Steric-effect-induced enhancement of electrical-double-layer overlapping phenomena

doi:10.1103/PHYSREVE.84.012501

PHYSICAL REVIEW E 84, 012501 (2011)

Siddhartha Das

1

and Suman Chakraborty

2

1

Physics of Fluids Group, J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217,

7500 AE Enschede, The Netherlands

2

Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, Kharagpur-721302, India

(Received 3 March 2011; revised manuscript received 18 May 2011; published 20 July 2011)

In this paper, we demonstrate that nontrivial interactions between steric effect and electrical-double-layer

(EDL) overlap phenomena may augment the effective extent of EDL overlap in narrow ﬂuidic conﬁnements

to a signiﬁcant extent by virtue of rendering the channel centerline potential tending to the ζ potential in a

limiting sense as the steric effect progressively intensiﬁes. Such a behavior may result in a virtually uniform

(undiminished) magnitude of the EDL potential across the entire channel height and may cause lowering of the

total charge within the EDL.

DOI: 10.1103/PhysRevE.84.012501 PACS number(s): 82.45.Gj

Recent advances toward understanding nanoscale elec-

trochemical hydrodynamics have led to the development of

several widely applicable miniaturized devices and systems

tuned for practical applications [1–8]. The intricate dynamics

within the surface-bound double layer of charge, popularly

known as the electric-double layer (EDL) [9], often dictates

the functionalities of these systems to an extremely signiﬁcant

extent. Theoretical electromechanical analysis of the EDL, on

the other hand, has attracted great interest for close to half

a century [10–12], and well known solutions exist for EDL

charge distribution in a ﬂuidic conﬁnement, obtained under

classical approximations, such as mean ﬁeld assumption,

assumption of nonoverlapping EDLs, assumption of EDL ions

as point charges, etc.

A growing need to understand the electromechanics of

systems with overlapping EDLs, and/or with very high

ionic concentration, and/or with very large ζ potential, has

necessitated the underlying analysis to be extended beyond

the above-mentioned simpliﬁed assumptions, even within the

framework of mean ﬁeld approximations. For example, several

recent investigations demonstrate remarkable inﬂuences of

EDL overlap on nanoscale electrokinetic transport [13–21],

and there has been considerable effort in the theoretical

modeling of the concerned phenomena [22–33], obeying

the governing principle of net charge electroneutrality of an

isolated channel.

Interestingly, independent of the considerations of EDL

overlap, there has been a continuous endeavor toward relaxing

the treatment of ionic species as point charges by aptly taking

the effects of their ﬁnite sizes into account. Such ﬁnite size

effects of ions have been shown to have tremendous inﬂuence

on different applications of nanoelectrokinetic phenomena

[34–42]. Stern, in his celebrated work [43], introduced cor-

rections to the Poisson-Boltzmann (PB) equation to account

for the unphysical divergences of the Gouy-Chapman model

of the double layer and indicated the volume constraints

of ions in the electrolyte phase [43,44]. However, the ﬁrst

complete ion-size-effect-induced modiﬁed Poisson Boltzmann

(mPB) model was developed by Bikerman [45]. Over the next

several years, Bikerman’s mPB equation was reformulated

by Grimley and Mott [46], Grimley [47], Dutta and Bagchi

[48], Bagchi [49,50], Dutta and Sengupta [51], Wicke and

Eigen [52–54], Wiegel and Strating [55], Strating and Wiegel

[56,57], and Bohinc and co-workers [58,59]. For a ﬂat double

layer, Kilic et al.[60] used the lattice approach to obtain

Bikerman’s mPB equation with a simple composite diffuse

layer model. Issues delineating the concerned consequences

on the resultant diffuse layer capacitances were also addressed

by some other researchers [44,60,61], in contrast with the

predictions obtained using the Gouy-Chapman theory [62].

A signiﬁcant step forward in the development of the effect

of ion size in PB description was achieved by the use of the

hard-sphere (HS) model for the ions, which Biesheuvel and

van Soestbergen, in their pioneering paper [63], showed to be

capable of addressing several inconsistencies and limitations

in the lattice model. Their approach is distinct from the

earlier approaches in the sense that, while Bikerman [45]

assumed the local available volume to be the total volume

minus the volume of all ions plus their hydration shell, and

Sparnaay [64] replaced the volume of the hydrated ions

by the excluded volume, Biesheuvel and van Soestbergen

empirically included an extended Carnahan-Starling equation

of state to describe HS interactions in EDLs containing ions

of different sizes and charges. In this way, they attempted

to overcome certain important limitations of the traditional

lattice based models, which invariably underestimated the

excluded volume effect as well as erroneously considered

the size of a lattice as equivalent to the size of a solvent

molecule and not the hydrated site. Their results were in

excellent agreement with experimental ﬁndings [65], and

their model was successfully extended to study the case of

mixture of ions of different sizes [66–69]. However, none of

the above studies have reported any analysis based on the

simultaneous considerations of steric effects and EDL overlap

phenomenon.

Here, we demonstrate that, by simultaneous considerations

of ﬁnite ion-size effect and EDL overlap, one can remarkably

have a completely geometric phenomenon (i.e., the EDL

overlap induced by the physical presence of a conﬁnement

having a length scale below a threshold limit) inﬂuenced by a

completely thermoelectrochemical transport phenomenon as

governed by the ionic-size effects. In this perspective, it is

important to pinpoint the distinctive fundamental contributions

from our Brief Report as compared to some statistical mechan-

ics based investigations (postulated on the density functional

formalism) of Urbanija et al. [70] and Perutkova et al. [71] in

012501-1

BRIEF REPORTS PHYSICAL REVIEW E 84, 012501 (2011)

which they have considered ﬁnite ion sizes and conﬁnement

simultaneously. In their studies, the ions considered are giant

macroions that are randomly placed inside a conﬁned channel.

Minimization of the free energy of the system results in govern-

ing integrodifferential equations for the potential distribution

within the conﬁnement. Such an approach, however, may be

intrinsically complex in addressing complicated scenarios, and

in many cases, reduces analytical tractability so that nontrivial

numerical treatment may be necessary to address even a simple

problem [60]. Moreover, their approach never considers the

condition of EDL overlap (realized when the conﬁnement

size goes below a threshold limit in relation to the Debye

length so that a nonzero centerline potential is realized that

cannot be prescribed apriori). In this context, it is important

to mention that mere consideration of a conﬁned geometry

does not implicate the consideration of an EDL overlap

phenomenon, since, if the characteristic length scale is beyond

a critical limit (typically on the order of the Debye length),

then far-stream boundary conditions on the electrostatic

potential can effectively be employed at the channel centerline,

rendering the situation qualitatively identical to the one with

a single charged plate, despite the physical existence of a

conﬁnement.

The mathematical model employed in the present Brief

Report follows from the modiﬁed space charge model de-

veloped by Cervera and co-researchers [72,73] in which the

ionic-size effect is explicitly incorporated in the ionic ﬂux

density expression in the governing species transport equation

as follows:



J

i

=un

i

− D

i

n

i



∇(ln a

i

+

z

i

Fφ

RT

), where u is the

advective velocity, F is the Faraday constant, z

i

is the valency

of the ith ionic species, T is the absolute temperature, D

i

is

the diffusivity of the ith ionic species, n

i

is the number density

of the ith ionic species, φ is the resultant electrical potential,

and a

i

is the activity of the ith ionic species and is expressed

in terms of the ionic concentrations n

i

as a

i

=

n

i

/n

r

1−v



k

n

k

/n

r

.

Here, n

r

is the reference ionic number density, and v is the

partial molal volume of the ionic species (may alternatively be

termed the steric factor). This model is applied to a geometrical

conﬁnement in which the characteristic EDL thickness is

speciﬁcally chosen to be larger than the channel half height.

The resulting EDL overlap phenomenon is assessed in terms

of the difference between the centerline potential and the

ζ potential, as a combined consequence of electrochemical,

thermodynamic, and a pure geometrical artifact of the system

conﬁguration. In fact, the channel centerline potential is not

aprioriknown in the present context, which needs to be

calculated in consistency with the ζ potential and the potential

gradient developed within the geometrical conﬁnement as a

function of the partial molal volume of the ionic species. Thus,

the channel centerline potential derived in a self-consistent

fashion, effectively represents the extent EDL overlap. The

central result of our calculation is that the ﬁnite ion-size

effects may remarkably enhance the effective extent of EDL

overlap, quantitatively delineated by the proximity of the

channel centerline potential with the ζ potential, as compared

to the cases without involving ﬁnite ion-size effects. The

resulting effective extent of the EDL overlap, under critical

circumstances, may be so signiﬁcantly enhanced that one may

recover the ζ potential virtually undiminished across the entire

channel height. This implies that one can completely ignore

the EDL and can end up, quite dramatically, with an ideally

homogeneously charged nanoﬂuidic system.

For the present Brief Report, we consider a nanoﬂu-

idic system (essentially a slit type nanochannel connected

with its end reservoirs) in which the steady state transport

phenomenon of each ionic species may be represented by

the following [72,73]:



∇·



J

i

= 0. In the absence of the

bulk advective transport of ions, the same consideration

implicates

k

B

T ln

(

a

i

)

+ ez

i

ψ = constant, (1)

where ψ is the EDL potential, e is the protonic charge, and k

B

is

the Boltzmann constant. Considering equilibrium between the

nanochannel and its connecting reservoirs (where n

+

= n

−

=

n

∞

and ψ = 0) and assuming a z : z symmetric electrolyte for

simplicity, one gets

n

±

=

n

∞

1 +

2vn

∞

n

r



cosh



ezψ

k

B

T



− 1



exp



∓

ezψ

k

B

T



. (2)

Substituting the above in the Poisson equation

d

2

ψ

dy

2

=

−ρ

e

ε

=

−ez(n

+

−n

−

)

ε

(where ε is the permittivity of the ionic

medium and y is the channel wall-normal coordinate) and

integrating the same from the Stern layer-diffuse layer inter-

face to the channel centerline (where

dψ

dy

= 0 and ψ = ψ

c

=

0 under overlapped EDL conditions), one gets

dψ

dy

=±

ψ

r

λ

√

v







ln

⎧

⎨

⎩

1 +



cosh



ψ

r



− 1



1 + 2v



cosh



ψ

c

ψ

r



− 1



⎫

⎬

⎭

, (3)

where λ is the Debye length (

1

λ

2

=

2n

∞

e

2

z

2

εk

B

T

) and ψ

r

=

k

B

T

ez

is a

reference potential.

In an effort to obtain the variations in ψ,weemployan

iterative solution technique, where we start with a guessed

value of ψ

c

and iteratively obtain a corrected value of the

same by using the EDL potential gradient as given by Eq. (3)

and a speciﬁed ζ potential, until convergence is achieved. It is

important to mention here that instead of a given ζ potential

boundary condition, one could as well employ a given surface

charge density boundary condition or, more appropriately, a

chemical equilibrium based interfacial treatment [74]. While

the later ones effectively include more parameters into the

iterative framework mentioned as above, the same does not

introduce any essential new physics in the context of the

physical issue addressed in this Brief Report. Accordingly, the

speciﬁed ζ potential boundary condition is used in the present

simulations for simplicity, without sacriﬁcing the essential

physics.

The central result of this Brief Report is the variation in

the ratio ψ

c

/ζ with the partial molal volume or the effective

steric factor ν, for different λ/H ratios (see Fig. 1). The

effective extent of EDL overlap is essentially governed by

the parameter ψ

c

/ζ (the closer its value to unity, the more

intense the EDL overlap). From Fig. 1, it is evident that an

increase in ν enhances ψ

c

/ζ , implicating a greater extent

012501-2

BRIEF REPORTS PHYSICAL REVIEW E 84, 012501 (2011)

10

−2

10

−1

10

0

0.2

0.4

0.6

0.8

1

ν

ψ

c

/ζ

−1 −0.8 −0.6 −0.4 −0.2 0

0

1

2

3

x 10

y/H

dψ/dy (V/m)

ν = 1

λ /H = 2

λ /H = 1

λ /H = 3/4

λ /H = 1/4

ν = 1/100

ν = 1/10

FIG. 1. (Color online) Variation of the ratio ψ

c

/ζ with steric factor

ν for different λ/H ratios. In the inset of the ﬁgure, we show the

transverse variation of dψ/dy for the channel bottom half for λ/H =

1 for different values of steric factor ν. Results are provided for H =

5nmandζ =−100 mV.

of EDL overlap (theoretically, the maximum extent of EDL

overlap occurs when ψ

c

/ζ →1). This enhancement is most

prominent for channels where the EDL overlap phenomenon

can be predicted even in the absence of steric effects. For such

systems, realizable values of the steric factor ensure that a state

is reached where ψ

c

/ζ →1. Such a prominent inﬂuence of the

steric effect in magnifying the extent of EDL overlap, however,

is not witnessed for channels that have nonoverlapping EDLs

in the absence of steric effects. To understand such a role of the

steric effect, we study the variation of the transverse gradient

of the EDL (see the inset of Fig. 1)asafunctionofν. Larger

ν leads to a weaker EDL potential gradient [this is clearly

noticeable from the analytical expression in Eq. (3)], causing

a much weaker decay (with height) of the wall conditions

and making it penetrate much deeper into the nanochannel

bulk. Consequently, the channel centerline potential has a

value signiﬁcantly closer to the ζ potential, which, in effect,

signiﬁes a more enhanced EDL overlap. The manifestation

of pronounced inﬂuence of the steric effect, based on the

steric-effect-independent extent of EDL interactions, can be

argued from steric-effect-driven weaker crowding of ions in

the EDL. For an overlapped EDL even in the absence of steric

effects, the EDL counterions are in a spars condition so that

the average attractive pull of the charged wall is substantially

weak. For such cases, the contribution of the entropic mixing

of the ions (caused by their ﬁnite sizes, s ee Ref. [60])

can easily overcome the electrostatic attractive pull of the

wall, ensuring a homogeneous mixing and, hence, a uniform

distribution (rather than a wall-potential-dictated distribution)

of the ions across the channel height. This effectively leads

to a signiﬁcantly enhanced value of the channel centerline

potential, implying a high effective degree of EDL overlap.

On the contrary, for an initially nonoverlapping system (even

for ﬁnitely large EDL thicknesses), the ions are always

−1 −0.8 −0.6 −0.4 −0.2 0

0.4

0.6

0.8

1

y/H

ψ/ζ

−1 −0.8 −0.6 −0.4 −0.2 0

10

y/H

n /n , n /n

n /n ,

ν=1/100

n

/n ,

ν=1/10

ν = 1/10

ν = 1

ν = 1/100

n /n ,

ν=1/100

n

/n ,

ν=1/10

FIG. 2. (Color online) Transverse variation of the EDL potential

(ψ) for the channel bottom half, corresponding to λ/H = 1for

different values of the steric factor ν. In the inset of the ﬁgure, we

show the corresponding transverse variation of the number density of

the counterions (n

+

/n

∝

) and coions (n

−

/n

∝

). Results are provided for

H = 5nmandζ =−100 mV.

tightly bound within the attractive range of the channel wall

so that the entropic considerations can never lead to uniform

distribution of the ions across the channel. Hence, for such

cases, only minor enhancement of the extent of EDL overlap

can be noted.

The consequences of such a steric-effect-induced en-

hancement of EDL overlap are r emarkable. For example,

the EDL potential, under such conditions, remains virtually

undiminished across the entire channel (see Fig. 2). This

implies that it is possible to achieve a uniformly charged

nanochannel where the behavior at any location in the conduit

is exactly identical to that interfacing the wall. Implicitly, this

signiﬁes that the role of t he added ions toward dictating how

the bulk sees and how the EDL screens the wall ceases to be

important. This can straightaway be related to the fact that there

is a signiﬁcant lowering of the number of ions in the EDL, as a

consequence of a lowering of the interfacial potential gradient

(see t he inset of Fig. 2).

To summarize, we have shown that the steric effect can

substantially enhance the extent of effective EDL overlap in

narrow conﬁnements. These enhancements are most promi-

nent for channels that have some degree of steric-effect-

independent EDL overlap (i.e., EDL overlap under hypotheti-

cal conditions of the ionic species taken as point charges). Such

a phenomenon may turn out to be immensely signiﬁcant, since

it leads to the paradigm of a homogeneously charged nanoﬂu-

idic system, where the concept of strong near-wall gradients

due to EDL phenomena virtually appears to be nonexistent.

This, in turn, may affect several electrokinetic phenomena that

depend on the extent of EDL interactions in nanochannels,

such as streaming potential and energy transfer effects in

narrow ﬂuidic conﬁnements [75– 80], conﬁnement-inﬂuenced

macromolecular separation [81–83], and many others.

[1] A. Watson, Science 289, 850 (2000).

[2] S. K. Grifﬁths and R. H. Nilson, Anal. Chem. 78, 8134 (2006).

[3] F. H. J. van der Heyden, D. Stein, K. Besteman, S. G. Lemay,

and C. Dekker, Phys.Rev.Lett.96, 224502 (2006).

[4] L. Bocquet and E. Charlaix, Chem.Soc.Rev.39, 1073

(2010).

[5] M. Napoli, J. C. T. Eijkel, and S. Pennathur, Lab Chip 10, 957

(2010).

012501-3

BRIEF REPORTS PHYSICAL REVIEW E 84, 012501 (2011)

[6] W. Sparreboom, A. van den Berg, and J. C. T. Eijkel, Nat.

Nanotechnol. 4, 713 (2009).

[7] W. Sparreboom, A. van den Berg, and J. C. T. Eijkel, New J.

Phys. 12, 015004 (2010).

[8] P. Abgrall and N. T. Nguyen, Anal. Chem. 80, 2326 (2008).

[9] R. J. Hunter, Zeta Potential in Colloid Science: Principles and

Applications (Academic, New York, 1981).

[10] D. Burgreen, and F. R. Nakache, J. Phys. Chem. 68, 1084 (1964).

[11] C. L. Rice and R. Whitehead, J. Phys. Chem. 69, 4017 (1965).

[12] S. Levine, J. R. Marriott, G. Neale, and N. Epstein, J. Colloid

Interface Sci. 52, 136 (1975).

[13] H. Ban, B. Lin, and Z. Song, Biomicroﬂuidics 4, 014104 (2010).

[14] R. Chein, C. Liao, and H. Chen, Nanoscale Microscale

Thermophys. Eng. 14, 75 (2010).

[15] J. Jamaati, H. Niazmand, and M. Renksizbulut, Int. J. Therm.

Sci. 49, 1165 (2010).

[16] X. Wang and W. Hong, Phys. Rev. E 81, 041803 (2010).

[17] E. C. Yusko, R. An, and M. Mayer, ACS Nano 4, 477 (2010).

[18] M. L. Kovarik, K. Zhou, and S. C. Jacobson, J. Phys. Chem.

113, 15960 (2009).

[19] R. Chein, H. Chen, and C. Liao, J. Electroanal. Chem. 1, 630

(2009).

[20] S. Chakraborty and A. K. Srivastava, Langmuir 23, 12421

(2007).

[21] S. Talapatra and S. Chakraborty, Eur. J. Mech. B 27, 297 (2008).

[22] W. Qu and D. Li, J. Colloid Interface Sci. 224, 397 (2000).

[23] C. L. Ren, Y. D. Hu, D. Q. Li, and C. Werner, J. Adhes. 80, 831

(2004).

[24] A. T. Conlisk, J. McFerran, Z. Zheng, and D. Hansford, Anal.

Chem. 74, 2139 (2002).

[25] Z. Zheng, D. J. Hansford, and A. T. Conlisk, Electrophoresis 24,

3006 (2003).

[26] S. Bhattacharyya, Z. Zheng, and A. T. Conlisk, J. Fluid Mech.

540, 247 (2005).

[27] D. Stein, M. Kruithof, and C. Dekker, Phys.Rev.Lett.93,

035901 (2004).

[28] E. H. Kwak and E. F. Hasselbrink,

J. Colloid Interface Sci. 284,

753 (2005).

[29] S. Kang and Y. K. Suh, J. Mech. Sci. Technol. 20, 2250

(2006).

[30] R. B. Schoch and P. Renaud, Appl. Phys. Lett. 86, 253111

(2005).

[31] R. B. Schoch, H. van Lintel, and P. Renaud, Phys. Fluids 17,

100604 (2005).

[32] F. Tessier and G. W. Slater, Electrophoresis 27, 686 (2006).

[33] F. Baldessari and J. G. Santiago, J. Colloid Interface Sci. 325,

526 (2008).

[34] A. Garai and S. Chakraborty, Electrophoresis 31, 843 (2010).

[35] C. K. Hua, I. S. Kang, K. H. Kang, and H. A. Stone, Phys. Rev.

E 81, 036314 (2010).

[36] H. J. Zhao, J. Phys. Chem. C 114, 8389 (2010).

[37] A. S. Khair and T. M. Squires, J. Fluid Mech. 343, 640 (2009).

[38] B. D. Storey, L. R. Edwards, M. S. Kilic, and M. Z. Bazant,

Phys.Rev.E77, 036317 (2008).

[39] L. H. Olesen, M. Z. Bazant, and H. Bruus, Phys. Rev. E 82,

011501 (2010).

[40] M. Z. Bazant, M. S. Kilic, B. D. Storey, and A. Ajdari, Adv.

Colloid Interface Sci. 152, 48 (2009).

[41] M. Z Bazant, M. S. Kilic, B. D. Storey, and A. Ajdari, New J.

Phys. 11, 075016 (2009).

[42] M. Z. Bazant, B. D. Storey, and A. A. Kornyshev, Phys. Rev.

Lett. 106, 046102 (2011).

[43] O. Stern, Z. Elektrochem. 30, 508 (1924).

[44] V. Freise, Z. Elektrochem. 56, 822 (1952).

[45] J. J. Bikerman, Philos. Mag. 33, 384 (1942).

[46] T. B. Grimley and N. F. Mott, Discuss. Faraday Soc. 1, 3 (1947).

[47] T. B. Grimley, Proc. R. Soc. London, Sec. A 201, 40 (1950).

[48] M. Dutta and S. N. Bagchi, Indian J. Phys. 24, 61 (1950).

[49] S. N. Bagchi, Indian J. Chem. Soc. 27, 199 (1950).

[50] S. N. Bagchi, Indian J. Chem. Soc. 27, 204 (1950).

[51] M. Dutta and M. Sengupta, Proc. Natl. Inst. Sci. India 20,1

(1954).

[52] E. Wicke and M. Eigen, Naturwiss. 38

, 453 (1951).

[53] E. Wicke and M. Eigen, Z. Elektrochem. 56, 551 (1952).

[54] E. Wicke and M. Eigen, J. Phys. Chem. 58, 702 (1954).

[55] F. W. Wiegel and P. Strating, Mod. Phys. Lett. B 7, 483 (1993).

[56] P. Strating and F. W. Wiegel, J. Phys. A 26, 3383 (1993).

[57] P. Strating and F. W. Wiegel, Physica A 193, 413 (1993).

[58] K. Bohinc, V. Kralj-Iglic, and A. Iglic, Electrochim. Acta 46,

3033 (2001).

[59] K. Bohinc, A. Iglic, T. Slivnik, and V. Kralj-Iglic,

Bioelectrochemistry 57, 73 (2002).

[60] M. S. Kilic, M. Z. Bazant, and A. Ajdari, Phys.Rev.E75,

021502 (2007).

[61] A. A. Kornyshev, J. Phys. Chem. B 111, 5545 (2007).

[62] D. L. Chapman, Philos. Mag. 25, 475 (1913).

[63] P. M. Biesheuvel and M. van Soestbergen, J. Colloid Interface

Sci. 316, 490 (2007).

[64] M. J. Sparnaay, Recl. Trav. Chim. Pays-Bas 77, 872 (1954).

[65] V. L. Shapovalov and G. Brezesinski, J. Phys. Chem. B 110,

10032 (2006).

[66] S. Lamperski and C. W. Outhwaite, J. Colloid Interface Sci. 328,

458 (2008).

[67] T. Goel, C. N. Patra, S. K. Ghosh, and T. M ukherjee, J. Chem.

Phys. 132, 194706 (2010).

[68] V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag, and S. Doniach,

Biophys. J. 93, 3202 (2007).

[69] J. G. Ibarra-Armenta, A. Mart

´

ın-Molina, and M. Quesada-P

´

erez,

Phys. Chem. Chem. Phys. 11, 309 (2009).

[70] J. Urbanija, K. Bohinc, A. Bellen, S. Maset, A. Iglic, V. Kralj-

Iglic, and P. B. S. Kumar, J. Chem. Phys. 129, 105101 (2008).

[71] S. Perutkova, M. Frank, K. Bohinc, G. Bobojevic, J. Zelko, B.

Rozman, V. Kralj-Iglic, and A. Iglic, J. Membr. Biol. 236,43

(2010).

[72] J. Cervera, V. Garc

´

ıa-Morales, and J Pellicer, J. Phys. Chem. B

107, 8300 (2003).

[73] J. Cervera, P. Ramirez, J. A. Manzanare, and S. Mafe, Microﬂuid.

Nanoﬂuid. 9, 41 (2010).

[74] M. Wang and A. R evil, J. Colloid Interface Sci.

343, 381 (2010).

[75] S. Chakraborty and S. Das, Phys.Rev.E77, 037303 (2008).

[76] S. Das and S. Chakraborty, Langmuir 26, 11589 (2010).

[77] T. Das, S. Das, and S. Chakraborty, J. Chem. Phys. 130, 244904

(2009).

[78] M. Wang and Q. Kang, Microﬂuid. Nanoﬂuid. 9, 181 (2010).

[79] C.-C. Chang and R.-J. Yang, Microﬂuid. Nanoﬂuid. 9, 225

(2010).

[80] P. Goswami and S. Chakraborty, Langmuir 26, 581 (2010).

[81] S. Das and S. Chakraborty, Electrophoresis 29, 1115 (2008).

[82] S. Das and S. Chakraborty, Langmuir 24, 7704 (2008).

[83] S. Das and S. Chakraborty, Langmuir 25, 9863 (2009).

012501-4

Steric-effect-induced enhancement of electrical-double-layer overlapping phenomena

Figures (2)

Citations

References

"Steric-effect-induced enhancement o..." refers background in this paper

Related Papers (5)