψ¯ 0 ψ¯ (1 + β) e − (1 − β) e d¯ y n ¯ 1 −1 E ¯ E= = P eR 1 + R 1 ,(13) E0 (1 + β1 ) e−ψ¯ + (1 − β1 ) eψ¯ d¯ y (1 + β) e−ψ¯ + (1 − β) eψ¯ d¯ y −1 −1 R1 n

R1

¯ y 4¯ u sinh (ψ)d¯ −1

where y¯ = y/h, E0 = kB T /(eL), u¯ = u/U ∗ (where U ∗ is the characteristic velocity scale that defines the Peclet number: P e = U ∗ L/(D+ + D− )), ψ¯ = eψ/(kB T ), β = n ¯ 01 =

D+ −D− , D+ +D−

and

L∇n∞ . n∞

Diffusioosmotic (DOS) Velocity Field In order to obtain the DOS velocity field, we shall employ the Navier-Stokes (NS) equation. The NS equation in y-direction yields the necessary pressure field, i.e., ∂p dψ ¯ ψ¯ ⇒ p = patm + 2kB T n∞ cosh (ψ) ¯ − 1 .(14) + 2e (n+ − n− ) = 0 ⇒ ∂p = 2kB T n∞ sinh (ψ)d ∂y dy

9



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Finally, the NS equation in the x-direction yields (considering only the nanochannel bottom half): u d2 u ∂p + η − e (n+ − n− ) (E − ∇ψ) f or − h ≤ y ≤ −h + d, = dy 2 ∂x kd d2 u ∂p η 2 = − e (n+ − n− ) (E − ∇ψ) f or − h + d ≤ y ≤ 0. dy ∂x

η

(15)

Eq.(15) can be expressed in dimensionless form as (see the Appendix for the detailed derivation): 2

d u¯ = d¯ y2

d2 u¯ = d¯ y2

( R1 ¯ 0 4¯ u sinh (ψ)dy −1 2 ¯ ¯ A n ¯ 1 cosh (ψ) − 1 + α u¯ + A sinh (ψ) P e R 1 (1 + β) e−ψ¯ + (1 − β) eψ¯ d¯ y −1 i ) R1 0 h ¯ ¯ n ¯ (1 + β) e−ψ − (1 − β) eψ d¯ y −1 1 ¯ + R1 f or − 1 ≤ y¯ ≤ −1 + d, ¯ ¯ − ψ ψ (1 + β) e + (1 − β) e d¯ y −1 ( R1 ¯ 0 4¯ u sinh (ψ)dy −1 ¯ − 1 + A sinh (ψ) ¯ P eR A n ¯ 1 cosh (ψ) 1 −ψ¯ + (1 − β) eψ¯ d¯ (1 + β) e y i ) −1 R1 0 h ¯ ¯ y n ¯ (1 + β) e−ψ − (1 − β) eψ d¯ −1 1 + R1 f or − 1 + d¯ ≤ y¯ ≤ 0, −ψ¯ + (1 − β) eψ¯ d¯ (1 + β) e y −1 (16)

h2 Πosm where d¯ = d/h, η is the dynamic viscosity, A = 2kBLT n∞ ηU , and α = h2 /kd [where ∗ = Π drive 2 kd = a2k σa3dNp φ is the permeability, with φ being the monomer distribution along the k

length of the PE brush]. It is obvious that we arrive at eq.(15) by using the expression of E [see eq.(13)]. Eq.(15) is an integro-differential equation in u¯, which is solved numerically in presence of the boundary conditions expressed below. (¯ u)y¯=−1 = 0, (¯ u)y¯=(−1+d) u)y¯=(−1+d) ¯ + = (¯ ¯− ,

d¯ u d¯ y

=

¯+ y¯=(−1+d)

d¯ u d¯ y

, ¯− y¯=(−1+d)

d¯ u d¯ y

= 0. y¯=0

(17) Subsequently, this u¯ is used to obtain the dimensionless electric field E¯ [see eq.(13)]. Obviously, both the solution for u¯ and E¯ will depend on ψ¯ – section IIA provides a method to 10


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¯ calculate ψ.

Results and Discussions

(a)

(b)

(c) Figure 2: Transverse variation of ψ¯ for (a) brush-free nanochannels, (b) nanochannels grafted with PE brushes with large grafting density (with ` = 88 nm, Np = 2000), and (c) nanochannels grafted with PE brushes with weak grafting density (with ` = 22 nm, Np = 2000). Results are shown for three different bulk salt concentrations, namely c∞ = 10−4 M, 10−3 M, 10−2 M for each of (a-c). Other parameters are σch = −0.0008C/m2 (this charge density is the charge density for brush-free nanochannel walls and the ends of the end-charged PE brushes for nanochannels with PE brushes), h = 100nm, χ = 0.4, ak = 1nm, kB = 1.38 × 10−23 J/K, T = 300 K, e = 1.6 × 10−19 C, 0 = 8.8 × 10−12 F/m. These figures are adapted with permission from Ref, 35 Copyright (2017) American Chemical Society.

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(a)

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(b)

(c) Figure 3: Variation of the diffusioosmotically induced electric field (expressed in dimensionless form) with bulk salt concentrations for (a) brush-free nanochannels, (b) nanochannels grafted with PE brushes with large grafting density (with ` = 88 nm, Np = 2000), and (c) nanochannels grafted with PE brushes with weak grafting density (with ` = 22 nm, Np = 2000).. Results are shown for three different values of β for each of (a-c). We consider ∞ = 0.1, while the other parameters are identical to those used in Fig. 2. L ∇n n∞

12


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Variation of the EDL Electrostatic Field ψ Fig. 2 provides the variation of the dimesnionless EDL electrostatic field ψ for nanochannels with and without the grafted PE brushes for different values of bulk salt concentration. We ensure that the charge density of the non-grafted ends of the end-charged brushes is identical to the charge density of the inner walls of the brush-free nanochannels. In our previous paper, we have presented the results for the ψ variation for nanochannels grafted with end-charged PE brushes. 35 We repeat them here for the sake of continuity and for the better understanding of the diffusioosmotic phenomena. For the nanochannel without ¯ max is at the nanochannel walls, given the fact that the the brushes the location for the |ψ| nanochannel walls are charged [see Fig. 2(a)]. This changes for the nanochannels grafted with the end-charged PE brushes. For these brushes the charges are localized at their non-grafted ¯ max at the location of these ends and as a consequence, one witnesses the attainment of |ψ| ends, i.e., significantly away from the nanochannel walls. Smaller ` or larger grafting density leads to a larger equilibrium brush height – as a consequence, we witness the attainment of ¯ max at a location that is further away from the nanochannel wall for ` = 22 nm [see Fig. |ψ| 2(c)] as compared to ` = 80 nm [see Fig. 2(a)]. Furthermore, regardless of the presence or ¯ at any transverse location the absence of the PE brush grafting, one witnesses a larger |ψ| for a smaller salt concentration (c∞ ). Smaller c∞ implies a larger value of the EDL thickness ¯ ∝ λσch , one should witness a larger |ψ| ¯ for a given σch . λ – therefore, given the fact that |ψ| These information on the variation on ψ¯ will help us to better explain the different DOS behaviors.

Variation of the Diffusioosmotically Induced Electric Field Fig. 3 shows the variation of the dimensionless diffusioosmotically induced electric field for the nanochannels with and without the brushes as a function of salt concentration and β. The electric field is a combination of the osmotic effects (implying the electric field caused by the downstream advective migration of the charge imbalance present within the EDL) 13



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(please see the first term on the right hand side in eq. 13) and the imposed concentration gradient (CG) effect (please see the second term on the right hand side in eq. 13). The osmotic contribution is relatively weak in comparison to that due to the CG effect. The exact extent of the relative contribution of this osmotic component, as will be discussed later, depends on the presence (and the grafting density) of the brushes as well as the salt concentration. On the other hand, the contribution of the imposed CG effect on the induced electric field is contributed both by the diffusivity difference between the cations and anions (dictated by the parameter β) and the very fact that there is a charge imbalance induced by the EDL. For β = 0, 0.5, −0.5, this CG-imposed contribution of the electric field can be ¯ and therefore using DH approximation in eq. 13, i.e., obtained as (considering small |ψ| ¯ ¯ expressing e±ψ = 1 ± ψ):

E¯

CG,β=0

n ¯ 01 =− 2

Z

1

−1

¯ y, ψd¯

E¯

CG,β=0.5

=

R1 (1 0 −1 n ¯1 R 1 (2 −1

¯ y − ψ)d¯ , ¯ y − ψ)d¯

E¯

CG,β=−0.5

=

R1 ¯ y (1 + 2ψ)d¯ 0 −1 −¯ n1 R 1 .(18) ¯ y (2 + ψ)d¯ −1

ψ¯ is always negative and n ¯ 01 is always positive. Accordingly, from eq.(18), we can easily identify why for β = 0 one witnesses a positive electric field for both brush-free and brushgrafted nanochannels (see Fig. 3). On the other hand, for β > 0, a negative ψ¯ would imply an even larger positive value of E for both brush-free and brush-grafted nanochannels (see Fig. 3). This effect of β > 0 can be understood by noting the fact that a larger mobility of the counterions (which corresponds to a positive β) augments the effect associated with the ¯ Accordingly for charge difference of the EDL. Smaller c∞ leads to a larger magnitude of |ψ|. both β = 0 and β = 0.5, the positive value of E will be larger for smaller c∞ for both brushfree and brush-grafted nanochannels (see Fig. 3). Completely contrary to these behaviors is the case corresponding to β = −0.5. This refers to a situation where the coions have higher mobilities, which in turn will nullify the effect associated with the EDL electrostatic potential. ¯ the electric field can indeed be negative (see eq. 18) Accordingly for weak enough value of |ψ| ¯ values. This also justifies a weaker and this negative magnitude is enhanced for weaker |ψ|

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¯ This negative magnitude of ψ for smaller c∞ that corresponds to a larger magnitude of |ψ|. is true for both nanochannels with and without grafted PE brushes. We next consider the influence of the osmotic component of the electric field. For the same DH consideration, ¯ i.e., proportional to we can use eq.(13) to show that this component is proportional to u¯ψ, the effects of advection of the EDL charge density. For the case of very densely grafted PE brushes, the significantly small velocity component (as evident in Fig. 6 later) would lead to a very weak contribution of the osmotic effect and accordingly, as yielded by our analysis (results not shown here) the DOS electric field is effectively the electric field due to the imposed CG. On the other hand, for cases of brush free nanochannels or nanochannels grafted with PE brushes with weak grafting density, the velocity field is significantly higher ensuring a finite (albeit less than 20%, as revealed by our analysis) variation of the overall electric field. However, this finite osmotic contribution mostly occurs for smaller salt concentration that yields a much larger value of ψ¯ (see Fig. 2). Finally, for the situations where the osmotic component has insignificant contribution to the overall electric field (namely for the case of nanochannels grafted with PE brushes with large grafting density or nanochannels with or without brushes operating at large salt concentration), we witness insignificant effect of the presence of brushes on the overall value of E for a given β and c∞ . For such cases where the effect of the imposed CG dominates, the electric field is solely dictated by the integral of ψ¯ across the channel height. This integral is proportional to the net surface charge density (at either the walls of the brush-free nanochannels or the non-grafted ends of the end-charged brushes decorating the nanochannel walls). This charge density being same for both brush free and brush-grafted nanochannels, we witness this nearly same E for a given c∞ and a given β for both brush-free and brush-grafted nanochannels.

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1

1 1

y¯

-1

0

u¯

-0.5 -1

1 ×10

c =0.0001 M

-3

∞

c ∞=0.01 M 0

0.1

0

0.3

-1

0.4

u¯

0

c ∞=0.0001 M

0

-1

-0.5 0.2

∞ 1 y¯

-1 -2

0

c =0.01 M

0.5

0

y¯

y¯

0.5

0

0.02 0.04 0.06

0.2

u¯

0.4

0.6

u¯ (b)

(a) 1 0.5

y¯

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 39

0

c =0.0001 M ∞

c ∞=0.01 M

-0.5 -1 -0.4

-0.3

-0.2

u¯

-0.1

0

(c)

(d)

Figure 4: Transverse variation of the dimensionless DOS velocity field (shown by bold lines) and EOS velocity field (obtained by switching off the contribution of the pressure gradient in eq. 15 and shown by dashed lines) for (a) β = 0, (b) β = 0.5, and (c) β = −0.5. In the insets of (a) and (b) the EOS and the DOS velocity fields for c∞ = 0.01 M are magnified. ¯ − 1 and (d) Transverse variation (for the nanochannel bottom half) of Ch = c∞ cosh (ψ) ¯ both of which are independent of β. The results are shown for the brushSh = c∞ sinh (ψ), free nanochannels and for two different salt concentrations (c∞ = 0.0001 M, 0.01 M ). Other parameters used in this study are same as that in Fig. 3.

16


0.8

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(b)

(a)

(d)

(c)

Figure 5: Transverse variation of the dimensionless DOS velocity field (shown by bold lines) and EOS velocity field (obtained by switching off the contribution of the pressure gradient in eq. 15 and shown by dashed lines) for (a) β = 0, (b) β = 0.5, and (c) β = −0.5. In the inset of (a) the EOS and the DOS velocity fields for c∞ = 0.01 M are magnified. (d) ¯ Transverse variation (for the nanochannel bottom half) of Ch = c∞ cosh (ψ) − 1 and Sh = ¯ both of which are independent of β. The results are shown for the brush-grafted c∞ sinh (ψ), nanochannels (with Np = 2000 and ` = 80 nm) and for two different salt concentrations (c∞ = 0.0001 M, 0.01 M ). Other parameters used in this study are same as that in Fig. 3.

17



Page 18 of 39

(b)

(a)

(d)

(c)

Figure 6: Transverse variation of the dimensionless DOS velocity field (shown by bold lines) and EOS velocity field (obtained by switching off the contribution of the pressure gradient in eq. 15 and shown by dashed lines) for (a) β = 0, (b) β = 0.5, and (c) β = −0.5. In the inset of (a) the EOS and the DOS velocity fields for c∞ = 0.01 M are magnified. (d) ¯ Transverse variation (for the nanochannel bottom half) of Ch = c∞ cosh (ψ) − 1 and Sh = ¯ both of which are independent of β. The results are shown for the brush-grafted c∞ sinh (ψ), nanochannels (with Np = 2000 and ` = 22 nm) and for two different salt concentrations (c∞ = 0.0001 M, 0.01 M ). Other parameters used in this study are same as that in Fig. 3.

18


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Variation of the Diffusioosmotic (DOS) Velocity Field Case of Brush-Free Nanochannels: In Fig. 4, we show the variation of the diffusioosmotically induced velocity field. This velocity field is a combination of the EOS velocity and the COS velocity fields. The COS velocity is triggered on account of the axial pressure gradient induced by the imposition of the axial CG. In order to distinguish between the relative contributions of the EOS and the COS velocity fields in deciding the overall DOS velocity field, in Figs. 4-6 we plot both the overall DOS velocity field as well as only the EOS velocity field (obtained by switching off the pressure gradient term in eq. 15). We first consider the case of brush free nanochannels with β = 0 [see Fig. 4(a)]. For this condition the induced DOS electric field (E) is positive for all values of c∞ [see Fig. 3(a)] with E increasing drastically with a decrease in c∞ . Consequently, the EOS velocity field is invariably positive (i.e., occurs from left to right) and increases significantly with a decrease in c∞ . However, the overall DOS velocity field bears the signature of the contribution of the COS component as well. Accordingly, there is a significant reduction in the net velocity (i.e, the DOS velocity) as compared to the pure EOS velocity stemming from the fact that the COS velocity being caused by a positive pressure gradient is invariably negative (i.e., opposes the EOS flow caused by a positive electric field). Dimensionless COS body force, for a given value of imposed

L∇n∞ , n∞

¯ − 1], arising from the fact that from eq.(16) we get is proportional to Ch = c∞ [cosh (ψ) the dimensionless pressure gradient (or the dimensionless per unit volume COS body force) L∇n∞ ¯ − 1] = 2kB T ∇n∗ ∞ h2 [cosh (ψ) ¯ − 1] = 2kB T∗ h2 ¯ − 1] = n∞ [cosh (ψ) as A¯ n01 [cosh (ψ) ηU ηU L n∞ 2 L∇n∞ BT h ¯ − 1] (where c∞ is the concentration in M and NA is the 103 NA 2kηU c∞ [cosh (ψ) ∗L n∞ Avogadro number). Fig. 4(d) provides the transverse variation of Ch , which is independent of β, for two different values of c∞ for the case of brush free nanochannels. Ch is present only within the EDL and hence non-existent at locations even slightly away from the nanochannel wall for large salt concentrations. This justifies a much lesser retarding contribution of the COS velocity field for larger c∞ . Hence we witness the overall DOS velocity field to be much 19



smaller than the EOS velocity field for smaller c∞ . However, the EOS velocity field is very small for a larger c∞ owing to a significantly weaker electric field; accordingly, we witness an actual inversion (i.e., a flow field from right to left) of the entire DOS velocity field on account of the retarding influence of the COS velocity. We next consider the case of β = 0.5 for the brush-free nanochannels [see Fig. 4(b)]. For such a positive value of β, the electric field is significantly enhanced [see Fig. 3(a)] causing a significantly larger induced EOS velocity field. Here too an increase in salt concentration lowers this electric field [see Fig. 3(a)] and hence the EOS velocity field and also lowers the retarding COS velocity component (as evident from the lowering of the corresponding COS body force, see Fig. 4(d)] thereby reducing the difference between the overall DOS velocity and the EOS velocity. Of course, for this case the electric field and the EOS velocity is significantly larger and positive even for large c∞ ensuring a net positive DOS velocity field for such c∞ . It is worthwhile to note here that the EOS body force, in addition to being propor¯ stemming from the fact tional to the electric field E, is also proportional to Sh = c∞ sinh (ψ) ¯ = that the EOS body force varies as A sinh (ψ)

2kB T h2 n∞ LηU ∗

¯ = 103 NA 2kB T h∗ 2 c∞ sinh (ψ) ¯ sinh (ψ) LηU

[see eq.(16)]. In Fig. 4(d), we also plot Sh , which being finite only within the EDL will be non-existent at locations slightly away from the nanochannels wall for large c∞ . Therefore, for β = 0, 0.5, the electroosmotic flow is enhanced for a smaller salt concentration due to a combined action of larger electric field [see Fig. 3(a)] and a larger Sh . Finally, we consider the case of β = −0.5 for brush-free nanochannels [see Fig. 4(c)]. Here the electric field is negative [see Fig. 3(a)], and consequently the EOS flow will also be negative, i.e., from right to left. However, as evident from Fig. 3(a), for β = −0.5, the negative magnitude of the electric field is larger for larger c∞ . Despite that, we witness a larger magnitude of the negative EOS velocity for weaker c∞ , stemming from the fact that the component Sh is significantly larger (particularly at locations slightly away from the wall) for smaller c∞ . Secondly, given the fact that the EOS velocity field is negative, the negative COS velocity will augment the net DOS flow rather than decreasing it. Obviously

20


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the COS velocity component is larger in magnitude for smaller c∞ , owing to a larger COS body force for smaller c∞ [see Fig. 4(d)]. Accordingly, one will witness a significantly larger (and significantly augmented as compared to the pure EOS velocity field) DOS velocity field occurring from right to left for such small c∞ values. In summary, therefore, we establish the manner in which parameters such as β and c∞ interplay to dictate the relative contribution of the EOS and the COS velocity fields to dictate the overall DOS velocity field in the brush-free nanochannel.

Case of Brush-Grafted Nanochannels (Np = 2000, ` = 80 nm): For the brush-grafted nanochannels, we first consider the case of long, weakly grafted brushes (quantified by Np = 2000, ` = 80 nm). We start with the case of β = 0 [see Fig. 5(a)]. As evident from Fig. 3(b), very much like the case of brush-free nanochannels, the electric field is positive and increases with a decrease in c∞ . Therefore, there will be a positive EOS flow that increases significantly with a decrease in c∞ . Also, very much like the case of the brush-free nanochannels, the variable Ch dictating the body force for the COS flow (which occurs from right to left) massively increases for smaller c∞ particularly at locations slightly away from the location of the maximum charge density (i.e., the non-grafted charged end of the brushes) [see Fig. 5(d)]. Therefore, here too the DOS velocity will be significantly weaker than the EOS velocity for smaller c∞ . However, given that the EOS velocity itself is distinctly small for a larger c∞ , the retarding influence of the COS velocity will eventually lead to a reversal in the direction of the overall DOS velocity for large c∞ . We next consider the case of β = 0.5 [see Fig. 5(b)]. The significantly large positive electric field [see Fig. 3(b)] will lead to a significantly larger positive EOS velocity that increases with a decrease in c∞ . Of course, here too, the retarding influence of the COS velocity reduces the flow strength and this reduction is much more prominent for weaker c∞ . It is worthwhile to compare the velocity profiles with those of the brush-free nanochannels at

21



this point. Despite the electric field values being similar for both brush-free and brush-grafted nanochannels, we witness (i) a distinctly higher magnitude of both the EOS and the DOS velocities for a given c∞ [compare Figs. 4(b) and 5(b)] and (ii) there is a much lesser difference between the DOS velocities at different salt concentrations [compare Figs. 4(b) and 5(b)]. We can ascribe both of these effects to the charge localization induced enhancement in the EDL-induced EOS transport. 35 The EDL is invariably localized around the charged ends of the PE brushes. The EOS body force is proportional to the charge density difference within the EDL and therefore would be maximum around these charged ends of the PE brushes. These charged ends being significantly away from the wall would imply the maximum drag force caused by the wall will be significantly away from the location of the maximum driving body force. This would imply that a same driving EOS body force would lead to a larger EOS velocity. Obviously this enhancement is countered by the additional drag induced by the presence of the brushes. This countering influence is overcome for the case of weak brush grafting density and significantly large Np (e.g., the present case of Np = 2000 and ` = 80 nm), leading to the most non-intuitive scenario where the EOS flow strength is actually enhanced by the presence of the brushes. 35 This justifies why one witnesses a much larger EOS velocity (and hence a larger DOS velocity) for nearly similar electric field (or similar combinations of β and c∞ ) for the brush-grafted nanochannels (with Np = 2000 and ` = 80 nm) as compared to the brush-free nanochannels. Of course, in Chen and Das, 35 the EOS flow was triggered by an applied electric field, while in the present case, it is induced due to diffusioosmosis. Secondly, this localization of the EDL around the charged end of the end-charged PE brushes is more prominent for larger salt concentration (which causes a smaller EDL). Accordingly, this EDL localization induced localization of the driving EOS body force is more prominent for larger c∞ , leading to a larger enhancement of the EOS (and hence DOS) velocity fields for larger c∞ . This justifies the second observation that for brush-grafted nanochannels the difference in the EOS (and hence DOS velocity fields) between large and small salt concentration values are much less, despite the electric field

22


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being much larger for smaller salt concentration (for β = 0, 0.5). Finally, we consider the case of β = −0.5 [see Fig. 5(c)]. Here the electric field is negative [see Fig. 3(b)] and the magnitude of the electric field increases with an increase in salt concentration. Coupled with this effect is the fact that the effect of EDL localization and the resulting enhancement in the EOS transport 35 is much more enhanced for the case of larger c∞ . These two effects overcome the effect of a reduced Sh for larger concentration, eventually ensuring a much larger negative magnitude of the EOS transport for larger c∞ . Of course, as with the case of brush-free nanochannels here the COS velocity, which is itself negative, will augment the EOS flow strength leading to a DOS flow that is more enhanced than the EOS flow. Comparing with the case of the brush-free nanocahnnels [see Fig. 4(c)], here we witness that the EOS body force localization effect associated with the EOS transport in nanochannels grafted with end-charged brushes not only enhnaces the magnitude of the EOS (and hence DOS) velocity fields, but also causes a reversal in the salt-concentrationdependence of the velocity field.

Case of Brush-Grafted Nanochannels (Np = 2000, ` = 22 nm): We finally consider the case of brush-grafted nanochannels with large grafting density (` = 22 nm). A large grafting density will impart such a large drag force on the flow that the effect of EDL (and hence the EOS body force) localization will be nullified leading to a decrease in the overall velocity field. Obviously the decrease is most stark at transverse locations deep within the brushes depicting a virtually non-existent velocity. Also, very much like the case of brush-free nanochannels and naochannels grafted with less dense brushes, we witness a negative DOS velocity for large c∞ and β = 0. On the other hand, the EDL charge density and EOS body force localization effect ensures very little difference in the EOS (and hence DOS) velocities between the cases of large and small salt concentrations for β = 0.5. Finally for β = −0.5, the negative electric field causes a negative EOS velocity field, which is much

23



larger for larger c∞ owing to the combined influence of enhanced magnitude of the negative electric field [see Fig. 3(c)] and the flow enhancement due to a stronger localization of the EOS body force away from the nanochannel walls. On the other hand, the significantly larger value of Ch [and hence the COS velocity, see Fig. 6(d)] for a smaller c∞ outweighs this enhanced EOS velocity and eventually ensures that the magnitude of the overall DOS velocity is very much similar.

Possible Significance of the Different Approximations Employed in this Model In this model, we have neglected the influence of the possible flow-induced tilt of the PE brushes. Following Kim et al., 65 one can employ mean-field approach to quantify the tilt angle θ(s) for the brushes (where θ is the angle made by the local tangent to the brushes with the y axis and s describes the contour). The analytical result for the tilt angle is (see 65 for detailed derivations): s2 s 1 [cosh (λθ s) − 1] − cosh (λθ `c ) + [λθ `c cosh (λθ `c ) − sinh (λθ `c )] , λ2θ 2 λθ (19) p where `c is the contour length of the PE brush, λθ = ζθ ρθ /(η`2c ) (ρθ = `c /` denotes the ζθ ξθ θ(s) = κθ λθ

number of grafted PE molecule per contour length and ζθ = 3πη), ξθ = (γ/λ ˙ θ ) cosh (λθ `c ) (γ˙ = u/`c is the average shear rate), and κθ is the bending rigidity of the PE brushes. For the present study, we consider the brush deformation for the case where the induced velocity is largest, namely the weak grafting density (` = 88 nm). The maximum possible deformation is possible when we replace `c and s by the equilibrium brush height d. Using the above formulation as well as the value of d (≈ 40 nm) at c∞ = 10−2 M (i.e., studied concentration where the EDL localization mediated enhancement of the DOS velocity is maximum), u ∼ 10−5 m/s (in nanochannel transport, this is typically the velocity value one witnesses; 66 in our simulations we considered the parameters that led to a much smaller value 24


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than this), κ = 3 × 10−28 J.m, 67 and η = 10−3 P a.s, we can obtain θmax = 0.79◦ . Therefore, under actual nanochannel flow scenarios, it is safe to neglect the tilt of the brushes. Secondly, we have employed Poisson-Boltzmann model to describe the EDL electrostatics. In other words, we have neither considered the mean-field based non-Poisson-Boltzmann effect (i.e., Steric effect or excluded volume effect of the ions 68–70 and the solvent polarization effect 71,72 ) nor have we considered the non-mean-field-based ion-ion and ion-PE-charge correlation effects. 73 The effects like ion Steric effect or the solvent polarization effect becomes important for cases when the EDL electrostatic potential is large (i.e., ψ¯ 1) and/or the salt concentration is large. 68 On the other hand, the ion-ion correlation effects become important for the cases of concentrated electrolyte, presence of multivalent cations, solvent-free ionic liquids, etc. 73 In the present scenario, we encounter none of these conditions. Therefore, we can safely neglect the influence of these different non-Poisson-Boltzmann effects. In this context, it is useful to pinpoint a recent paper that have studied ionic DOS transport in presence of very large ion concentration. 74 The paper showed that during the mineral replacement reactions, a large gradient in the concentration of salts like KBr and KCl is induced, which in turn induces a DOS transport that plays a central role in the overall ion transport process. This paper considered a large value (several molar) of the salt concentration, which necessitated the consideration of the finite ion size effect in the modelling of the EDL electrostatics. In fact, in another study the same group had experimentally demonstrated the most remarkable charge inversion at very large concentration of monovalent salt and proposed a Monte Carlo simulation method to establish their findings. 75 For the PEbrush-grafted nanochannel systems, for cases where the salt concentration is very high one would definitely need to include the different non-PB components in the EDL electrostatic free energy to obtain the thermodynamically consistent description of the PE brush configuration and the EDL electrostatics. Consideration of these non-PB elements lead to an increase of the effective EDL thickness. 72,76 Given that this EDL electrostatics would then be used to compute the DOS transport, the EDL localization at the charged, non-grafted

25



end of the PE brushes would be less severe. On the other hand, such enhanced effective EDL would imply that for the brush-free nanochannels, the EDL charge density is less confined at near-wall locations. Under these conditions, therefore one would anticipate a lesser difference between the DOS transport in brush-free and end-charged-brush-grafted nanochannels. On the other hand, given that all the non-PB elements are captured in a continuum framework (without requiring any molecular scale information on the ion behavior), this approach will only suffice to account for the effects that can be described in a continuum framework. In other words, the effect such as the charge inversion at large concentration of the monovalent salt, 75 which necessitated the use of Monte Carlo simulation for the analysis, would not be captured by considering the different non-PB components in the analysis of the ionic DOS transport. For such cases, one would need to carry out atomistic simulations with explicit water (SPC/E) that first reproduces the charge inversion phenomenon in presence of the end-charged PE brushes and subsequently quantify the ionic nanochannel DOS transport by appropriate modification of the force-balance approach as proposed for non-ionic diffusioosmosis previously. 59

Conclusions We develop theory to study the problem of IDO in end-charged brush-grafted nanochannels. We show that the weird fluid physics of augmented EOS transport in such systems, owing to the localization of the EDL charge density and the EOS body force away from the nanochannel wall, can be successfully leveraged to develop an extremely augmented DOS fluid transport in nanochannels grafted with long and weakly dense brushes. This augmentation is with respect to brush-free nanochannels and depend on the exact extent of the influence of the COS transport. Depending on the relative magnitude of the counterion and coion diffusivities, this COS transport may augment or retard the EOS tranpsort in deciding the overall DOS transport. The effect of end-charge localization in augmenting the

26


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flow is found to be the most prominent for the case where the COS transport aids the EOS transport. We anticipate that this paper will bring to light a new mechanism of electrokinetic transport in functionalized nanochannels that can be successfully used for a myriad of flow-field-dependent and hitherto unconceived applications in such nanochannels.

Appendix A: Derivation of eq.(13) Eq.(13) expresses the diffusioosmotically induced electric field obtained by making the net Rh current equal to zero, i.e., −h (J+ − J− ) dy = 0. The flux J± can be expressed as (considering ¯ ¯ ¯ ¯ n± = n∞ e∓ψ so that ∇n± = ∇n∞ e∓ψ ∓ n∞ e∓ψ ∇ψ):

e n± (∇ψ − E) + n± u = J± = −D± ∇n± ± kB T e ∓ψ¯ ∓ψ¯ ¯ −D± ∇n∞ e ∓ n∞ e n± (∇ψ − E) + n± u = ∇ψ ± kB T e e ∓ψ¯ −D± ∇n∞ e ∓ n± ∇ψ ± n± (∇ψ − E) + n± u = kB T kB T e ∓ψ¯ n± E + n± u −D± ∇n∞ e ∓ kB T

(A1)

We can calculate the electric field by equating the net current to zero, i.e., Z

h

(J+ − J− ) dy = 0 ⇒ Z h Z h e −ψ¯ −D+ ∇n∞ e − n+ E dy + n+ udy = kB T −h −h Z h Z h e ψ¯ −D− ∇n∞ e + n− E dy + n− udy ⇒ kB T −h −h Z h e E (D+ n+ + D− n− ) dy = kB T −h Z h Z h ¯ ¯ u (n− − n+ ) dy + D+ ∇n∞ e−ψ − D− ∇n∞ eψ dy −h

−h

−h

27


(A2)


Also, considering β =

D+ −D− , D+ +D−

D+ =

Page 28 of 39

we can write:

(1 + β) (D+ + D− ) (1 − β) (D+ + D− ) , D− = . 2 2

(A3)

Using eqs.(A3), we can write:

D+ n+ + D− n− =

−ψ¯

D+ ∇n∞ e

h i D+ + D− ¯ ¯ n∞ (1 + β) e−ψ + (1 − β) eψ . 2

i D+ + D− n ¯ 01 n∞ h ψ¯ −ψ¯ − D− ∇n∞ e = (1 + β) e − (1 − β) e , 2 L ψ¯

(A4)

(A5)

where

n ¯ 01 =

L∇n∞ . n∞

(A6)

Finally we can write: ¯ u (n− − n+ ) = 2n∞ u sinh (ψ).

(A7)

Using eqs.(A4,A5,A7), we can finally obtain from eq.(A2) as: Rh ¯ 4u sinh (ψ)dy kB T −h R e (D+ + D− ) h (1 + β) e−ψ¯ + (1 − β) eψ¯ dy −h i Rh 0 h −ψ¯ ψ¯ ¯ 1 (1 + β1 ) e − (1 − β1 ) e dy kB T −h n + ⇒ Rh −ψ¯ + (1 − β) eψ¯ dy eL (1 + β) e −h R1 ¯ 4¯ u sinh (ψ)dy E −1 = P eR 1 −ψ¯ + (1 − β) eψ¯ d¯ E0 (1 + β) e y −1 i R1 0 h ¯ ¯ n ¯ (1 + β) e−ψ − (1 − β) eψ d¯ y −1 1 + R1 , −ψ¯ + (1 − β) eψ¯ d¯ (1 + β) e y −1

E=

E¯ =

where E0 =

kB T , eL

u¯ =

u , U∗

P e = U ∗ L/ (D+ + D− ).

28


(A8)

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Appendix B: Derivation of eq.(16) For the bottom half of the nanochannel, the Navier Stokes equation for the x-momentum for the location within the PE brushes can be expressed as [using eq.(14) to express p so that ∂p ¯ − 1 + 2kB T n∞ sinh (ψ)∇ ¯ ψ¯ ]: = 2k T ∇n cosh ( ψ) B ∞ ∂x d2 u u ∂p η 2 = + η − e (n+ − n− ) (E − ∇ψ) dy ∂x kd

f or − h ≤ y ≤ −h + d ⇒

d2 u ¯ − 1 + 2kB T n∞ sinh (ψ)∇ ¯ ψ¯ + η u = 2kB T ∇n∞ cosh (ψ) 2 dy kd ¯ − 2kB T n∞ sinh (ψ)∇ ¯ ψ¯ f or h ≤ y ≤ −h + d ⇒ +2en∞ sinh (ψ)E η

n∞ L∇n∞ ηU ∗ ηU ∗ d2 u¯ ¯ = 2k T cosh ( ψ) − 1 + u¯ B h2 d¯ y2 L n∞ kd n∞ ¯ E¯ f or − 1 ≤ y¯ ≤ −1 + d¯ ⇒ +2eE0 L sinh (ψ) L ( R1 ¯ 0 4¯ u sinh (ψ)dy d2 u¯ −1 ¯ − 1 + α2 u¯ + A sinh (ψ) ¯ P eR = A n ¯ cosh ( ψ) 1 1 d¯ y2 (1 + β) e−ψ¯ + (1 − β) eψ¯ d¯ y −1 h i R1 0 ) ¯ ¯ n ¯ (1 + β) e−ψ − (1 − β) eψ d¯ y −1 1 ¯ + R1 f or − 1 ≤ y¯ ≤ −1 + d. ¯ ¯ − ψ ψ (1 + β) e + (1 − β) e d¯ y −1 (B1)

Of course, by the same procedure as employed above, we can show:

η

d2 u = dy 2 d2 u¯ = d¯ y2

∂p u + η − e (n+ − n− ) (E − ∇ψ) f or − h + d ≤ y ≤ 0 ⇒ ∂x kd ( R1 ¯ 0 4¯ u sinh (ψ)dy −1 ¯ − 1 + A sinh (ψ) ¯ P eR A n ¯ 1 cosh (ψ) 1 −ψ¯ + (1 − β) eψ¯ d¯ (1 + β) e y i ) −1 R1 0 h ¯ ¯ n ¯ (1 + β) e−ψ − (1 − β) eψ d¯ y −1 1 + R1 f or − 1 + d¯ ≤ y¯ ≤ 0. −ψ¯ + (1 − β) eψ¯ d¯ (1 + β) e y −1 (B2)

29



Notes: The authors declare no competing financial interest.

References (1) Gao, W.; Chan, J. M.; Farokhzad, O. C. pH-Responsive Nanoparticles for Drug Delivery. Mol. Pharmaceutics 2010, 7, 1913–1920. (2) Meng, F.; Zhong, Y.; Chen, R.; Deng, C.; Zhong, Z. pH-Sensitive Polymeric Nanoparticles for Tumor-Targeting Doxorubicin Delivery: Concept and Recent Advances. Nanomedicine 2014, 9, 487–499. (3) Matricardi, P.; Meo, C. D.; Coviello, T.; Alhaique, F. Recent Advances and Perspectives on Coated Alginate Microspheres for Modified Drug Delivery. Expert Opin. Drug Delivery 2008, 5, 417–425. (4) Barati, B.; Johnson, S. J.; McCool, S.; Green, D.W.; Wilhite, G. P.; Liang, J.-T. Polyelectrolyte Complex Nanoparticles for Protection and Delayed Release of Enzymes in Alkaline pH and at Elevated Temperature During Hydraulic Fracturing of Oil Wells. J. Appl.Polym. Sci. 2012, 126, 587–592. (5) Liu, G.; Cai, M.; Wang, X.; Zhou, F.; Liu, W. Core–Shell–Corona-Structured Polyelectrolyte Brushes-Grafting Magnetic Nanoparticles for Water Harvesting.ACS Appl. Mater. Interfaces 2014, 6, 11625–11632. (6) Siwy, Z.; Trofin, L.; Kohli, P.; Baker, L. A.; Trautmann, C.; Martin, C. R. Protein Biosensors Based on Biofunctionalized Conical Gold Nanotubes. J. Am. Chem. Soc. 2005, 127, 5000–5001. (7) Ali, M.; Yameen, B.; Neumann, R.; Ensinger, W.; Knoll, W.; Azzaroni, O. Biosensing and Supramolecular Bioconjugation in Single Conical Polymer Nanochannels. Facile

30


Page 30 of 39

Page 31 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Incorporation of Biorecognition Elements into Nanoconfined Geometries. J. Am. Chem. Soc. 2008, 130, 16351–16357. (8) Xie, G.; Tian, W.; Wen, L.; Xiao, K.; Zhang, Z.; Liu, Q.; Hou, G.; Li, P.; Tian, Y.; Jiang, L. Chiral Recognition of L-Tryptophan with Beta-Cyclodextrin-Modified Biomimetic Single Nanochannel. Chem. Commun. 2015, 51, 3135–3138. (9) Ali, M.; Tahir, M. N.; Siwy, Z.; Neumann, R.; Tremel, W.; Ensinger, W. Hydrogen Peroxide Sensing with Horseradish Peroxidase-Modified Polymer Single Conical Nanochannels. Anal. Chem. 2011, 83, 1673–1680. (10) Xia, F.; Guo, W.; Mao, Y.; Hou, X.; Xue, J.; Xia, H.; Wang, L.; Song, Y.; Ji, H.; Ouyang, Q.; Wang, Y.; Jiang, L. Gating of Single Synthetic Nanopores by ProtonDriven DNA Molecular Motors. J. Am. Chem. Soc. 2008, 130, 8345–8350. (11) Yameen, B.; Ali, M.; Neumann, R.; Ensinger, W.; Knoll, W.; Azzaroni, O. Single Conical Nanopores Displaying pH-Tunable Rectifying Characteristics. Manipulating Ionic Transport With Zwitterionic Polymer Brushes. J. Am. Chem. Soc. 2009, 131, 2070–2071. (12) Ali, M.; Yameen, B.; Cervera, J.; Ramorez, P.; Neumann, R.; Ensinger, W.; Knoll, W.; Azzaroni, O. Layer-By-Layer Assembly of Polyelectrolytes into Ionic Current Rectifying Solid-State Nanopores: Insights from Theory and Experiment. J. Am. Chem. Soc. 2010, 132, 8338–8348. (13) Zhulina, Y. B.; Pryamitsyn, V. A.; Borisov, O.V. Structure and Conformational Transitions in Grafted Polymer Chain Layers. A New Theory. Pol. Sci. U.S.S.R. 1989, 31, 205–216. (14) Zhulina, Y. B.; Borisov, O. V. Structure and Interaction of Weakly Charged Polyelectrolyte Brushes: Self-Consistent Field Theory. J. Chem. Phys. 1997, 107, 5952–5967.

31



(15) Zhulina, Y. B.; Borisov, O. V. PoissonBoltzmann Theory of pH-Sensitive (Annealing) Polyelectrolyte Brush. Langmuir 2011, 27, 10615–10633. (16) Alexander, S. Polymer Adsorption on Small Spheres. A Scaling Approach. J. Phys. 1977, 38, 977–981. (17) de Gennes, P.-G. Scaling Theory of Polymer Adsorption. J. Phys. 1976, 37, 1445–1452. (18) de Gennes, P.-G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069–1075. (19) Milner, S. T. Compressing Polymer Brushes: a Quantitative Comparison of Theory and Experiment. Europhys. Lett. 1988, 7, 695. (20) Milner, S. T. Polymer Brushes. Science 1991, 251, 905–914. (21) Milner, S. T.; Witten, T. A.; Cates, M. E. Theory of the Grafted Polymer Brush. Macromolecules 1988, 21, 2610–2619. (22) Chanda, S.; Sinha, S.; Das, S. Streaming Potential and Electroviscous Effects in Soft Nanochannels: Towards Designing More Efficient Nanofluidic Electrochemomechanical Energy Converters. Soft Matter 2014, 10, 7558–7568. (23) Chen, G.; Das, S. Electroosmotic Transport in Polyelectrolyte-Grafted Nanochannels with pH-Dependent Charge Density. J. Appl. Phys. 2015, 117, 185304. (24) Chen, G.; Das, S. Streaming Potential and Electroviscous Effects in Soft Nanochannels Beyond DebyeHckel Linearization. J. Colloid Interface Sci. 2015, 445, 357–363. (25) Poddar, A.; Maity, D.; Bandopadhyay, A.; Chakraborty, S. Electrokinetics in Polyelectrolyte Grafted Nanofluidic Channels Modulated by the Ion Partitioning Effect. Soft Matter 2016, 12, 5968–5978.

32


Page 32 of 39

Page 33 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(26) Jian, Y.; Li, F.; Liu, Y.; Chang, L.; Liu, Q.; Yang, L. Electrokinetic Energy Conversion Efficiency of Viscoelastic Fluids in a Polyelectrolyte-Grafted Nanochannel. Colloid. Surf. B 2017, 156, 405–413. (27) Milne, Z.; Yeh, L. H.; Chou, T. H.; Qian, S. Tunable Donnan Potential and Electrokinetic Flow in a Biomimetic Gated Nanochannel with pH-Regulated Polyelectrolyte Brushes. J. Phys. Chem. C 2015, 118, 19806–19813. (28) Yeh, L.-H.; Zhang, M.; Hu, N.; Joo, S. W.; Qian, S.; Hsu, J-P. Electrokinetic Ion and Fluid Transport in Nanopores Functionalized by Polyelectrolyte Brushes. Nanoscale 2012, 4, 5169–5177. (29) Yeh, L.-H.; Zhang, M.; Hu, N.; Joo, S. W.; Qian, S.; Hsu, J-P. Controlling pH-Regulated Bionanoparticles Translocation Through Nanopores with Polyelectrolyte Brushes. Anal. Chem. 2012, 84, 9615-9622. (30) Benson, L.; Yeh, L.-H.; Chou, T.-H.; Qian, S. Field Effect Regulation of Donnan Potential and Electrokinetic Flow in a Functionalized Soft Nanochannel. Soft Matter 2013, 9, 9767–9773. (31) Zhou, C.; Mei, L.; Su, Y-S.; Yeh, L-H.; Zhang, X.; Qian, S. Gated Ion Transport in a Soft Nanochannel with Biomimetic Polyelectrolyte Brush Layers. Sens. Actuat. B 2016, 229, 305–314. (32) Li, F.; Jian, Y.; Chang, L.; Zhao, G.; Yang, L. Alternating Current Electroosmotic Flow in Polyelectrolyte-Grafted Nanochannel. Colloid. Surf. B 2016, 147, 234–241. (33) Cao, Q.; You, H. Electroosmotic Flow in Mixed Polymer Brush-Grafted Nanochannels. Polymers 2016, 8, 438. (34) Li, F.; Jian, Y.; Xie, Z.; Liu, Y.; Liu, Q. Transient Alternating Current Electroosmotic

33



Flow of a Jeffrey Fluid through a Polyelectrolyte-Grafted Nanochannel. RSC Adv. 2017, 7, 782–790. (35) Chen, G.; Das, S. Massively Enhanced Electroosmotic Transport in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes. J. Phys. Chem. B 2017, 121, 31303141. (36) Tsori, Y.; Andelman, D.; Joanny, J.-F. Interfacial Instability of Charged-End-Group Polymer Brushes. Europhys. Lett. 2008, 82, 46001. (37) Chen, G.; Das, S. Anomalous ShrinkingSwelling of Nanoconfined End-Charged Polyelectrolyte Brushes: Interplay of Confinement and Electrostatic Effects. J. Phys. Chem. B 2016, 120, 6848-6857. (38) Chen, G.; Das, S. Thermodynamics, Electrostatics, and Ionic Current in Nanochannels Grafted with pHResponsive EndCharged Polyelectrolyte Brushes. Electrophoresis 2017, 38, 720-729. (39) Sparreboom, W.; van den Berg, A.; Eijkel, J. C. T. Principles and Applications of Nanofluidic Transport. Nature Nanotech. 2009, 4, 713. (40) Dukhin, S. S.; Derjaguin, B. V. Electrokinetic Phenomena, in Surface and Colloid Science, Vol. 7, E. Matijevic. Wiley, New York 1974. (41) Prieve, D.C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. Motion of a Particle Generated by Chemical Gradients. Part 2. Electrolytes. J. Fluid Mech. 1984, 148, 247–269. (42) Anderson, J. L. Colloid Transport by Interfacial Forces. Annu. Rev. Fluid Mech. 1989, 21, 61–99. (43) Keh, H. J. Diffusiophoresis of Charged Particles and Diffusioosmosis of Electrolyte Solutions. Curr. Opin. Colloid Interface Sci. 2016, 24, 13–22.

34


Page 34 of 39

Page 35 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(44) Keh, H. J.; Ma, H. C. Diffusioosmosis of Electrolyte Solutions along a Charged Plane Wall. Langmuir 2005, 21, 5461–5467. (45) Hsu, L. Y.; Keh, H. J. Diffusioosmosis of Electrolyte Solutions around a Circular Cylinder at Arbitrary Zeta Potential and Double-Layer Thickness. Ind. Eng. Chem. Res. 2008, 48, 2443–2450. (46) Ma, H. C.; Keh, H. J. Diffusioosmosis of Electrolyte Solutions in a Fine Capillary Slit. J. Colloid Interface Sci. 2006, 298, 476–486. (47) Ma, H. C.; Keh, H. J. Diffusioosmosis of Electrolyte Solutions in a Capillary Slit with Adsorbed Polyelectrolyte Layers. J. Colloid Interface Sci. 2007, 313, 686–696. (48) Qian, S.; Das, B.; Luo, X. Diffusioosmotic Flows in Slit Nanochannels. J. Colloid Interface Sci. 2007, 315, 721–730. (49) Keh, H. J.; Ma, H. C. Diffusioosmosis of Electrolyte Solutions in Fine Capillaries. Colloid. Surf. A 2004, 233, 87–95. (50) Hoshyargar, V.; Ashrazadeh, S. N.; Sadeghi, A. Drastic Alteration of Diffusioosmosis due to Steric Effects. Phys. Chem. Chem. Phys. 2015, 17, 29193–29200. (51) Hoshyargar, V.; Ashrazadeh, S. N.; Sadeghi, A. Diffusioosmotic Flow in Rectangular Microchannels. Electrophoresis 2016, 37, 809–817. (52) Keh, H. J.; Ma, H. C. Diffusioosmosis of Electrolyte Solutions in a Fine Capillary Tube. Langmuir 2007, 23, 2879–2886. (53) Cho, I.; Kim, W.; Kim, J.; Kim, H-Y.; Lee, H.; Kim, S. J. Non-Negligible DiffusioOsmosis Inside an Ion Concentration Polarization Layer. Phys. Rev. Lett. 2016, 116, 254501. (54) Keh, H. J.; Ma, H. C. Diffusioosmosis of Electrolyte Solutions in Fibrous Porous Media. Microfluid. Nanofluid. 2008, 5, 347–356. 35



(55) Anderson, J. L.; Lowell, M. E.; Prieve, D. C. Motion of a Particle Generated by Chemical Gradients Part 1. Non-electrolytes. J. Fluid Mech. 1982, 117, 107–121. (56) Lee, C.; Cottin-Bizonne, C.; Fulcrand, R.; Joly, L.; Ybert, C. Nanoscale Dynamics Versus Surface Interactions: What Dictates Osmotic Transport?. J. Phys. Chem. Lett. 2017, 8, 478–483. (57) Shen, M.; Ye, F.; Liu, R.; Chen, K.; Yang, M.; Ripoll, M. Chemically Driven Fluid Transport in Long Microchannels. J. Phys. Chem. Lett. 2016, 145, 124119. (58) Marbach, S.; Yoshida, H.; Bocquet, L. Osmotic and Diffusio-Osmotic Flow Generation at High Solute Concentration. I. Mechanical Approaches. J. Chem. Phys. 2017, 146, 194701. (59) Yoshida, H.; Marbach, S.; Bocquet, L. Osmotic and Diffusio-Osmotic Flow Generation at High Solute Concentration. II. Molecular Dynamics Simulations. J. Chem. Phys. 2017, 146, 194702. (60) Ajdari, A.; Boccquet, L. Giant Amplification of Interfacially Driven Transport by Hydrodynamic Slip: Diffusio-Osmosis and Beyond. Phys. Rev. Lett. 2006, 96, 186102. (61) Lee, C.; Cottin-Bizonne, C.; Biance, A.-L.; Joseph, P.; Bocquet, L.; Ybert, C. Osmotic Flow through Fully Permeable Nanochannels. Phys. Rev. Lett. 2014, 112, 244501. (62) Ortiz-Riveraa, I.; Shumb, H.; Agrawal, A.; Sen, A.; Balazs, A. C. Convective Flow Reversal in Self-Powered Enzyme Micropumps. Proc. Natl. Acad. Sci. USA 2016, 113, 2585–2590. (63) Feldmann, D.; Maduar, S. R.; Santer, M.; Lomadze, N.; Vinogradova,O. I.; Santer, S. Manipulation of Small Particles at Solid Liquid Interface: Light Driven Diffusioosmosis. Sci. Rep. 2016, 6, 36443.

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Page 36 of 39

Page 37 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(64) Hoshyargar, V.; Ashrazadeh, S. N.; Sadeghi, A. Mass Transport Characteristics of Diffusioosmosis: Potential Applications for Liquid Phase Transportation and Separation. Phys. Fluid. 2017, 29, 012001. (65) Kim, Y. W.; Lobaskin, V.; Gutsche, C.; Kremer, F.; Pincus, P.; Netz, R. R. Nonlinear Response of Grafted Semiflexible Polymers in Shear Flow. Macromolecules 2009, 42, 3650–3655. (66) Pennathur, S.; Santiago, J. G. Electrokinetic Transport in Nanochannels. 2. Experiments. Anal. Chem. 2005, 77, 67826789. (67) Wynveen, A.; Likos, C. N. Interactions between Planar Stiff Polyelectrolyte Brushes. Phys. Rev. E 2009, 80, 010801. (68) Kilic, M. S.; Bazant, M. Z.; Ajdari, A. Steric Effects in the Dynamics of Electrolytes at Large Applied Voltages. I. Double-Layer Charging. Phys. Rev. E 2007, 75, 021502. (69) Chanda, S.; Das, S. Effect of Finite Ion Sizes in an Electrostatic Potential Distribution for a Charged Soft Surface in Contact with an Electrolyte Solution. Phys. Rev. E 2014, 89, 012307. (70) Andrews, J. E.; Das, S. Effect of Finite Ion Sizes in Electric Double Layer Mediated Interaction Force between two Soft Charged Plates. RSC Adv. 2015, 5, 46873–46880. (71) Misra, R. P.; Das, S.; Mitra, S. K. Electric Double Layer Force between Charged Surfaces: Effect of Solvent Polarization. J. Chem. Phys. 2013, 138, 114703. (72) Sinha, S.; Myers, L.; Das, S. Effect of Solvent Polarization on Electroosmotic Transport in a Nanofluidic Channel. Microfluid. Nanofluid. 2016, 20, 119. (73) Storey, B. D.; Bazant, M. Z. Effects of Electrostatic Correlations on Electrokinetic Phenomena. Phys. Rev. E 2016, 86, 056303.

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(74) Kar, A.; McEldrew, M.; Stout, R. F.; Mays, B. E.; Khair, A.; Velegol, D.; Gorski, C. A. Self-Generated Electrokinetic Fluid Flows during Pseudomorphic Mineral Replacement Reactions. Langmuir 2016, 32, 5233–5240. (75) Garg, A.; Cartier, C. A.; Bishop, K. J. M.; Velegol, D. Particle Zeta Potentials remain Finite in Saturated Salt Solutions. Langmuir 2016, 32, 11837–11844. (76) Das, S.; Chakraborty, S. Steric-Effect-Induced Enhancement of Electrical-Double-Layer Overlapping Phenomena.

Phys. Rev. E 2011, 84, 012501.

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TOC Graphic

∇n∞ > 0

uEOS uCOS

Einduced

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Ionic Diffusoosmosis in Nanochannels Grafted with End-Charged

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