Massively Enhanced Electroosmotic Transport in Nanochannels

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Massively Enhanced Electroosmotic Transport in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes Guang Chen, and Siddhartha Das J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b00493 • Publication Date (Web): 21 Mar 2017 Downloaded from http://pubs.acs.org on March 26, 2017

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The Journal of Physical Chemistry

Massively Enhanced Electroosmotic Transport in Nanochannels Grafted with End-charged Polyelectrolyte Brushes 1

Guang Chen and Siddhartha Das∗ Department of Mechanical Engineering, University of Maryland, College Park, MD-20742, USA E-mail: [email protected];Phone:+1-301-405-6633

2

Abstract

3

We establish that nanochannels grafted with pH-responsive, end-charged polyelec-

4

trolyte (PE) brushes demonstrate a massive augmentation in the strength of the elec-

5

troosmotic (EOS) transport in presence of an external electric field. This contradicts

6

the existing understanding that the EOS transport is severely retarded in channels

7

grafted with the PE brushes due to the brush-induced enhanced drag force. Our math-

8

ematical model, developed on the basis of the fact that the ion concentration polar-

9

ization (ICP) effect can be neglected, explains this enhancement of the EOS transport

10

by noting that the end-charged PE brushes demonstrate a unique ability of localizing

11

the electric double layer (or the EDL) or equivalently localizing the maximum charge

12

density of the electrolyte ions at the location of its end, i.e., away from the grafting

13

surface. Accordingly, the maximum EOS driving force on the liquid, which is propor-

14

tional to this charge density, can be maximum at a location far away from the wall.

15

As a consequence, the resulting local EOS velocity suffers very little retardation due

16

to the wall shear stress enabling such massive augmentation of the EOS transport.

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We anticipate that the present paper will unravel a completely new paradigm in the

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employment of functionalized interfaces in regulating the nanofluidic transport for a

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myriad of applications.

Page 2 of 38

4

Introduction

5

Functionalization of nanochannels by grafting them with charged polyelectrolyte (PE) molecules

6

in a fashion such that these molecules attain a brush like architecture 1–3 has received wide

7

attention for applications ranging from sensing of biomolecules and analytes 4–6 to developing

8

nanofluidic current rectifiers, 7,8 ionic diodes, 9,10 and flow valves. 11 It is a standard knowledge

9

that nanochannels and microchannels grafted with PE brushes will invariably lower the flow

10

strength (e.g., electric-field-driven electroosmotic or EOS flow) on account of an enhanced

11

drag on the flow by the polymer molecules. 12–14 Studies have proposed to use this flow-

12

strength reduction for designing EOS flow suppressors 13,14 that can be useful for a number

13

of applications (e.g., bio-analyte sensing, current rectification, etc.) where appropriate and

14

highly sensitive measurement of the ionic current becomes necessary.

15

In this study, we report our discovery of a completely opposite effect of the PE-brush-

16

grafted nanochannels on the EOS transport – we demonstrate that nanochannels grafted

17

with PE-brushes containing charges only at their non-grafted ends may actually experience

18

a massive augmentation of the strength of the EOS flow. These end-charged PE brushes are

19

characterized by density of their end charges – such brushes are typically synthesized by end-

20

functionalizing long polymer chains (e.g., poly-ethylene glycol or PEG chains) with charged

21

terminal groups 15,16 and has been used for a variety of applications such as disassembling

22

amyloid fibrils, 15 targeted pH-responsive drug delivery, 17 etc. In the present paper, therefore,

23

we consider a design where the EOS transport is studied in a nanochannel whose walls

24

are grafted with such end-charged PE brushes. We show that particularly for large salt

25

concentration, the EOS flow in a PE-brush-free nanochannel, having the same surface charge

26

density (as the charge density of the ends of the end-charged PE brushes) at its rigid walls, is 2


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1

significantly weaker than the strength of the EOS flow in this end-charged PE-brush-grafted

2

nanochannels (see Fig. 1). We attribute this highly non-intuitive effect to the fact that the

3

end-charged brushes, particularly for large salt concentration (or thin electric double layer

4

or EDL), allow a localization of the EDL or equivalently a localization of the maximum

5

charge density of the electrolyte ions (i.e., the maximum difference between the number

6

densities of counterions and coions forming the EDL) at a location that is far from the

7

grafting surface. As a consequence, the driving EOS body force on the liquid, which is

8

proportional to this charge density gradient, can be highest (or most concentrated) at a

9

location where the drag on the flow caused by the wall shear stress is very small. This, in

10

turn, ensures a massive augmentation of the EOS flow strength. For charged nanochannels

11

with no brushes, this maximum driving force (or equivalently, the maximum difference in

12

the charge density) is at the wall where the no-slip condition retards the flow. On the

13

other hand, for the PE brush with backbone charges, the body force is distributed along

14

the brush height and therefore gets dispersed and does not remain localized away from the

15

wall (unlike the end-charged PE brushes). As a consequence, impact of the EOS body force

16

in driving the flow is somewhat reduced in both brush-free nanochannels and nanochannels

17

grafted with backbone-charged brushes. This justifies this rather remarkable finding where

18

the EOS flow gets enhanced in nanochannels functionalized by end-charged PE brushes. In

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this context, it is useful to discuss the study by Yeh et al., 18 which showed that the EOS

20

transport in a nanopore is significantly augmented in case the entire membrane (containing

21

the nanopore) is end-grafted with PE brushes, as compared to the case where only the inner

22

walls of the nanopore are grafted with the PE brushes. Of course, Yeh et al. considers

23

a uniformly-charged PE brush without pH regulation. But the most important difference

24

of the findings of our paper with respect to the paper by Yeh et al. is that we show that

25

such velocity augmentation is possible in PE-brush grafted nanochannels as compared to

26

nanochannels that do not contain any brush at all and this is extremely significant given the

27

standard notion of velocity reduction due to the presence of the brushes.

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In addition to discovering this highly non-trivial behavior of the PE-brush-grafted nanochan-

2

nels in context of the EOS transport, the present paper provides the first model that accounts

3

for the thermodynamics and the configuration of the PE brushes in a mean field setting for

4

modelling the EOS flow in the PE-grafted-nanochannels. There have been a considerable

5

number of studies on the EOS transport in PE-grafted nanochannels ranging from probing

6

of the streaming current generation 12,19–21 and suppressed EOS transport 13,14 to quantifying

7

the influence of the Donnan potential. 22,23 However, in all of these studies the height of the

8

PE brush layer has been assumed to be constant with no dependence on the salt concentra-

9

tion or the pH. On the contrary, in this present study we account for the effect of the salt

10

concentration and the pH on the brush configuration by describing the brush height explicitly

11

as an interplay of the elastic, excluded volume, and electrostatic energies of the brush and

12

the EDL energy. As a consequence, we embark upon a more complete model where the EOS

13

transport depends not only on the EOS body force resulting from the ion concentrations

14

but also on the specificities (e.g., strength of the drag force, location of the end charge, etc.)

15

associated with the ion concentration regulated brush height. The second critical issue is the

16

description of the monomer distribution along the PE brush. While these monomers being

17

uncharged (for the present problem) do not contribute to the overall electrostatics, they

18

indeed dictate the drag coefficient that affects the EOS transport. Previous studies on the

19

fluid flow in the PE brush grafted channels have considered a uniform or a parabolic or a cu-

20

bic monomer distribution. 13,19–23 For the present study, we shall simply consider a uniform

21

monomer distribution that is equivalent to the consideration of the well-known monomer

22

step profile proposed by Alexander 24 and de Gennes. 25,26 Therefore, in summary the mean

23

field model developed here is coupled with the appropriate description of the fluid flow in

24

order to propose the nanofluidic-transport-enhancing capabilities of the PE-brush-grafted

25

nanochannels that can be potentially employed for engineering novel nanofluidic concepts.

4


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Figure 1: Schematic depicting the electroosmotic transport in (a) nanochannels free of PE brushes and (b) nanochannels grafted with end-charged PE brushes. The schematic categorically represents the enhancement of the strength of the EOS transport in nanochannels grafted with end-charged PE brushes under the condition that the charge density at the walls (of brush-free nanochannels) is identical to that of the end charges of the end-charged-brush grafted nanochannels.

5



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1

Brush Configuration: Thermodynamics

2

We consider an EOS transport in presence of an axial electric field in nanochannels (of height

3

2h) grafted with pH-responsive, end-charged PE brushes, as shown in Fig. 1. End-charged

4

brushes contain charges at their non-grafted ends. This end-charging depends on the pH-

5

dependent ionization of the functional group that ionizes to produce the end charge. For

6

example, consider end-charged PE brushes made up of PEG chains end-functionalized with a

7

sulphonic acid 15 or a carboxylic acid group 16 – such end-charge brushes will demonstrate pH-

8

dependent ionization and charging due to the pH-dependent charging of these end groups. In

9

this section, we shall discuss the thermodynamics that will provide the brush configuration

10

and the resulting EDL electrostatics. This model has already been discussed in details in

11

our previous papers; 27,28 therefore, we shall only provide the key ideas of the model for the

12

sake of the continuity.

13

We start by expressing the free energy (F ) of the PE-brush-surrounding-electrolyte system:

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

(1)

14

where FB is the total energy associated with the brush with FB,els , FB,EV , and FB,elec being

15

the elastic, excluded volume and electrostatic components of that energy, and FEDL is the

16

energy associated with the surrounding EDL. Here we consider an Alexander-de Gennes

17

model of “stepped” brush profile (i.e., the monomer concentration φ = 1 along the chain

18

and φ = 0 outside the chain) in order to quantify the elastic and the excluded volume energies

19

of the brush. Accordingly, FB,els 3 d2 , = kB T 2 Np a2k

(2)

ωNp2 σ FB,EV = , kB T d

(3)

20

21

where σ ∼ 1/ℓ2 is the grafting density (ℓ is the lateral separation between grafted PE

22

brushes), ak is the Kuhn length of the polymer, Np is the polymer size, kB T is the thermal 6


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(1−2χ) 3 ak 2

(where χ is the Flory exponent).

1

energy, d is the equilibrium brush height, and ω =

2

Finally, the electrostatic energy of the PE brush can be expressed as 27–29 1 (ψs σch,pH ) , σ

FB,elec =

(4)

3

where ψs = ψ (ye ) (here ye is the location of the charged end of the PE brush and ψ is the

4

electrostatic potential) and σch,pH = σch K ′ +nKa+ (ye ) is the charge density (in units of C/m2 ) of

′

a

H

5

the charged end of the PE-brush (see the appendix for the detailed derivation of this expres-

6

sion of σch,pH ). Here σch represents the maximum possible charge density, nH + = nH + (ye )

7

is the local hydrogen ion number density at the location of the end of the PE brush, and

8

Ka′ = 103 NA Kaa (where NA is the Avogadro number and Ka is the ionization constant of

9

the group located at the PE brush end whose ionization causes the charging of the brush

10

11

12

end). Eq.(4) has been derived by Tsori et al.; 29 we briefly re-derive it here for the sake of continuR ity. The electrostatic energy of a standard PE brush can be expressed as FB,elec = ρe ψd3 r

13

(where ρe is the charge density or charge per unit volume of the PE brush). Since for the

14

present case the brush is end-charged, we can write ρe = σch,pH δ(y − ye ) (where δ is the delta

15

function). Consequently, considering average area associated with each brush as ℓ2 or 1/σ,

16

we can obtain eq.(4) as the expression for the electrostatic energy of the end-charged PE

17

brush from the above expression of FB,elec .

18

19

This charge on the brush end will trigger an EDL, whose energy (for the nanochannel bottom

20

half) can be expressed as:

FEDL

1 = σ

Z

0

fEDL (ψ, ψ ′ , ni )dy. −h

7


(5)


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Here fEDL is the EDL energy per unit volume expressed as: 28 ε0 εr dψ 2 fEDL = − | | + eψ (n+ − n− + nH + − nOH − ) 2 dy n− n+ − 1 + n− ln −1 +kB T n+ ln n+,∞ n−,∞ nOH − nH + − 1 + nOH − ln −1 . +kB T nH + ln nH + ,∞ nOH − ,∞

(6)

2

In the right hand side of eq.(6), the first to the third terms respectively represent the self-

3

electrostatic energy of the EDL, the electrostatic energy due to the charges on the electrolyte

4

ions, and the mixing entropy of the ions. Here ni (i = ±, H + , OH − ) is the number density

5

of ion of type i, ni,∞ is the bulk number density of ion of type i, kB T is the thermal energy,

6

ψ is the EDL electrostatic potential, ǫ0 is the permittivity of free space, ǫr is the relative

7

permittivity of the medium (assumed identical for the media both inside and outside the PE

8

layer), and e is the electronic charge. Also the above equation assumes that the electrolyte

9

ions are monovalent and the non-Poisson-Boltzmann effects (e.g., finite ion size effect, sol-

10

vent polarization effect, and ion-ion correlation effects) have been neglected. Please note

11

that here we are considering a pH-responsive ionization and charging of the PE brush tip

12

– accordingly, we have considered the contributions of the hydrogen and the hydroxyl ions

13

in the EDL free energy. Also, similar to our previous studies, 27,28 we assume that we add

14

a salt of concentration c∞ and an acid (furnishing same anions as the electrolyte salt) of

15

concentration cH + ,∞ . Under these conditions, n+,∞ = n∞ (where n∞ = 103 NA c∞ , with NA

16

being the Avogadro number) and n−,∞ = n∞ + nH + ,∞ (where nH + ,∞ = 103 NA cH + ,∞ ).

17

18

Under these circumstances, we can now express the overall free energy of PE-brush-surrounding-

19

electrolyte system by using eqs.(2,3,4,5,6) in eq.(1): ωNp2 σ 1 3kB T d2 1 F = + k T + (ψs σch,pH ) + B 2 2 Np ak d σ σ

8

Z

0

fEDL (ψ, ψ ′ , ni )dy. −h


(7)

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1

In order to obtain the equilibrium, the minimization of this resulting free energy F [see

2

eq.(7)] will be carried out with respect to the variables ψ, n± , nH + , nOH − , and d.

3

4

Minimization of F with respect to ψ yields: d2 ψ e (−n+ + n− − nH + + nOH − ) = . dy 2 ε0 εr

5

(8)

On the other hand, the minimization of F with respect to ni (i = ±, H + , OH − ) will yield zi eψ , ni = ni,∞ exp − kB T

(9)

6

where zi is the valence of the ionic species i.

7

Of course, eqs.(8, 9) are valid both inside and outside the PE brushes and employment of

8

eq.(9) in eq.(8) will yield the well-known Poisson-Boltzmann equation in terms of the EDL

9

potential ψ.

10

The necessary boundary conditions for solving this equation in ψ are:

dψ dy

= 0, y=−h

dψ dy

= 0, y=0

(ψ)(y=−h+d)− = (ψ)(y=−h+d)+ .

(10)

11

Finally, we shall require a fourth condition that will connect the gradients of electrostatic

12

potentials inside and outside the brush in terms of the PE end charge density σch,pH . This

13

condition is obtained by minimizing the overall free energy F with respect to ψs (see our

14

previous paper 27 for the detailed procedure):

dψ dy

y=(−h+d)+

σch,pH δF = 0 ⇒ ψs′ = − ⇒ δψs ε0 εr σch,pH dψ . − =− dy y=(−h+d)− ε0 εr

9


(11)


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Page 10 of 38

1

Solution of eqs.(8,9) in presence of conditions expressed by eqs.(10,11) will provide ψ and

2

ni . This, in turn, when employed in eqs.(5,6) will help us quantify the overall EDL energy.

3

Of course, this overall EDL energy is the equilibrium EDL energy that can be employed in

4

eq.(7) in order to obtain the total energy F . Subsequently, we shall minimize this F [see

5

eq.(7)] with respect to d and obtain the equilibrium value of the brush height d.

6

Calculation of the EOS Transport

7

The main purpose of this paper is to study the EOS transport triggered in presence of an

8

externally applied axial electric field (of magnitude E) in such pH-responsive, end-charged

9

(with charge density σch,pH ) PE-brush-grafted nanochannels [see Fig. 1(b)] and demonstrate

10

that this EOS transport is significantly stronger than the EOS transport ensued in a brush-

11

free nanochannel with a wall charge density of σch,pH [see Fig. 1(a)]. This EOS flow in

12

either of the systems (with and without the wall-grafted PE brushes) is triggered due to

13

the interaction of the applied electric field E and the imbalance of the ionic number density

14

within the EDL. Considering the EOS flow to be steady, hydrodynamically fully-developed,

15

and uni-directional, we can express the governing equation dictating the flow as (considering

16

the nanochannel bottom half): d2 u η + e (n+ − n− + nH + − nOH − ) E − u = 0 [for − h ≤ y ≤ −h + d] , 2 dy k 2 du [for − h + d ≤ y ≤ 0] , η 2 + e(n+ − n− + nH + − nOH − )E = 0 dy η

17

18

(12)

where u is the uni-directional velocity, η is the dynamic viscosity of the liquid, e is the elec 2 d 2 tronic charge, ni is the number density of the ionic species i, and k = ak σa3 Np φ is the k

σa3k Np φ d

is the volume fraction of the PE brush layer. For the present study,

19

permeability and

20

we consider a stepped profile of the end-charged PE brush – hence we have φ = 1.

21

10


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1

Using eq.(9) and expressing the bulk number densities of the ions in a manner identical

2

to that described after eq.(6), we can express eq.(12) in dimensionless form as E¯ d2 u¯ ¯ + (1 + n ¯ −n ¯ +n ¯ ] + ,∞ ) exp ψ + ,∞ exp −ψ − ,∞ exp ψ − [− exp − ψ ¯ ¯ ¯ H H OH ¯2 d¯ y 2 2λ 2 2 ak σNp φ ¯ , u ¯ = 0 for − 1 ≤ y ¯ ≤ −1 + d − d¯ E¯ d2 u¯ ¯ + (1 + n ¯ −n ¯ +n ¯ ] + ,∞ ) exp ψ + ,∞ exp −ψ − ,∞ exp ψ − [− exp − ψ ¯ ¯ ¯ H H OH 2 2 ¯ d¯ y 2λ for − 1 + d¯ ≤ y¯ ≤ 0 . (13)

3

4

where ψ¯ = eψ/(kB T ), u¯ = u/u0 (u0 is the velocity scale), E¯ = E/E0 (u0 and E0 are p ¯ = λ/h (λ = ǫ0 ǫr kB T /(2n∞ e2 ) ¯ i,∞ = ni,∞ /n∞ , y¯ = y/h, λ connected as u0 = kBeT ε0 εηr E0 ), n

5

is the EDL thickness), and d¯ = d/h.

6

With the knowledge of the dimensionless EDL electrostatic potential ψ¯ (solution procedure

7

of ψ has been discussed in the previous subsection), eq.(13) is finally solved numerically in

8

the presence of the following dimensionless boundary conditions:

d¯ u d¯ y

= 0, (¯ u)y¯=−1 = 0, y¯=0

(¯ u)y¯=(−1+d)− = (¯ u)y¯=(−1+d)+ ,

d¯ u d¯ y

= y¯=(−1+d)−

d¯ u d¯ y

.

(14)

y¯=(−1+d)+

9

It is worthwhile to note here that our model does not consider the nanochannel ion

10

concentration polarization (ICP) effect, which refers to the simultaneous depletion and ac-

11

cumulation (of both coions and counterions) at axially separated locations (typically at the

12

entry and at the exit) in a nanochannel. ICP effect invariably arises when there is an axial

13

jump in the ion concentration. Therefore, one witnesses such ICP at the junction of mi-

14

crochannel and a nanochannel (such a nanochannel may or may not be grafted with PE

15

brushes), 18,30–32 for translocation of a DNA through a nanochannel or a nanopore (since the

16

DNA adds its own EDL distribution in a charged nanochannel), 33 etc. On the other hand,

11



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1

the present study considers a parallel-plate nanochannel with uniformly grafted end-charged

2

polyelectrolyte (PE) brushes. Therefore, unlike the examples discussed above there is no pos-

3

sibility of any axial variation/jump in the ion concentration – accordingly, we have neglected

4

the ICP effect in our modeling. Of course, there is a possibility of such an ICP effect at

5

the junction of this end-charged PE-brush grafted nanochannel with a possible microchannel

6

reservoir; but we consider the nanochannel to be sufficiently long so that such nanochannel

7

end-effects (or equivalently the ICP generated at the nanochannel ends) can be neglected in

8

our calculations.

9

Results

10

Variation of the brush height

11

In our recent papers, 27,28 we have provided a detailed analysis of the variation of the height

12

of the nanoconfined, end-charged, pH-responsive PE brushes as a function of degree of con-

13

finement (or equivalently, short and tall brushes), pH and salt concentration. We do not

14

repeat those results here, but rather simply summarize the key issues that will be needed to

15

explain the subsequent results on the variation of the EDL electrostatics, ion concentration,

16

and the EOS transport.

17

A key parameter that dictates the brush height is the degree of confinement, quantified

18

by whether d0 < h/2 (weak confinement) or d0 > h/2 (strong confinement). Here d0 refers to

19

the height of the uncharged polymer brushes having identical expressions of the elastic and

20

excluded volume energies as the end-charged PE brushes. For brushes with d0 > h/2, the

21

electrostatic energy contributes to the lowering of the brush height d as compared to d0 , while

22

for brushes with d0 < h/2, the electrostatic energy ensures that d > d0 . The electrostatic

23

effects are maximum for pH = 7, for which the ionization of the end-charge-producing group

24

is maximum. Consequently, at pH = 7 there is a maximum difference between d and d0 –

25

this difference ensures that for a given salt concentration d has the largest (smallest) value 12


Page 12 of 38

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1

for pH = 7 for brushes with d0 < h/2 (d0 > h/2). For such large pH values, increase in salt

2

concentration increases (decreases) the impact of the electrostatic free energy (in altering

3

the brush height) for small (large) values of salt concentration. Accordingly, we witness that

4

d first decreases (increases), attains a minimum (maximum), and then increases (decreases)

5

with the salt concentration for brushes with h > d0 > h/2 (d0 < h/2). It is worthwhile

6

to mention here that the height d0 of the uncharged brushes can be increased by either

7

increasing the grafting density σ or the polymer size Np [since d0 ∼ Np (a5k σ)

8

that d0 ∼ σ 1/3 and d0 ∼ Np , one will require a much larger (smaller) increase in σ (Np ) in

9

order to ensure a given increase in the brush height. Of course, as will be explained later, the

10

EOS transport in such end-charged PE brush grafted nanochannels will be severely affected

11

depending on whether d0 increased by increasing σ or by increasing Np .

12

EDL electrostatic potential and the ion number density distribution

13

In Fig. 2, we study the transverse variation of the magnitude of the dimensionless EDL

14

¯ = |eψ/(kB T )|] (top panel) and the number densities of the counelectrostatic potential [|ψ|

15

terions (n+ ) (middle panel) and coions (n− ) (bottom panel). Significance of this figure is

16

that it provides a critical understanding about the electrostatics of the problem, which is

17

central to correctly interpret the EOS flow fields. We study all the quantities (ψ¯ and n± )

18

for pH = 7 for five different nanochannel systems: a) nanochannels grafted with short-loose

19

brushes (Np = 2000, ℓ = 80 nm, d0 = 34 nm), b) nanochannels grafted with short-loose,

20

but slightly taller brushes (Np = 2800, ℓ = 80 nm, d0 = 49 nm), c) nanochannels grafted

21

with short-dense brushes (Np = 846, ℓ = 22 nm, d0 = 34 nm), d) nanochannels grafted with

22

tall-dense brushes (Np = 2000, ℓ = 22 nm, d0 = 82 nm), and (e) nanochannels grafted with

23

no brush. This variation of the configuration of the grafted brushes (or the complete absence

24

of the brushes) dictates the specific location of the EDL within the nanochannel. As will be

25

discussed later, this location of the EDL along with the EDL thickness (or equivalently, bulk

26

ionic concentration) play vital roles in dictating the overall EOS transport. For example, for 13


1/3

]. 27 Given


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

Figure 2: Variation of the magnitude of the dimensionless transverse EDL electrostatic ¯ (top panel), transverse counterion number density n+ (middle pannel), and potential |ψ| transverse coion number density n− (bottom pannel) as functions of the bulk salt concentrations for pH=7. Results are shown for a) nanochannels grafted with short-loose brushes (Np = 2000, ℓ = 80 nm, d0 = 34 nm), b) nanochannels grafted with short-loose, but slightly taller brushes (Np = 2800, ℓ = 80 nm, d0 = 49 nm), c) nanochannels grafted with shortdense brushes Np = 846, ℓ = 22 nm, d0 = 34 nm, d) nanochannels grafted with tall-dense brushes Np = 2000, ℓ = 22 nm, d0 = 82 nm, and (e) nanochannels grafted with no brush. Other parameters are σch = −0.0008 C/m2 (this is the charge density of the end charges of the brushes for nanochannels grafted with brushes; this is also the charge density of bare walls of nanochannels without the brushes), pKa = 4, h = 100 nm, χ = 0.4, ak = 1 nm, kB = 1.38 × 10−23 J/K, T = 300 K, e = 1.6 × 10−19 C, ǫ0 = 8.8 × 10−12 F/m, ǫr = 79.8.

1

the PE-brush-free nanochannel [see Fig. 2(e)], the EDL is obviously confined at the wall,

2

whereas for the nanochannels grafted with the PE-brushes [see Fig. 2(a-d)], the EDLs are lo-

3

cated at the brush tip. This can be understood by noting the location of the maximum value

4

¯ for all of these five cases. Variation in the salt concentration has profound influence of |ψ|

5

¯ for all the five cases. Smaller salt concentration on the magnitude and the distribution of |ψ|

6

¯ is implies a larger EDL thickness. As a consequence, the extent of the spatial decay of |ψ|

7

¯ even at locations further away from weak ensuring that there is a distinctly finite value of |ψ| 14


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1

¯ is highest. Consequently for weak salt concentration, one witnesses the locations where |ψ|

2

¯ at locations far away from the wall for the PE-brush-free nanochannels, as well as finite |ψ|

3

far away on either sides of the charged end of the PE brush for nanochannels grafted with

4

¯ ensures a loend-charged PE brushes. This specific salt-concentration-driven variation of |ψ|

5

calization of the maximum charge density and maximum EOS body force at the brush tip for

6

large salt concentration, whereas a dispersion (and hence non-localization) of the body force

7

away from the brush tip for small salt concentration. As will be illustrated subsequently,

8

such force localization for large salt concentration is key to attributing ascertaining EOS

9

¯ increases in magnitude for velocity for nanochannels with end-charged brushes. Finally, |ψ|

10

smaller salt concentration. It can be easily explained by noting that |ψ| ∼ σch,pH λ (where

11

λ is the EDL thickness), thereby justifying enhanced |ψ| for weaker salt concentration (i.e.,

12

enhanced λ).

13

¯ distribuNumber density distributions of the ions (n± ) are commensurate with this |ψ|

14

tion. Consequently, we witness (a) largest (smallest) values of the counterion (coion) number

15

densities at the wall (for PE-brush-free nanochannels) and at the location of the end charge

16

(for end charged PE brush grafted nanochannels) and (b) the ion concentrations demonstrat-

17

ing weakest (strongest) decays for smallest (largest) bulk electrolyte salt concentrations.

18

Electroosmotic velocity field

19

In Fig. 3, we show the transverse variation of the EOS velocity field for different salt

20

concentrations at pH = 7 for these five different nanochannel systems. This is the central

21

result of this paper. The key features of these EOS flow fields are as follow. Firstly, for

22

brush-free nanochannels [see Fig. 3(e)], the least magnitude of the EOS flow strength as

23

well as the flattest EOS velocity profile (or the profile that has closest resemblance to the

24

classical “plug”-shaped EOS flow profile) is witnessed for the largest salt concentration.

25

Also for this case, a decrease in the salt concentration leads to a progressive increase in the

26

magnitude of the EOS flow strength and a progressive decrease in the “plug”-like nature 15



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

Figure 3: Variation of the transverse electroosmotic velocities as functions of the bulk salt concentrations for pH=7. Results are shown for a) nanochannels grafted with short-loose brushes (Np = 2000, ℓ = 80 nm, d0 = 34 nm), b) nanochannels grafted with short-loose, but slightly taller brushes (Np = 2800, ℓ = 80 nm, d0 = 49 nm), c) nanochannels grafted with short-dense brushes (Np = 846, ℓ = 22 nm, d0 = 34 nm), d) nanochannels grafted with tall-dense brushes (Np = 2000, ℓ = 22 nm, d0 = 82 nm), and (e) nanochannels grafted with no brush. We represent the loose brushes as light green [see (a) and (b)] and dense brushes as pink [see (c) and (d)]. Other parameters for all the figures are identical to that in Fig. 2.

1

of the flow field. It is trivial to infer that the corresponding variation in ψ [see Fig. 2(e),

2

top panel], demonstrating the weakest value and the steepest variation for this largest salt

3

concentration, is responsible for this velocity profile. For nanochannels with short-loose

4

brushes (by “loose” we imply less dense) [see Fig. 3(a), for which Np = 2000, ℓ = 80 nm,

5

d0 = 34 nm], we witness, most remarkably, a significantly larger velocity (as compared to

6

the brush-free nanochannels) for large salt concentration. Additionally, we witness that the

7

decrease in the salt concentration first decreases and then increases the strength of the EOS

8

transport particularly in the vicinity of the nanochannel centreline. Further, a decrease

9

in the salt concentration (or equivalently, an increase in the EDL thickness) enforces the

10

velocity profile to suffer a large deviation from its “classical” plug shape. Obviously, the

11

most surprising of all the results is this remarkable enhancement (in particular, for large salt

12

concentration) in the EOS flow strength for nanochannels grafted with the PE brushes, as

13

compared to the EOS flow in brush-free nanochannels. This is completely contrary to the

14

classical understanding that the presence of the brushes lining the nanochannel walls will

15

invariably lower the flow speed on account of PE-induced enhanced drag. 12,13 For the end

16

charged brushes, the EDL is localized at the brush tip that is substantially away from the 16


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1

wall. For larger salt concentration, the EDL thickness being very small, this localization is

2

so strong that the EDL ceases to exist well before the wall. The driving force for the EOS

3

transport comes from the difference between the counterions (n+ ) and coion (n− ) number

4

densities – this difference is maximum within the EDL. Therefore for end-charged brushes,

5

in presence of large bulk salt concentration (i.e., small EDL), this driving force is primarily

6

localized at the location substantially away from wall. As a consequence, the liquid is

7

subjected to a maximum localized body force at a location where the wall induced retarding

8

shear stress is very minimal. This, in turn, ensures a significantly large influence of the

9

EOS body force, enforcing a large EOS velocity. Therefore, we can infer that this most

10

remarkable scenario where the presence of the end-charged PE brushes enhances the EOS

11

flow strength is triggered by the localization of the EDL and the corresponding localization

12

of the body force at a distance far away from the wall. It is clear from the above discussions

13

that the key to the development of this augmented EOS transport is the preferential location

14

of the maximum EOS body force away from the wall. This is quantified in Fig. 4, where

15

the electroosmotic body force (quantified by those f¯, corresponding to which f¯ > 0) and

16

the drag force (quantified by those f¯, corresponding to which f¯ < 0) for each of the five

17

different nanochannel systems and for different salt concentrations are elucidated. One can

18

straightaway compare the case of no brush [see Fig. 4(e)] and the case of short brushes

19

[see Fig. 4(a)] – the results establish that the localization of the maximum EOS body

20

force is substantially away from the wall for the nanochannel with brushes as compared

21

to the nanochannel without the brushes. Also, it is trivial to identify from Fig. 4 that

22

the maximum value of the EOS body force occurs for large salt concentration for both

23

nanochannels with and without the brushes. Of course, for small salt concentrations the

24

EDL thickness becomes large – as a consequence, the EOS body force is no longer localized

25

and get dispersed ensuring that a part of the EOS body force is also present near the wall

26

(as the EDLs extend away from the brush tip). Therefore, for such a salt concentration

27

the impact of the localization of the EOS body force at the brush tip gets severely reduced,

17



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

1

ensuring minimal (if any) enhancement of the EOS flow speed in brush-grafted nanochannels

2

as compared to the brush-free nanochannels.

Figure 4: Variation of the dimensionless per unit volume forces (f¯) in the transverse direction as functions of the bulk salt concentrations for pH=7. Here f¯ > 0 represents the dimensionless per unit volume EOS body force, while f¯ < 0 represents the retarding PEbrush-layer-induced per unit volume drag force. Results are shown for a) nanochannels grafted with short-loose brushes (Np = 2000, ℓ = 80 nm, d0 = 34 nm), b) nanochannels grafted with short-loose, but slightly taller brushes (Np = 2800, ℓ = 80 nm, d0 = 49 nm), c) nanochannels grafted with short-dense brushes (Np = 846, ℓ = 22 nm, d0 = 34 nm), d) nanochannels grafted with tall-dense brushes (Np = 2000, ℓ = 22 nm, d0 = 82 nm), and (e) nanochannels grafted with no brush. Other parameters are identical to those in Fig. 2.

3

A very important question in this context is: For a given E and σch , does the EOS

4

transport depend solely on the value of the brush height or on the parameters that dictate the

5

brush height? In order to resolve this issue, we probe the EOS transport for the nanochannel

6

system grafted with end-charged PE brushes of same d0 as that of Fig 3(a) (i.e., d0 = 34nm)

7

but having a larger grafting density (ℓ = 22 nm) and a smaller Np (Np = 846) (i.e., we study

8

the case of nanochannels grafted with short and dense brushes) [see Fig. 3(c)]. Here, we

9

clearly see that the velocity for the case with larger σ is distinctly smaller (particularly for a

10

large salt concentration), on account of the larger drag force (since drag coefficient ∼ σ 2/3 ).

11

This becomes plainly evident from the comparison of the drag forces (represented by those

12

f¯ values for which f¯ < 0) for these two cases [please see Figs. 4(a) and 4(c)].

13

Finally, we are interested to probe the effect of the brush height in the augmentation of

14

the EOS flow strength. For that purpose, we probe the EOS flow behavior for the brushes

15

with heights greater than d0 = 34 nm. We consider a taller brush (as compared to one 18


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1

where d0 = 34 nm) with a large σ (Np = 2000, ℓ = 22 nm, d0 = 84 nm) [see Fig. 3(d)

2

for the corresponding EOS flow profile] and another taller brush (as compared to one where

3

d0 = 34 nm) with a small σ (Np = 2800, ℓ = 80 nm, d0 = 49 nm) [see Fig. 3(b) for the

4

corresponding EOS flow profile]. For the taller brush with large σ, the influence of larger σ

5

in augmenting the drag overwhelms the effect associated with the localization of the EOS

6

body force at the brush tip, resulting in a much smaller velocity (particularly at large salt

7

concentration) [please compare the EOS flow profiles of Figs.3(a) and (d)]. Such profound

8

influence of the drag force for taller and more dense brushes can be confirmed by noting

9

the significantly large drag force in Fig. 4(d). On the other hand, for taller but less dense

10

brushes [the corresponding EOS flow profile is shown in Fig. 3(b)], it is the localization

11

of the EDL at the brush tip that governs the flow, leading to an EOS flow strength that

12

is more (particularly at large salt concentration) than that for nanochannels grafted with

13

shorter and less dense brushes [the corresponding EOS flow profile is shown in Fig. 3(a)].

14

Of course, this issue gets substantiated when the corresponding forces are compared [i.e.,

15

compare the forces shown in Fig. 4(a) and (b)] – the drag force is similar for both these

16

cases, but the maximum of the EOS body is located at a greater distance from the wall for

17

the taller brush.

18

For all the five nanochannel systems studied here, the smallness of the EDL thickness for

19

large salt concentrations will ensure that the velocity profile remains very flat (or “plug”-

20

shaped) outside the brush, while a larger EDL (or equivalently, a larger spatial distribution

21

of the charge density gradient) for smaller salt concentration ensures a velocity profile far

22

deviated from the “plug”-shaped profiles.

23

pH-responsive electrostatics and EOS velocity fields

24

¯ u¯, f¯, In Figs. 5,6,7,8, we show the effect of the variation of pH and salt concentration on |ψ|,

25

n+ ,n− , and nH + for the four different kinds of PE-grafted nanochannel systems. For small

26

pH, the concentration of H + ions in the electrolyte solution is so high that the ionization of 19



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

¯ (magnitude of dimensionless electrostatic potential), u¯ Figure 5: Transverse variation of |ψ| (dimensionless EOS velocity), f¯ (f¯ > 0 represents the dimensionless per unit volume EOS body force, while f¯ < 0 represents the retarding PE-brush-layer-induced per unit volume drag force), n+ (counterion number density), n− (coion number density), and nH + (number density of hydrogen ions). Results are shown for nanochannels grafted with short-loose brushes (Np = 2000, ℓ = 80 nm, d0 = 34 nm) for (a) pH=3, (b) pH=4, and (c) pH=5. Other parameters are identical to those in Fig. 2.

20


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¯ (magnitude of dimensionless electrostatic potential), u¯ Figure 6: Transverse variation of |ψ| (dimensionless EOS velocity), f¯ (f¯ > 0 represents the dimensionless per unit volume EOS body force, while f¯ < 0 represents the retarding PE-brush-layer-induced per unit volume drag force), n+ (counterion number density), n− (coion number density), and nH + (number density of hydrogen ions). Results are shown for nanochannels grafted with short-loose, but slightly taller brushes (Np = 2800, ℓ = 80 nm, d0 = 49 nm) for (a) pH=3, (b) pH=4, and (c) pH=5. Other parameters are identical to those in Fig. 2.

21



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

¯ (magnitude of dimensionless electrostatic potential), u¯ Figure 7: Transverse variation of |ψ| (dimensionless EOS velocity), f¯ (f¯ > 0 represents the dimensionless per unit volume EOS body force, while f¯ < 0 represents the retarding PE-brush-layer-induced per unit volume drag force), n+ (counterion number density), n− (coion number density), and nH + (number density of hydrogen ions). Results are shown for nanochannels grafted with short-dense brushes (Np = 846, ℓ = 22 nm, d0 = 34 nm) for (a) pH=3, (b) pH=4, and (c) pH=5. Other parameters are identical to those in Fig. 2.

22


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¯ (magnitude of dimensionless electrostatic potential), Figure 8: Transverse variation of |ψ| u¯ (dimensionless EOS velocity), f¯ (f¯ > 0 represents the dimensionless per unit volume EOS body force, while f¯ < 0 represents the retarding PE-brush-layer-induced per unit volume drag force), n+ (counterion number density), n− (coion number density), and nH + (number density of hydrogen ions). Results are shown for nanochannels grafted with talldense brushes (Np = 2000, ℓ = 22 nm, d0 = 82 nm) for (a) pH=3, (b) pH=4, and (c) pH=5. Other parameters are identical to those in Fig. 2.

23



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

1

the end-charge-producing group (this ionization produces the H + ions) is severely retarded.

2

Accordingly, the magnitude of the end-charge is substantially lowered. This is reflected in

3

¯ and |¯ the substantially reduced value of |ψ| u| for cases with small pH (pH = 3, 4) for all the

4

four different PE-brush-grafted nanochannel systems. On the other hand, for nanochannels

5

grafted with both short and tall as well as loose and dense brushes, only for pH ≥ 5 we

6

¯ and u¯ that have been witness the salt concentration dependent qualitative variation in |ψ|

7

described in Figs. 2 and 3.

Figure 9: Phase space for iso-volume-flow-rate. The lines with numbers denote the ℓ − Np combinations that ensure identical flow rates with the numbers on the lines denoting the dimensionless flow rate, defined in eq.(15). The shaded region shows the ℓ−Np combinations that ensure that the grafted PE molecules form brushes whose heights are less than the nanochannel half height. Below this shaded zone the brushes are taller than the nanochannel half height, while above the shaded zone the grafting is so weak that the grafted polymer molecules no longer form the brushes.Results here are shown for c∞ = 10−2 M , σch = −8 × 10−4 C/m2 , and pH = 7.

24


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1

Phase space for the volume flow rate ratio in nanochannels grafted

2

with end-charged PE brushes

3

The interplay of the grafting density and the monomer size in dictating the strength of the

4

overall EOS transport in nanochannels grafted with the end-charge brushes is summarized

5

in the iso-volume-flow rate (or iso-Q) phase-space plot shown in Fig. 9. This phase space

6

is provided for large salt concentration (c∞ = 10−2 M ) and large pH (pH = 7) (i.e., the

7

conditions corresponding to which the EOS transport in PE-brush-grafted nanochannels is

8

most augmented). In this figure, the lines with numbers denote the ℓ − Np combinations

9

that ensure identical dimensionless volume flow rate ratios, expressed as: R1

u¯ d¯ y ¯ = R −1 B , Q 1 u¯ d¯ y −1 N B

(15)

10

where u¯B and u¯N B are the dimensionless EOS velocity profiles in nanochannels with and

11

without the brushes, respectively. Of course, in Fig. 9, we also show (see the shaded region)

12

the ℓ − Np combination that ensure that the grafted polymer molecules form brushes that

13

have height less than the nanochannel half height (so that there is no interpenetration of the

14

brushes grafted on the two opposing walls of the nanochannel). Therefore, basically we focus

15

on the iso-Q lines that are within this shaded zone. The first and the most important finding

16

is the massive increase in the volume flow rate for the nanochannels with brushes, evident

17

by the large magnitudes of the volume flow rate ratios [see Fig. 9 and eq.(15). As discussed

18

previously, combination of ℓ and Np dictate the brush height as well as the drag force and

19

the interplay of these two factors eventually decide EOS flow profile. This flow profile along

20

with the space available for the flow (i.e., the transverse section of the nanochannel not

21

occupied by the brushes) dictate the consequent volume flow rate. For very small ℓ/ak (or

22

equivalently for a very large grafting density) at a given Np (Np ≥ 2000) the brush height

23

is very large so that the major part of the nanochannel is blocked by the brush, causing a

24

relatively small volume flow rate. Of course, for such a large grafting density, the brush25



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

1

induced drag force is also substantially high, which contributes to ensure a very small flow

2

rate. On a subsequent increase of ℓ/ak (or equivalently, a decrease in the grafting density)

3

three things happen: first a slit-like gap opens due to the reduction of the brush height,

4

second the brush-induced drag reduces, and third the effect of the localization of the EOS

5

body force (on account of the localization of the EDL at the brush end, particularly for

6

large salt concentration as has been studied in Fig. 9) away from the wall starts becoming

7

important. All these three factors contribute to enhancing the flow rate. How these three

8

factors pan out and how they interplay to influence the final flow rate with a further increase

9

of ℓ/ak decides the above iso-Q phase diagram. A further increase in ℓ/ak leads to a greater

10

gap opening and a lesser effect of the localization of the EOS body force away from the

11

wall (both caused by a reduction in the brush height) and a further weakening of the drag

12

force – the consequence is a further increase in the flow rate. Here the central factor is this

13

localization of the body force away from the wall – while an increase in ℓ/ak and a consequent

14

lowering of the brush height makes this localization closer to the wall, its influence coupled

15

with an augmented flow-passage area and a substantially reduced (or even non-existent)

16

brush-induced drag ensures this augmented flow rate. However, beyond a certain value of

17

ℓ/ak , the brush height becomes so small that this localization effect is virtually non-existent

18

and this, in turn, ensures a net reduction in the flow rate. In this way, we can explain why

19

the flow rate first increases and then decreases with an increase in ℓ/ak for a given Np .

20

Discussions

21

Impact of the choice of the monomer profile

22

Quantification of the monomer distribution along the brush length is critical for quantifying

23

the drag coefficient used in the modelling of the EOS transport. In the present study, we have

24

considered a uniform monomer profile (or equivalently, a stepped monomer profile where φ =

25

1) as proposed by Alexander 24 and de Gennes 25,26 and subsequently used in a large number 26


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1

of studies on electrokinetic transport in PE-grafted nanochannels. 12,19,21,23 Of course, a more

2

complete model would have necessitated a more appropriate form of the monomer profile that

3

is self-consistent with the presence of PE electrostatic contribution due to the presence of the

4

charge only at the PE tip. Mishra et al. 34 and Zhulina and Borisov 35 respectively proposed

5

a parabolic and an exponentially decaying monomer distribution for the case where the PE

6

brushes contained charges along its entire backbone. However, for the present case where the

7

PE electrostatics is due to the presence of only PE end charge, monomer distribution remains

8

unknown. While developing this present model on EOS transport by first estimating the

9

appropriate monomer distribution for the end-charged PE brushes is indeed a possibility, we

10

refrain from doing that and provide our results for the simplistic case of uniform monomer

11

profile. The reason is that we believe that the consideration of a non-uniform, decaying

12

(along the brush height) monomer profile, if at all, will only strengthen the central message

13

of the present paper, which is the enhancement of the EOS flow by end-charged PE brushes.

14

This stems from the fact that for such a monomer distribution, the monomer density will

15

be least near the brush tip and accordingly the PE-induced drag force will be much smaller

16

(as compared to the case of φ = 1) near the brush tip. Consequently, the retardation to the

17

flow near the PE-tip (i.e., where the EOS body force is localized) due to the wall shear and

18

the PE drag will reduce even further, thereby ensuring an even further augmentation of the

19

EOS transport.

20

Choice of the parameters

21

The present study involves various physiochemical parameters. All of them have been mo-

22

tivated by realistic examples of polyelectrolyte and nanochannel systems as illustrated below:

23

24

Choice of pKa :

25

We consider pKa = 4. This is motivated by the fact that we consider the end-charged group

26

to be similar to a pH-responsive acrylic acid group that has a pKa value of 4.26. 36 27



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

1

2

Choice of ℓ:

3

Here ℓ defines the grafting density σ, i.e., σ ∼ 1/ℓ2 . We have used ℓ = 22 nm, 80 nm. This

4

implies a grafting density range of σ = 10−4 − 102 nm−2 . While σ = 10−4 nm−2 is a slightly

5

weak value of the grafting density, σ values in the order of 10−2 nm−2 and 10−3 nm−2 has

6

been routinely reported in the experiments involving polymer brushes. 37,38

7

8

Choice of Np :

9

Np is the number of monomers in a given PE brush chain. In the present study, we con-

10

sider two values of Np : 2000, 2800. These numbers lie within the typical average values

11

of the number of monomers in a polymer brush molecule experimentally studied. Exam-

12

ples include studies that have considered polymer brushes with very similar, 40 or larger, 40,41

13

or smaller 40–42 number of monomers per polymer chain (for these experimental studies 40–42

14

number of monomers Np are calculated as the ratio of the total molecular weight of the

15

polymer chain to the molecular weight of the monomer).

16

17

Choice of d0 :

18

d0 is the height of the brushes without the electrostatic effects. Our calculations consider a

19

relatively wide range of d0 values ranging from 34 nm to 82 nm, resulted by selection of ℓ and

20

Np . There are several experimental evidences of similar values of d0 – e.g., d0 = 30 nm, 38

21

d0 = 20 nm − 116 nm. 39

22

23

Choice of h:

24

Here we have consider the nanochannel half height h = 100 nm.

25

Such nanochannel/nanocapillary/nanocone dimensions have been routinely witnessed for ex-

26

perimental studies involving nanochannels/nanocpillaries/nanocones grafted with polymer

27

or polyelectrolyte brushes. 8,43–48

28


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Page 29 of 38

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1

Conclusions

2

In this study, we describe the most remarkable EOS flow enhancement in nanochannels

3

grafted with the end-charged PE brushes, completely contrary to the findings of the existing

4

studies demonstrating substantial EOS flow reduction in PE-grafted nanochannels. 12–14 Of

5

course, the critical difference between these previous studies and the present study is that

6

for all these previous studies the PE brush is charged along its entire backbone, while in

7

the present case the PE molecule contains charge only at its non-grafted end located far

8

away from the grafting surface. As a consequence, for the end-charged brushes, the EDL

9

and the EDL-induced maximum body force for driving the EOS-flow is localized at the

10

charged end of the brush (i.e., far away from the wall), ensuring that the resulting flow

11

is subjected to very weak wall-induced retarding shear force enabling a very large EOS

12

transport. We witness a much larger EOS flow strength as compared to nanochannels without

13

the PE brushes; however, an increase in the brush height enhances the EOS flow strength

14

progressively only when the parameters that dictate the increase in the brush height does

15

not additionally increase the drag coefficient. We believe that the present study, which is

16

based on a mean-field-based model that neglects the possible ICP effect, proposes a hitherto

17

unknown mechanism by which the EOS transport gets augmented in the PE-brush-grafted

18

nanochannels – we anticipate that this finding will open up new avenues for manipulating

19

the nanofluidic transport using nano-functionalized interfaces.

20

Appendix: Derivation of the expression of σch,pH

21

We consider a pH-responsive, negatively charged end-charged PE brush. Consequently, the

22

brush attains the charge by ionization of a weakly acidic group HA, which ionizes as: HA ⇌ H + + A− (ionization constant Ka ),

29


(16)


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

1

Page 30 of 38

so that Ka =

cH + cA − . cHA

(17)

2

Here ci (i = HA, A− , H + ) is the concentrations in moles/litre at the location of the end

3

charge (i.e., y = ye ) and Ka is the ionization constant in moles per litre. Consequently, Ka′ =

nH + nA− , nHA

(18)

4

i.e., the concentrations are expressed as number densities (having units of 1/m3 ) and Ka′ =

5

NA 103 Ka has a unit of 1/m3 (NA is the Avogadro number).

6

Following the large volumes of literature relating the charge density (σch ) of an ionizable

7

surface to the concentration of species in the reaction that produces this surface charging, 49–51

8

we can express −eΓ0 = −e (ΓHA + ΓA− ) = σch ,

9

(19)

where Γ0 is the total surface site density of the end charges, and ΓHA and ΓA− are the

10

corresponding values associated with HA and A− (see Behrens and Grier 49 for details).

11

For the present case, we are interested to obtain the pH-dependent charge density – therefore

12

we would need σA− , i.e., σch,pH = −eΓA−

(20)

13

Of course, following Behrens and Grier 49 [kindly see eq.(4) in Behrens and Grier and note that

14

the condition expressed in eq.(21) below allows one to obtain eq.(4) in Behrens and Grier]

15

the site density corresponding to the individual species can be related to their corresponding

16

concentrations (or number densities) as: nHA ΓHA = Γ A− nA−

30


(21)

Page 31 of 38

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

1


Using eq.(21), we can reduce eq.(18) to:

Ka′ =

nH + Γ A− , ΓHA

(22)

2

which is identical to eq.(4) in Behrens and Grier. Therefore, using eq.(19,20,22), we can

3

obtain: σch,pH = σch

Ka′ , nH + + Ka′

(23)

4

where nH + is the hydrogen ion number density at the location of the end charge, i.e., nH + =

5

nH + (ye ).

6

Acknowledgement

7

The authors gratefully acknowledge the Teaching Assistantship Program of The Department

8

of Mechanical Engineering, University of Maryland for providing financial support to Guang

9

Chen when she conducted this research. There was no external funding supporting this

10

work.

11

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Massively Enhanced Electroosmotic Transport in Nanochannels

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