Download The Effect of Axial Concentration Gradient on

Microgravity Sci. Technol. (2010) 22:329–338
DOI 10.1007/s12217-010-9195-8
ORIGINAL ARTICLE
The Effect of Axial Concentration Gradient
on Electrophoretic Motion of a Charged
Spherical Particle in a Nanopore
Sang Yoon Lee · Sinan E. Yalcin · Sang W. Joo ·
Ashutosh Sharma · Oktay Baysal · Shizhi Qian
Received: 30 November 2009 / Accepted: 3 April 2010 / Published online: 24 April 2010
© Springer Science+Business Media B.V. 2010
Abstract The electrophoretic motion of a charged
spherical nanoparticle along the axis of a nanopore
connecting two fluid reservoirs, subjected to an axial
electric field and electrolyte concentration gradient, has
been investigated using a continuum model. The model
consists of the Poisson and Nernst–Planck equations for
the electric potential and ionic concentrations and the
Stokes equations for the hydrodynamic field with zero
gravity. In addition to the electrophoresis generated
by the externally imposed electric field, the particle
also experiences diffusiophoresis arising from the externally imposed concentration gradient. The effects of
the diffusiophoresis on the axial electrophoretic motion
are examined with changes in the ratio of the particle
size to the thickness of the electric double layer (EDL),
and the imposed concentration gradient. Since the EDL
thickness, the particle size, and the nanopore size are
of the same order of magnitude, the diffusiophoresis
is dominated by the induced electrophoresis driven by
the generated electric field arising from the doublelayer polarization (DLP). For a relatively small κa p ,
the ratio of the particle size to the EDL thickness,
the diffusiophoresis is dominated by the induced elec-
S. Y. Lee · S. W. Joo · A. Sharma · S. Qian
School of Mechanical Engineering, Yeungnam University,
Gyongsan 712-749, South Korea
S. E. Yalcin · O. Baysal · S. Qian (B)
Department of Aerospace Engineering,
Old Dominion University, Norfolk, VA 23529, USA
e-mail: [email protected]
A. Sharma
Department of Chemical Engineering,
Indian Institute of Technology, Kanpur 208016, India
trophoresis from the type II DLP, which propels the
particle toward regions with lower salt concentration.
Depending on the magnitude and direction of the
externally imposed concentration gradient, the electrophoretic motion can be accelerated, decelerated, and
even reversed by the diffusiophoresis.
Keywords Electrophoresis · Diffusiophoresis ·
Electrical double layer · Nanopore
Introduction
In recent years, there has been a growing interest in
developing nanopore-based nanofluidic devices with
features comparable in size to DNA, proteins and
other biological molecules for biological and chemical analysis (Branton et al. 2008; Lemay 2009; Clarke
et al. 2009). In nanofluidic devices, it is necessary to
manipulate fluids or nanoparticles such as DNA for
various applications. Since the sample volume is extremely small, the gravitational body force is negligible.
The interfacial electrokinetic phenomena such as electroosmosis and electrophoresis are thus widely used
to manipulate fluids and/or particles in microfluidic
and nanofluidic applications (Li 2004; Masliyah and
Bhattacharjee 2006; Branton et al. 2008; Lemay 2009;
Clarke et al. 2009; Ai et al. 2009a, b, 2010a, b).
When a charged particle is immersed in an electrolyte solution, the accumulation of a net electric
charge near its surface leads to the formation of an
EDL. In the presence of an external electric field,
both the charges on the particle and the ions in the
EDL interact with the overall electric field near the
particle, resulting in electrostatic forces acting on both
330
the particle and the fluid and resulting in simultaneous
electrophoretic and electroosmotic motions. The electrophoresis of charged and non-charged particles that
acquire charge by polarization has been widely utilized
in characterizing, separating, and purifying colloidal
particles and macromolecules, such as DNA fragments,
proteins, drugs, viruses, and biological cells (Li 2004).
For example, when a DNA molecule is electrophoretically driven through a nanopore, nucleobases would
modify the ionic current through the nanopore, and
thus the sequence of bases in DNA might be recorded
by monitoring the current modulations (Branton et al.
2008; Lemay 2009; Clarke et al. 2009). This nanoporebased DNA sequencing method is called the third generation DNA sequencing, and its cost is believed to be
sufficiently low. Therefore, this new technology might
potentially revolutionize genomic medicine (Branton
et al. 2008; Clarke et al. 2009).
In the existing study on nanoparticle translocation
through a nanopore, the particle motion is driven by
the externally imposed electric field, while the electrolyte concentrations in the two fluid reservoirs are the
same. In the present study, electrophoretic motion of
a charged nanoparticle in a nanopore connecting two
fluid reservoirs filled with different electrolyte concentrations is studied for the first time. Since the electrolyte concentrations on both sides of the nanopore
are different, diffusiophoretic motion is also induced in
addition to the electrophoresis (Anderson and Prieve
1984; Keh and Li 2007; Qian et al. 2007; Lou and Lee
2008a, b; Keh and Wan 2008; Abecassis et al. 2008,
2009; Prieve 2008; Lou et al. 2009; Hsu et al. 2009, 2010;
Hsu and Keh 2009; Zhang et al. 2009). Depending on
the magnitude and direction of the imposed concentration gradient, the induced diffusiophoretic motion
can enhance the particle’s electrophoretic motion or
slow down nanoparticle translocation in a nanopore,
and thus can be used to regulate the nanoparticles
translocation process to achieve a nanometer-scale spa-
Fig. 1 Schematic of a
nanopore of length L and
radius a connecting two
identical reservoirs on either
side. A concentration
gradient of electrolyte
solution and an electric field
are applied across the two
reservoirs. A charged
spherical particle of radius ap
bearing uniform surface
charge density, σ p , is
positioned at the center of the
nanopore
Microgravity Sci. Technol. (2010) 22:329–338
tial accuracy for DNA sequencing. In particular, it is
conceivable that slowing down the motion of DNA in a
nanopore by diffusiophoretic control seems especially
attractive as it would offer a greater window of opportunity for enhanced spatio-temporal resolution.
In the following section, a “Mathematical Model” is
introduced based on the continuum hypothesis for the
fluid motion and the ionic mass transport. The former
is induced by the externally imposed electric field and
concentration gradient and the latter accounts for the
polarization of the EDL and is valid for any thickness
of the EDL. The effect of the imposed concentration
gradient on the electrophoretic motion of a nanoparticle along the axis of a nanopore is presented in “Results
and Discussion”, followed by concluding remarks in
“Conclusions”.
Mathematical Model
We consider an uncharged nanopore of length L and
radius a connecting two identical reservoirs, as shown
in Fig. 1. An axisymmetric cylindrical coordinate system (r, z) with the origin located at the center of the
nanopore is used. A charged spherical nanoparticle of
radius ap and surface charge density σ p is submerged
in an electrolyte solution in the nanopore. We assume
that the nanoparticle is initially positioned with axis
coinciding with the nanopore’s axis, and the location of
the particle’s center of mass coincides with the origin.
The axisymmetrical model geometry is represented by
the region bounded by the outer boundary ABCDEFGH, the line of symmetry HI, the particle’s surfaces
IJ and JK, and the symmetry line KA. The dashed
line segments, AB, BC, FG, and GH represent the
regions in the reservoirs. The lengths L R and radius
b of the reservoirs are sufficiently large to ensure that
the electrochemical properties at the locations of AB,
BC, FG and GH are not influenced by the charged
Microgravity Sci. Technol. (2010) 22:329–338
331
nanoparticle. We assume that the rigid walls of the two
reservoirs (line segments CD and EF) are electrically
neutral surfaces. The left and right reservoirs are filled
with two identical electrolyte solutions with different
bulk concentrations, C L and C R . The segments AB and
GH are borders with the reservoirs, between which a
potential difference, φ0 , is applied. We also assume that
there is no externally applied pressure gradient across
the two reservoirs.
Considering its good agreement with experimental
observations for nanoparticle’s electrophoretic motion
through a nanopore (Qian et al. 2006, 2008; Liu et al.
2007; Qian and Joo 2008) and with the molecular
dynamics simulation of a liquid flowing through a
nanopore of 2.2 nm in diameter and 6 nm in length
(Huang et al. 2007), we adopt a continuum model
that consists of the Poisson and Nernst-Planck (PNP)
equations for the electrical potential and ionic concentrations, and the Navier–Stokes equations for the
flow field to study the combined electrophoresis and
diffusiophoresis in a nanopore.
The Reynolds number of electrokinetic flows in
nanopores is typically very small, and so we neglect
the inertial terms in the Navier–Stokes equation, and
model the fluid motion with the Stokes equations. The
flow of an incompressible binary electrolyte solution
then is described by
∇ •u=0
(1)
and
−∇ p + μ∇ 2 u − F(z1 c1 + z2 c2 )∇V = 0,
(2)
where u = uer + vez is the velocity (er and ez are unit
vectors in the radial and axial directions, respectively),
p is the pressure, V is the electric potential in the
electrolyte solution, c1 and c2 are, respectively, the
molar concentrations of the positive and negative ions
in the electrolyte solution, z1 and z2 are, respectively,
the valences of the positive and negative ions, F is the
Faraday constant, and μ is the electrolyte solution’s
dynamic viscosity. The last term on the left hand side
of Eq. 2 represents the electrostatic force acting on the
fluid through the interactions between the electric field
and the net charge density in the electrolyte solution.
The infinitesimal contribution of the body force due to
gravitational acceleration is neglected.
A non-slip boundary condition (i.e., u = v = 0) is
specified at the rigid walls of the nanopore and the
reservoirs (line segments CD, DE, and EF in Fig. 1). On
the planes AB and GH of the reservoirs, since they are
far away from the nanopore and there is no externally
applied pressure gradient across the two reservoirs,
normal flow with pressure p = 0 is used. Symmetric
boundary condition is used along the lines of symmetry,
HI and KA. Slip boundary conditions are used on the
segments BC and FG which represent the regions in the
reservoirs and are far away from the entrances of the
nanopore. Finally, along the surface of the particle (arc
segment IJK in Fig. 1) translating with an electrokinetic
velocity u p , we neglect the thickness of the adjacent
Stern layer, and impose the no-slip condition as
u(r, z) = u p ez , on IJK.
(3)
The particle’s electrokinetic velocity u p is determined
by requiring the total force in the z direction (FT )
acting on the particle
FT = F E + F D = 0,
(4)
where
E FE =
T · n · ez dS
(5)
S
and
FD =
D T · n · ez dS
(6)
S
are, respectively, the electrostatic and hydrodynamic
forces acting on the particle. S is the particle’s
surface; T E = εEE − 12 ε(E • E)I and T D = − pI+
μ(∇u + ∇uT ) are, the Maxwell stress tensor and the
hydrodynamic stress tensor, respectively; ε is the permittivity of the electrolyte solution; E = −∇V is the
electric field; and I is the unit tensor.
The flux density of each aqueous species due to
convection, diffusion, and migration is given by
Dk
Fck ∇V, k = 1 and 2. (7)
RT
In the above, Dk is the diffusion coefficient of the
kth ionic species; T is the absolute temperature of the
electrolyte solution; and R is the universal gas constant.
Under steady state, the concentration of each species
is governed by the Nernst–Planck equation
Nk = uck − Dk ∇ck − zk
∇ • Nk = 0 (k = 1 and 2)
(8)
and the electric potential, V, in the electrolyte solution
is governed by the Poisson equation
−ε∇ 2 V = F(z1 c1 + z2 c2 ).
(9)
On the plane AB which is sufficiently far away from the
nanopore, the ionic concentrations are the same as the
bulk concentration of the electrolyte solution present in
the left reservoir:
c1 = c2 = C L on AB
(10)
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Microgravity Sci. Technol. (2010) 22:329–338
Similarly, the ionic concentrations on the plane GH are
the same as the bulk concentration of the electrolyte
solution in the right reservoir:
c1 = c2 = C R
on GH
(11)
Along the rigid walls of the reservoirs and the
nanopore, and the surface of the nanoparticle, since
the solid surfaces are impervious to ions, the net ionic
fluxes normal to the rigid surfaces satisfy:
n • Nk = n • (uck ) (k = 1 and 2)
on CD, DE, EF, and IJK.
(12)
In the above, n is the unit vector normal to the corresponding surface.
The boundary conditions on the segments BC and
FG are defined with the assumption that these surfaces
are in the bulk electrolyte reservoirs. Accordingly, zero
normal flux is used for the Nernst–Planck equations:
n • N1 = n • N2 = 0 on BC and FG.
(13)
Along the segments HI and KA, symmetric boundary
condition is used for the Nernst–Planck equations:
n • N1 = n • N2 = 0 on HI and KA.
(14)
Symmetric boundary condition for the electric potential
in the electrolyte solution is used on the planes HI and
KA:
n • ∇V = 0 on HI and KA.
(15)
An external potential, φ0 , is applied along the plane
AB:
V = φ0
on AB.
(16)
Along the plane GH, the boundary condition for the
electric potential is
V=0
on GH.
(17)
Since the surfaces of BC and FG are far away from the
nanopore and are in the bulk electrolyte reservoirs, no
charge boundary condition for the potential is used:
n • ∇V = 0 on BC and FG.
(18)
Since the rigid walls of the reservoirs (planes CD and
EF) and the nanopore (plane DE) do not carry fixed
charge, we use
n • ∇V = 0 on CD, DE, and EF
(19)
Along the particle’s surface (line segment IJK), surface
charge boundary condition is used:
n • (−ε∇V) = σ p
on IJK
(20)
Results and Discussion
Note that the above model neglects the particle
Brownian motion and assumes that the particle only
translates along the axis of the nanopore. When the
particle is off the centerline of the pore, particle translation and rotation should be considered. The model
also neglects the effect of the finite ion size on the ionic
mass transport. However, a good agreement of the predicted DNA translocation velocity with experimental
measurements (Liu et al. 2007) suggests that the above
model is able to capture the essential physics of the
nanoparticle translocation process.
The commercial package COMSOL version 3.5a
(http://www.comsol.com), installed in a workstation
with 96 GB RAM, is chosen to integrate the system with a finite-element method. Quadratic triangular elements with variable sizes are used to accommodate finer resolutions near the particle surface IJK
where EDL is present. Solution convergence is guaranteed through mesh-refinement tests on conservation
laws. The Mathematical Model and its implementation
with COMSOL have been validated by many benchmark tests. For example, the predictions of the electrophoretic motion of a particle through a nanopore
driven by an imposed electric field are in good agreement with the approximate analytical solution and experimental results obtained from the literature (Qian
et al. 2006, 2008; Liu et al. 2007; Qian and Joo 2008).
The predictions of the diffusio-osmotic flow in a slit
nanochannel, driven by a concentration gradient, agree
with the results obtained from the literature (Qian et al.
2007). These good agreements of various benchmark
problems under either electric field or concentration
gradient make us confident of our following computational results.
In this section, we present a few numerical results
of the electrokinetic motion of a charged spherical
nanoparticle along the axis of a nanopore filled with
KCl electrolyte. We focus on the effects of the induced diffusiophoresis on the particle’s electrophoretic
motion by varying the concentration ratio α = C R /C L
for different values of κa p , ratio of particle radius to
the EDL thickness (κ −1 = λ D = ε RT/2F 2 C0 is the
dimensional EDL thickness with C0 = (C L + C R )/2).
The diffusion coefficients of the ions K+ and Cl −
are, respectively, 1.95 × 10−9 m2 /s and 2.03 × 10−9 m2 /s
(page 195 in Masliyah and Bhattacharjee 2006). The
temperature of the electrolyte solution in the reservoirs
and the nanopore is maintained at 300 K. In the numerical simulations, the following parameters are used:
L = 0.5 μm, a p = 5 nm, a/a p = 4, L R = 0.15 μm, b =
0.15 μm, φ0 = 100 mV, and σ p = −0.1 C/m2 .
Microgravity Sci. Technol. (2010) 22:329–338
333
Figures 2 and 3 depict, respectively, the dimensionless particle velocity u p *, normalized by U 0 =
ε R2 T 2 /(μa p F 2 ), as a function of the imposed concentration ratio, α, for κa p = 1 and 3. When α = 1, there
is no external concentration gradient imposed, and the
negatively charged particle moves along the opposite
direction of the applied electric field. For the case of
α = 1, a negative axial concentration gradient is imposed for α < 1 and vice-versa.
For κa p = 1 and α < 1 (Fig. 2), as α gradually decreases from α = 1, the magnitude of the particle’s
velocity decreases. When α is smaller than a threshold
value, αc , at which the particle velocity is zero, the
particle’s motion reverses, and the negatively charged
particle migrates along the same direction of the imposed electric field. The particle moves towards lower
salt concentration, and its motion is dominated by the
diffusiophoretic motion. As α further decreases, the
particle velocity first increases and eventually saturates and becomes independent of α as α → 0. On
the contrary, as α gradually increases from α = 1, the
magnitude of the particle’s velocity increases. When α
is above a certain critical value, the particle’s velocity
also becomes independent of the imposed concentration ratio. Therefore, one can accelerate, slow down or
even reverse the particle’s electrophoretic motion by
controlling the magnitude and direction of the imposed
concentration gradient. For the case of κa p = 3 (Fig. 3),
the induced diffusiophoresis basically slows down the
particle’s electrophoretic motion for both α < 1 and
α > 1, and the particle’s motion does not reverse in the
range of 10−3 ≤ α ≤ 103 . Obviously, the diverse effects
of the induced diffusiophoresis on the particle’s elec-
0.015
up*
0.01
0.005
0
-0.005
-0.01
-0.015 -4
10
10
-2
10
0
α
10
2
10
4
Fig. 2 Dimensionless particle velocity as a function of the concentration ratio, α, when κa p = 1
0
x 10
-3
up*
-0.5
-1
-1.5
-2
-2.5
-4
10
10
-2
10
0
α
10
2
10
4
Fig. 3 Dimensionless particle velocity as a function of the concentration ratio, α, when κa p = 3
trophoretic motion depend on the ratio of the particle
size to the EDL thickness, κa p , which will be elaborated
later.
Figure 4 depicts the fluid flow field near the negatively charged particle when κa p = 1 under the conditions of α = 0.3, 1, and 5. There is an obvious recirculating flow in the gap between the particle and
the nanopore wall. A clockwise circulation is generated
when the particle migrates upwards (i.e., the case of
α = 0.3), and a counter-clockwise eddy is generated
when the particle moves downwards (i.e., the cases of
α = 1 and 5). Since the wall of the nanopore is assumed
uncharged in this study, the fluid motion is very weak
in the region far away from the charged particle. In
the region closed to the charged particle, the fluid’s
motion is induced by the particle’s motion and the
electroosmotic flow (EOF) in the EDL surrounding
the negatively charged particle. The negatively charged
particle attracts positive ions and repels negative ions,
resulting in higher concentration of the positive ions
(Fig. 5) and lower concentration of the negative ions
(Fig. 6) within the EDL surrounding the negatively
charged particle. The net charge in gap between the
particle and the nanopore wall is positive for different
values of α. For α = 1, the interactions between the
externally imposed, positive axial electric field and the
net charge induces EOF directed towards the cathode,
and drives the fluids in the gap upwards; The fluids
near the particle moves downwards due to the particle’s
motion. Consequently, a counter-clockwise motion is
induced in the gap for the case of α = 1. For the case
of α = 1, the fluid motion in the gap results from the
interactions between the positive net charge and the
overall electric field which includes both the externally
334
Microgravity Sci. Technol. (2010) 22:329–338
Fig. 4 Flow field in the gap
between the particle and the
wall of the nanopore for
different concentration ratios
when κa p = 1
imposed electric field and the generated electric field
by the imposed concentration gradient.
In the absence of the imposed electric field, the
diffusiophoretic motion is generated by two mechanisms: one is induced electrophoresis generated by
the induced electric fields arising from the difference
of ionic diffusivities and DLP, and the other is
Fig. 5 Distribution of the
dimensionless ionic
concentration of K+ near the
negatively charged particle
for different concentration
ratios when κa p = 1
chemiphoresis generated by the induced osmotic pressure gradient around the particle (Lou and Lee 2008a,
b; Hsu et al. 2009, 2010; Zhang et al. 2009; Dukhin 1993,
1995). The induced chemiphoresis always propels the
particle towards higher salt concentration, regardless
of the sign of charge on the particle. Figure 2 depicts
that the resulting diffusiophoretic motion drags the
α=1
α=0.1
α=5
1
1.1
20
5
1.1
1.1
20
2
1.2
5
20
2
1.2
5 2
1.2
1.1
1.1
1.1
1
Microgravity Sci. Technol. (2010) 22:329–338
Fig. 6 Distribution of the
dimensionless ionic
concentration of Cl− near the
negatively charged particle
for different concentration
ratios when κa p = 1
335
α=1
α=0.1
0.9
α=5
1
0.9
0.8
0.9
0.5
0.8
0.1
0.1
0.5
0.1 0.5
0.8
0.9
0.9
1
particle towards lower salt concentration; therefore the
diffusiophoresis is mainly controlled by the induced
electrophoresis instead of chemiphoresis under the
considered conditions. Since the diffusion mobilities of
the anions and cations in KCl electrolyte are almost
the same order of magnitude, the induced electric field
arising from the difference in the ionic diffusivities,
Ediffusivity , is very small, the induced electrophoresis
driven by the generated electric field, Ediffusivity , thus
is negligible. In addition to the generated electric field
due to the difference in the ionic diffusivities, an electric
field, leading to electrophoresis, is induced by the induced dipole moment, a consequence of DLP (Dukhin
1993, 1995). Due to the imposed concentration gradient, the double layer is polarization resulting in higher
concentrations of counterions and coions and thinner
EDL on the high-concentration side and lower concentrations of counterions and coions and thicker EDL on
the low-concentration side of the particle. Note that
the DLP stated here refers to the concentration polarization only by the imposed concentration gradient,
and did not include the relaxation effect arising from
the imposed and generated electric field, movement of
the particle, and the fluid convection. Due to the DLP,
the center of charge inside the EDL shifts away from
the particle’s center. Together with the charge of the
particle and the charge of the EDL, a dipole moment
is induced, which induces an electric field reaching
0.9
beyond the limits of the EDL (Dukhin 1993, 1995).
The concentration polarization inside the EDL by the
imposed concentration gradient is named as the type I
DLP, and its resulting electric field is named as EI−DLP ,
the direction of which is opposite to that of the applied
concentration gradient when the particle is negatively
charged (Lou and Lee 2008a, b; Hsu et al. 2009, 2010;
Zhang et al. 2009). For example, the induced electric
field due to the type I DLP, EI−DLP , is directed from
higher salt concentration towards lower salt concentration (i.e., upward for α < 1 and downward for α > 1).
Meanwhile, the concentration of the coions near the
outer boundary of the EDL on the high-concentration
side is higher than that on the low-concentration side
of the particle (i.e., the cases of α = 0.1 and 5 in Fig. 6),
which is called the type II DLP, generating an electric
field, EII−DLP , the direction of which is opposite to
that established by the counterions inside the EDL,
EI−DLP (Lou and Lee 2008a, b; Hsu et al. 2009, 2010;
Zhang et al. 2009). Therefore, when a concentration
gradient is imposed, an electric field is generated by
three mechanisms: the first one, Ediffusivity , is established
by the difference in the ionic diffusivities; the second
one, EI−DLP , is established by the type I DLP inside
the EDL; and the third one,EII−DLP , is generated by
the type II DLP near the outer boundary of the EDL.
The induced EI−DLP by the type I DLP always propels
the particle towards higher salt concentration, while the
336
EII−DLP generated by the type II DLP always drags the
particle towards lower salt concentration, regardless of
the sign of charge on the particle. For thin EDL (i.e.,
κa p is large), usually, the electric field generated by the
type I DLP, EI−DLP , is stronger than that from the type
II DLP, EII−DLP , if the ratio of the nanopore size to
the particle size, a/ap , is very large (the boundary effect
arising from the nanopore wall is insignificant), and the
magnitude of the particle’s surface charge is relatively
low (Hsu et al. 2010); however, the electric field arising
from the type II DLP dominates over that from the
type I DLP if the surface charge or surface potential
of the particle is relatively high (Hsu et al. 2010). If κa p
is small (thick EDL), usually the electric field induced
by the type II DLP dominates over that by the type I
DLP. Depending on the difference of the diffusivities
of the cations and anions which depends on the type
of salt used, and the sign of the surface charge of the
particle, the induced electrophoretic motion generated
by the induced Ediffusivity might drive the particle towards either low- or high-concentration side. Usually,
the induced electrophoretic effect by Ediffusivity is more
significant than that driven by EI−DLP and EII−DLP if
the boundary effect is insignificant (i.e., large a/ap and
κa p ). In the present study, since Ediffusivity is very small
due to the almost identical diffusivities of ions K+ and
Cl− , the induced electrophoresis driven by the electric
fields EI−DLP and EII−DLP arising from DLP dominates
since the EDL thickness, the particle size and the pore
size are of the same order of magnitude.
Since the particle’s surface charge density is relatively high (i.e., σ p = −0.1 C/m2 ), the diffusiophoresis
is dominated by the induced electrophoresis driven by
the electric field generated by the type II DLP, EII−DLP ,
which propels the particle towards lower salt concentration. For α < 1, the salt concentration beneath the
particle is higher than that above the particle, therefore,
the generated electric field, EII−DLP , by the type II
DLP is opposite to the imposed electric field, consequently, the induced electrophoretic motion generated
by EII−DLP is opposite to the electrophoretic motion
driven by the externally imposed electric field. Since
the generated EII−DLP increases with the imposed concentration gradient, EII−DLP increases as α decreases
(or 1 − α increases). Therefore, as α decreases from
α = 1, the particle’s electrophoretic motion is slowed
down by the opposite diffusiophoretic motion, leading
to the decrease in the particle velocity, as shown in
Fig. 2. As α further decreases, the induced EII−DLP
exceeds the imposed electric field, and the particle
motion is then reversed, as shown in Fig. 2. After the
reversion, as α further decreases, the particle velocity
increases due to the increase in the generated electric
Microgravity Sci. Technol. (2010) 22:329–338
field EII−DLP . When the concentration ratio α is less
than a certain value, the dimensionless ionic concentrations, normalized by C0 , in the left and right reservoirs
shown in Fig. 1 are, respectively, c∗1 = c∗2 = 2/(1 + α) →
2 and c∗1 = c∗2 = 2α/(1 + α) → 0, and the concentration
gradient saturates as α further decreases, resulting in
a saturated particle velocity, as shown in Fig. 2. Since
the magnitude of the generated electric field EII−DLP
dominates over the imposed electric field, the overall
electric field in the gap between the particle and the
nanopore wall is directed opposite to that of the imposed one, and drags the positive net charges in the gap
downward, as shown in Fig. 4 for α = 0.3. For α > 1,
the salt concentration above the particle is higher than
that below the particle, resulting in EII−DLP , and the
direction of which is the same as the imposed electric
field, leading to the enhancement of the electrophoretic
motion as α increases. When α is above a certain value,
the dimensionless concentrations in the left and right
reservoirs shown in Fig. 1 are, respectively, c∗1 = c∗2 =
2/(1 + α) → 0 and c∗1 = c∗2 = 2α/(1 + α) → 2, and the
concentration gradient saturates as α further increases
leading to the saturation of the particle’s velocity shown
in Fig. 2. Since the EDL surrounding the particle is relatively thick under the condition of κa p = 1, the induced
electrophoresis driven by the generated EI−DLP is not
significant. Since the overall electric field is directed
upward, the positive net charge in the gap between
the particle and the nanopore wall thus moves upward,
shown in Fig. 4 for the case of α = 5.
For κa p = 3 (Fig. 3), the effect of the diffusiophoretic
motion on the electrophoretic motion becomes more
complicated. Comparing to the case of κa p = 1, the
EDL is thinner, and the induced electrophoresis driven
by EII−DLP becomes smaller. The net particle motion
will be the result of the competing three driving forces
generated by the imposed electric field and the generated electric fields including EI−DLP and EII−DLP . For
α < 1, the decrease of the particle’s velocity is primarily
due to the induced electrophoresis driven by EII−DLP ,
which slows down the particle motion. The particle’s
motion is not reversed since the opposite driving force
from EII−DLP is smaller comparing to that of κa p = 1.
For α > 1, the decrease of the particle’s velocity is
mainly due to the induced electrophoresis driven by
EI−DLP , which propels the particle towards higher salt
concentration. In the absence of the imposed electric
field, the same magnitude but opposite sign of the
diffusiophoretic velocity is obtained when the direction
of the imposed concentration gradient is reversed. Similarly, in the absence of the imposed concentration gradient, we obtained the same magnitude with opposite
sign of electrophoretic velocity when the direction of
Microgravity Sci. Technol. (2010) 22:329–338
the imposed electric field is reversed. However, since
the hydrodynamic field, electric field and ionic mass
transport are strongly coupled, as shown in the mathematical model described in “Mathematical Model”,
the net electrokinetic motion driven by the combined
electric field and concentration gradient is not simply
the superposition of the electrophoresis solely driven
by the imposed electric field and the diffusiophoresis
solely driven by the imposed concentration gradient.
The equilibrium double layer surrounding a spherical
particle is spherically symmetrical. The double layer
changes to a non-equilibrium state under the effect of
any action which results in its deformation (Dukhin
1993, 1995). For example, in diffusiophoresis, the thickness of the EDL near the high-concentration side of the
particle is thinner while the EDL is thicker near the
low-concentration side of the particle. In electrophoresis, the externally imposed electric field directed from
bottom to top (i.e., the positive axial direction) displaces the mobile positive counter-ions within the double layer to the top surface of the particle resulting in
thinner EDL with higher concentration of counterions
near the top and thicker EDL with lower concentration
of counterions near the bottom of the particle. For
α > 1, the salt concentration near the top surface of the
particle is higher leading to higher concentration of the
counterions within the EDL near the top surface of the
particle. When an electric field directed from bottom
to top is imposed, since the counterions are displaced
along the double layer from bottom to top by the imposed electric field, more counterions appear on the top
hemisphere, resulting in the enhancement of the type I
DLP effect. Using the similar analysis, the type II DLP
effect is reduced by the DLP arising from the imposed
electric field. Therefore, the driving force arising from
the type I DLP is stronger than that of the type II
DLP, which reduces the particle’s velocity, as shown in
Fig. 3 in the range of α > 1. For α < 1, the salt concentration near the bottom of the negatively charged
particle is higher than that near the top surface of the
particle, leading to higher concentration of counterions
near the bottom and lower concentration of counterions near the top of the particle. The imposed electric
field displaces the positive, counterions within the EDL
towards the top surface of the particle, thus reduces
the counterions’ concentration difference between bottom and top EDL and consequently reduces the type
I DLP effect. Meanwhile, the type II DLP effect is
enhanced by the DLP arising from the imposed electric
field. Therefore, the induced electrophoresis driven by
EII−DLP is stronger than that driven by EI−DLP for α <
1, leading to the reduction of the particle’s velocity, as
shown in Fig. 3.
337
Conclusions
The effects of the induced diffusiophoresis by an imposed concentration gradient on the electrophoretic
motion of a charged particle along the axis of an uncharged nanopore has been numerically investigated
using a continuum model, which consists of Poisson
and Nernst-Planck equations for the potential and ionic
concentrations, and Stokes equations for the flow field.
The model accounts for double layer polarization induced by the imposed electric field and concentration gradient and the compression of the EDL by
the impervious nanopore wall, and is valid for any
thickness of the EDL and the imposed external fields.
Since the EDL thickness, the nanoparticle size, and
the nanopore size are of the same order of magnitude,
the induced diffusiophoresis is dominated by the induced electrophoresis driven by the induced electric
field arising from the double layer polarization. When
the EDL is thick, the diffusiophoretic motion is dominated by the induced electrophoresis driven by EII−DLP ,
which propels the particle towards lower salt concentration, regardless of the sign of the particle’s surface
charge. The diffusiophoretic motion can be used to
enhance, slow down or even reverse the electrophoretic
motion depending on the magnitude and direction of
the imposed concentration gradient. When the EDL
is relatively thin, the induced diffusiophoretic motion
slows down the electrophoretic motion, regardless of
the direction of the imposed concentration gradient.
For a negatively charged particle, the imposed electric
field enhances the type I DLP and reduces the type
II DLP effects if the imposed concentration gradient
and electric field are in the same direction. Otherwise,
the imposed electric field enhances the type II DLP
effect and reduces the type I DLP effect when the
imposed concentration gradient is opposite to the direction of the imposed electric field. It is conceivable that
diffusiophoresis can be used to regulate the nanoparticles translocation process to achieve a nanometer-scale
spatial accuracy for DNA sequencing by controlling
both the electric field and the concentration gradient.
Acknowledgement This work is supported by the World Class
University Grant No. R32-2008-000-20082-0 of the Ministry of
Education, Science and Technology of Korea.
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