Article
pubs.acs.org/Langmuir
Spatiotemporal pH Dynamics in Concentration Polarization near IonSelective Membranes
Mathias B. Andersen,† David M. Rogers,‡ Junyu Mai,§ Benjamin Schudel,§ Anson V. Hatch,§
Susan B. Rempe,*,‡ and Ali Mani*,†
†
Mechanical Engineering Department, Stanford University, Stanford, California 94305, United States
Center for Biological and Material Sciences, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States
§
Biological Sciences and Engineering Department, Sandia National Laboratories, Livermore, California 94551, United States
‡
S Supporting Information
*
ABSTRACT: We present a detailed analysis of the transient pH dynamics for
a weak, buffered electrolyte subject to voltage-driven transport through an ionselective membrane. We show that pH fronts emanate from the concentration
polarization zone next to the membrane and that these propagating fronts
change the pH in the system several units from its equilibrium value. The
analysis is based on a 1D model using the unsteady Poisson−Nernst−Planck
equations with nonequilibrium chemistry and without assumptions of
electroneutrality or asymptotically thin electric double layers. Nonequilibrium
chemical effects, especially for water splitting, are shown to be important for the dynamical and spatiotemporal evolution of the
pH fronts. Nonetheless, the model also shows that at steady state the assumption of chemical equilibrium can still lead to good
approximations of the global pH distribution. Moreover, our model shows that the transport of the hydronium ion in the
extended space charge region is governed by a balance between electromigration and water self-ionization. On the basis of this
observation, we present a simple model showing that the net flux of the hydronium ion is proportional to the length of the
extended space charge region and the water self-ionization rate. To demonstrate these effects in practice, we have adopted the
experiment of Mai et al. (Mai, J.; Miller, H.; Hatch, A. V. Spatiotemporal Mapping of Concentration Polarization Induced pH
Changes at Nanoconstrictions. ACS Nano 2012, 6, 10206) as a model problem, and by including the full chemistry and transport,
we show that the present model can capture the experimentally observed pH fronts. Our model can, among other things, be used
to predict and engineer pH dynamics, which can be essential to the performance of membrane-based systems for biochemical
separation and analysis.
■
INTRODUCTION
Micro- and nanofluidic lab-on-a-chip systems have attracted
considerable attention in the past decade due to their favorable
properties for small-scale biochemical analysis including sample
separation, preconcentration, and detection.1 Around a decade
ago, it was realized that the interface region at an ion-selective
element (such as a nanopore or a membrane) is effective for
biochemical separation and analysis when subjected to an
electrical bias.2−5 However, even though the initial studies were
promising, the fundamental understanding of basic physicochemical transport phenomena at such interfaces was lacking
and is still not fully developed. This understanding not only is
sought in the context of biochemical analysis, but also is
essential in other applications relying on ion-selective interfaces
such as electrodialysis and the production of chemicals,6
microfluidic pumps,7 and soil remediation.8 Depending on the
geometric confinement and fluidity of the electrolyte, several
phenomena can play a role in the transport characteristics
spanning concentration polarization,9 deionization shocks,10,11
electrohydrodynamic instabilities,12 and enhanced chemical
reactions.13 In this work, we focus on elucidating the role
played by chemical interactions in the electrolyte within and
© 2014 American Chemical Society
outside of a membrane and their consequences in biochemical
transport. Our work is distinct from previous work in a number
of aspects, including nonequilibrium chemical effects, pH
transport, buffering weak acids/bases, considering nonelectroneutrality effects, and numerically resolving all electrochemical
boundary layers. Hydrodynamic effects are assumed to be
mostly negligible due to the presence of immobilizing
polymers.
Figure 1 shows a schematic of the prototypical microchannel−membrane system studied here. Such a system was
recently used to elucidate aspects of chemical effects in
experiments by Mai et al.,14 in which a ratiometric dye was used
to make concentration-independent measurements of pH.
These experiments allow for a much more rigorous comparison
to theory due to the observations in both space and time,
whereas previous studies have been restricted to only one of
these dimensions, mostly the temporal one.9,15−18 Here, we
develop a model that captures the observed pH variations in
Received: April 14, 2014
Revised: June 2, 2014
Published: June 3, 2014
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of the microchannel and nanopores, passivation of the
microchannel surface charge by covalent polyacrylamide
grafting, and the addition of viscosity-enhancing agents to the
solution.14
There has been less attention to how ionic transport in
porous media couples with chemical interactions. The majority
of previous studies have assumed either steady state,30−32
electroneutrality, equilibrium chemistry, fixed pH values, binary
electrolytes, or combinations thereof.16,23,33−38 All of these
assumptions are relaxed in the present work. A key point to
emphasize in the context of chemical interactions is the
presence of both homogeneous and heterogeneous reactions,
i.e, reactions both in the bulk solution and at the surface groups
on the pore wall.39
Water splitting was one of the first mechanisms proposed to
explain the ionic transport properties during strong electrical
forcing at ion-selective membranes. Simons40 considered the
nonequilibrium of the water splitting reaction and argued that
the large fluxes of hydronium and hydroxyl ions could be
attributed to catalyzing effects of the protonation and
deprotonation reactions of water with the surface groups on
the membrane. This catalysis hypothesis, which essentially
corresponds to larger numerical values of the water-splitting
rate constants, has been supported in several other studies.9,17,18,25,35,36,41 The catalysis effect would also provide an
explanation of the difference in water splitting between
membranes constituting different surface groups per the
observation that water splitting in many cases is faster at
anion-selective membranes relative to that at cation-selective
membranes.9,17,18,41 A systematic review of the water-splitting
mechanism was given early on by Zabolotskii et al.,25 who
stressed the importance of including water splitting in the
modeling of membrane systems in the overlimiting-current
regime. In many studies,9,17,18,41 temporal pH changes up to
several units have been measured on both sides of anion- and
cation-selective membranes, but without accounting for the
spatial distribution, as done here. Measurements of the partial
currents have also shown that the increased flux of H+ and OH−
in general is too small to account for the observed overlimiting
current.9,41 Recently, bipolar membranes, essentially two
membranes of opposite surface charge in close contact, have
been used as effective water splitters as a means of regulating
the pH in microfluidic channels.42,43
Previous work on systems related to the one studied here has
been done in the group of Tallarek, where several studies of
preconcentration in microfluidic channels have been performed.44−46 However, where our focus is on the coupling of
chemical effects with drift diffusion, their focus has primarily
been on the coupling of advection with drift diffusion.
The detailed model developed in the present work can be
used to map out a reduced-order (or lumped parameter/shortcut) model,43 which could be more feasible for engineering
design and optimization studies. In particular, the present
model has the capability to study membranes of lower acidity
(higher pK value). Membrane acidity is another feature that can
be leveraged in the engineering of biochemical separation and
analysis.47−50 Furthermore, the present model can be used to
explore possible strategies to avoid unwanted concentration
polarization and pH variations. These strategies include pulsed
electrical forcing51 and biasing the porous membrane with a
gating voltage.47
The paper is organized in the following format. In the
Mathematical Model section, we introduce the system
Figure 1. Schematic of the microfluidic membrane system. At t = 0, a
potential difference ΔV is applied between the inlets (x1 and x6),
leading to concentration polarization around the membrane. Moreover, as predicted in this study, pH fronts emanate from the membrane
and lead to a variation in the pH of up to several units. The positive
direction is from left to right.
both space and time, thereby achieving heretofore unprecedented prediction capabilities. Some of the key features of the
model are nonequilibrium chemical effects, pH transport, and
directly capturing charge dynamics without assuming electroneutrality.
The ability to predict and engineer the local pH at
nanoporous interfaces is crucial to the use of such systems in
the detection and analysis of biological tracers, which are often
extremely sensitive to the local pH environment. Furthermore,
the mesoscopic continuum model developed in the present
work is envisioned to inform more detailed nanoscale models,
such as molecular dynamics simulations of ionic transport into
nanopores,19 with the local spatiotemporal pH, electric field,
and ionic strength. Such a combined approach of multiscale
modeling is a necessary predictive tool for the systematic
development of microfluidic membrane systems for the
detection and analysis of biomolecules.
Initial feasibility studies of microfluidic membrane systems
for biochemical analysis and preconcentration have already
been demonstrated. For instance, Hatch et al.20 used a
membrane for the preconcentration of proteins prior to
electrophoretic separation in a sieving gel and showed the
detection of proteins down to a concentration of 50 fM. It was
also found that the concentration polarization has a detrimental
effect on the reproducibility, which may be understood by
improved and more accurate models such as the one pursued in
the current work. Additionally, portable devices for the
detection of biological toxins and pathogens were demonstrated
by Meagher et al.,21 who showed the viability of integrating
microfluidic membrane systems, similar to the one considered
in this work, into a portable, self-contained device.
General reviews of the various transport mechanisms at ionselective membranes subject to high electrical forcing were
recently given by Nikonenko et al.13,22 In these reviews,
overlimiting current, i.e., the transport of charge at rates beyond
diffusion limitation, is generally attributed to four distinct
mechanisms: (i) water splitting,23−25 (ii) current-induced
convection such as electro-osmotic instability,26,27 (iii) co-ion
leakage, and (iv) current-induced membrane discharge.28
Furthermore, the reviews point out that there is a strong
need for additional understanding of the fundamental
phenomena including how water splitting proceeds at the
membrane−solution interface and how weak acids and bases
influence transport in the system. In the present work, we focus
on such chemical interactions and assume that convection
phenomena, such as the electro-osmotic instability,27,29 can be
ignored due to the confinement effects by the small dimensions
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Table 1. Overview of the Ionic Species and Their Electrochemical Propertiesa
fixed ions
mobile ions
symbol
description
i
zi
ci
Ki
ki
species index
valence
concentration
equilibrium constant
reaction rate constant
total concentration
salt ions
1
zcat
[Czcat]
buffer ions
2
zan
[Azan]
3
zbuf
[BHzbuf]
Kbuf
kf,buf, kb,buf
cbuf
water ions
4
zbuf − 1
[Bzbuf−1]
5
−1
[OH−]
Kw
kf,w, kb,w
membrane ions
6
+1
[H+]
7
zmem
[MHzmem]
Kmem
kf,mem, kb,mem
cmem
8
zmem − 1
[Mzmem−1]
a
Throughout the text, we will interchangeably use index notation (i = 1, 2,..., 8) and name notation (cation, cat; anion, an; buffer, buf; water, w;
membrane, mem) as well as denote concentration either by brackets [Czcat] or by ci. Note that as the unit of concentration is mM, the unit of the
equilibrium constant and the reaction rate constant is mM as well. However, the pH value is based on the concentration of hydronium with a unit of
M, i.e., pH = −log10([H+]/1 M).
reactions, namely, the buffer reaction, the water-splitting
reaction, and the protonation/deprotonation reaction inside
the membrane. These chemical reactions are not assumed to be
in equilibrium, but are modeled directly using their respective
forward kf,i and backward kb,i reaction rate constants as well as
their equilibrium dissociation constants Ki. As indicated in
Figure 1, for the purpose of comparing to the experimental
data, the ionic species are identified to be sodium Na+, chloride
Cl−, and phosphate ions H2PO4− and HPO42−. Also, since the
membrane used in Mai et al.14 was negatively charged, the
surface groups in the experiment are assumed to titrate between
negative and neutral forms.
Governing Equations. We employ the standard continuum formulation of ionic mass conservation in the limit of an
ideal dilute electrolyte,53
geometry and definitions. The governing equations with
boundary and initial conditions are developed. A short
description of the numerical model is given. Then we present
the Results and Discussion. We start by giving a thorough
motivation of the parameters used in the model. Then we
compare the theoretical predictions of pH dynamics to the
experimental observations. Potential causes for discrepancies
are discussed. After the benchmark, the model is used to
investigate the detailed dynamics of the system, including an
analysis of the much-debated assumption of chemical
equilibrium. We then give a summary and conclusions.
■
MATHEMATICAL MODEL
System Geometry and Definitions. As shown in Figure
1, we employ a 1D model of the system, with the axial
coordinate being x. Thus, the model assumes uniformity in the
transverse directions and neglects fluid flow. An ion-selective
membrane, which spans the entire cross section, is located
between x3 and x4. The inlets of the microchannel, between
which a voltage ΔV is applied at time t = 0, are located at
positions x1 and x6. There is a large separation of length scales
in the system, ranging from the small Debye screening length of
O(1 nm) up to the microchannel length of ∼O(1 cm), a ratio
of 107. That large range of length scales makes modeling the
system challenging since we do not resort to any asymptotic
approximations of the thin boundary layers such as the electric
double layer (EDL) and the extended space charge layer
(ESC).52 However, during the time of interest, which is O(10
s), large parts of the microchannel remain unchanged. We can
model these sections (from x1 to x2 and from x5 to x6) as ohmic
resistors of fixed concentration. This confinement of the
variable regions close to the membrane is sketched in Figure 1,
where the pH profile outside the membrane is uniform at
equilibrium but nonuniform between x2 and x5 during voltage
bias.
The model is verified by comparison to the experimental data
from Mai et al.,14 in which phosphate-buffered saline was used
as an electrolyte. Thus, as indicated in Table 1, the model is
generalized to take into account multiple ionic species. Those
ionic species are a fully dissociated salt cation Czcat of valence
zcat, a fully dissociated salt anion Azan of valence zan, a buffer
consisting of a weak acid BHzbuf of valence zbuf and its conjugate
base Bzbuf−1 of valence zbuf − 1, the hydroxyl ion OH− and
hydronium ion H+, and inside the membrane a fixed weak acid
surface group MHzmem of valence zmem and its conjugate base
surface group Mzmem−1 of valence zmem−1. We assume that the
concentration of the dye that tracks the pH is sufficiently small
to be neglected. The model accounts for three chemical
∂j
∂ci
+ i = ri
∂x
∂t
(1)
where ci, ji, and ri are the concentration (with units of mM =
mol/m3), flux, and source term, respectively, of species i. As we
have neglected fluid flow, the flux is due solely to diffusion and
electromigrative drift and, continuing with the assumption of an
ideal dilute electrolyte, is given by the Nernst−Planck equation,
⎛ ∂c
z ∂ϕ ⎞
ji = −αDi⎜ i + i ci ⎟
VT ∂x ⎠
⎝ ∂x
(2)
in which Di, zi, VT = RT/F, R, F, and ϕ are the diffusion
constant, ionic valence, thermal voltage, gas constant, Faraday
constant, and electrostatic potential, respectively. The membrane factor α is introduced to account for effects such as the
porosity, tortuosity, and constrictivity incurred by the
membrane. α has a value of unity in the microchannel (x1 <
x < x3 and x4 < x < x6) and a value below unity in the
membrane (x3 < x < x4),
⎧1,
in the microchannel
α=⎨
⎩ αmem , in the membrane
⎪
⎪
(3)
Flux matching is always ensured at the microchannel−
membrane interfaces (at x3 and x4). We emphasize our
assumption of a dilute electrolyte, which corresponds to all
activity coefficients being unity. Some nonideal effects at high
concentrations are therefore not captured, but we sacrifice this
accuracy to allow a more direct probing of effects of interest,
i.e., the importance of nonequilibrium chemistry and pH
dynamics.
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species i = 8 from the system of equations, whereby the source
term for species i = 7 becomes
The source term ri in eq 1 is due to chemical reactions
between the ionic species. Our model accounts for three
reactions of the protonation/deprotonation type: (i) a buffer
reaction, (ii) the water-splitting reaction, and (iii) the chargeregulation reaction in the membrane,
r7 = k b,mem[(cmem − c 7)c6 − K memc 7]
Electrostatic interactions are governed by the Poisson
equation,
k f,buf
z buf
z buf − 1
+
BH(aq)
HooooI B(aq)
+ H(aq)
k b,buf
8
∂ ⎛ ∂ϕ ⎞
⎜−ε
⎟ = ρ = F ∑ zc
i i
e
∂x ⎝ ∂x ⎠
i=1
(4a)
k f,w
H 2O(aq) HooI OH−(aq) + H+(aq)
k b,w
k b,mem
(4c)
Here, kf,i and kb,i are the forward and backward reaction rate
constants, respectively, with their unit based on mM and not M.
The phase of the species has been explicitly indicated in eq 4 to
emphasize that while the buffer and water-splitting reactions
occur exclusively between dissolved species, the charge
regulation reaction occurs between the solid-phase membrane
mem−1
species (MHz(s)mem and MHz(s)
) and the dissolved hydronium
+
ion (H (aq)). Following classical chemical kinetics,54 the source
term ri is the difference between the forward and backward
reaction rates, found as the product of the rate constant and the
concentrations (with r1 = r2 = 0 since the salt ions are
completely dissociated),
λD =
(6b)
r8 = −r7
(6c)
εRT
2F 2I
(9)
in which
I=
1
2
6
∑ zi 2ci
(10)
i=1
is the ionic strength, essentially the effective concentration of
the electrolyte. Note that our definition of the ionic strength
accounts only for the mobile ions, i.e., it does not include the
fixed ionic surface groups in the membrane. The Debye length
is the characteristic width of the EDLs at the microchannel−
membrane interfaces at x3 and x4 and is around 1 nm for an
electrolyte with an ionic strength of 100 mM.
Boundary and Initial Conditions. Equilibrium Ionic
Composition. We assume that the system is fully equilibrated
at t = 0 right before the voltage is applied. Thus, we find the
initial ionic composition by solving the governing equations
while neglecting unsteady and gradient terms, i.e., essentially
solving ri = 0 subject to the electroneutrality condition,
where Ki = kf,i/kb,i is the equilibrium constant with unit based
on mM and not M. In the special case of the water splitting
reaction, Kw = kf,w[H2O]/kb,w with the concentration of water
[H2O] being assumed constant. We note again, that an
overview of the ionic species and their indices is given in Table
1. The remaining source terms are expressed in terms of the
ones in eq 5
r4 = −r3
(6a)
r6 = −r3 + r5 − r7
(8)
where ε = εrε0 is the dielectric permittivity in which εr and ε0
are the relative and vacuum permittivities, respectively, and ρe is
the free charge density. Note that all ionic species are included
in the sum of the charge density in eq 8, including the
membrane ions that are responsible for the fixed charge in the
membrane. It is this smeared-out, or volume-averaged, charge
from the membrane ions that leads to membrane selectivity and
thus concentration polarization and other interesting effects to
be analyzed in this work. The length scale over which
electrostatic relaxation occurs is the Debye length
(4b)
k f,mem
z mem
z mem − 1
MH(s)
HoooooI M(s)
+ H+(aq)
(7)
6
∑ zici = 0
(11)
i=1
Furthermore, ri = 0 corresponds to chemical equilibrium
whereby it is possible to express through eq 5 all ionic
concentrations in terms of the hydronium concentration,
The membrane ions (i = 7, 8) constitute a special case for
which the modeling can be immediately reduced to a single
variable in the following way. We add the conservation
equation (eq 1) whereby, by virtue of eq 6c, the source
terms cancel. Additionally, since the membrane ions are fixed,
their flux is identical to zero. The resulting equation states that
the sum of the concentrations of the membrane ions is constant
in time,
c 7 + c8 = cmem
[B zbuf − 1] = cbuf
Kbuf
Kbuf + [H+]
(12a)
[BH zbuf ] = cbuf
[H+]
Kbuf + [H+]
(12b)
[OH−] =
where cmem is the total concentration of fixed surface groups in
the membrane. We make use of this condition to eliminate
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Kw
[H+]
(12c)
[M z mem − 1] = cmem
K mem
K mem + [H+]
(12d)
[MH z mem] = cmem
[H+]
K mem + [H+]
(12e)
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Table 2. List of Parametersa
symbol
description
value
unit
ref
symbol
m2 s−1
m2 s−1
m2 s−1
56
56
56
zcat
zan
zbuf
cation valence
anion valence
buffer top valence
1
−1
−1
m2 s−1
56
zmem
membrane top valence
0
OH− diffusivity
H+ diffusivity
Na+ concentration
1.3 × 10−9
2.0 × 10−9
0.96 ×
10−9
0.76 ×
10−9
5.3 × 10−9
9.3 × 10−9
160
m2 s−1
m2 s−1
mM
56
56
14
pKbuf
pKw
pKmem
7.2
14
−2
can
Cl− concentration
150
mM
14
kb,buf
cbuf
total concentration of buffer ions
6
mM
14
kb,w
cmem
total volume concentration of surface groups in 300
membrane
applied voltage
100
width of membrane
5 × 10−5
width of resolved parts of microchannel
7.5 × 10−4
buffer reaction equilibrium constant
water splitting equilibrium constant
membrane reaction equilibrium
constant
backward rate constant of buffer
reaction
backward rate constant of water
splitting reaction
backward rate constant of membrane
reaction
membrane factor
relative permittivity
temperature
D1
D2
D3
Na+ diffusivity
Cl− diffusivity
H2PO4− diffusivity
D4
HPO42− diffusivity
D5
D6
ccat
ΔV
Lmem
L23,
L45
L12,
L56
width of remaining parts of microchannel
2 × 10−2
kb,mem
mM
V
m
m
14
14
m
14
αmem
εr
T
description
value
107
109
107
0.2
78
293
unit
ref
56
56
mM−1
s−1
mM−1
s−1
mM−1
s−1
K
a
The specific choice of some parameter values is discussed in the main text. Note that due to tradition in the literature, pK values are computed
based on an equilibrium constant K in units of M and not mM, i.e., pK = −log10(K/1 M). Not listed in the table are the vacuum permittivity ε0 =
8.854 × 10−12 F m−1, the gas constant R = 8.314 J mol−1 K−1, and the Faraday constant F = 96 485 C mol−1.
ϕ|x2 − ϕ|x1
where cbuf is the total concentration of buffer ions. Using these
relations in the electroneutrality condition (eq 11) yields a
nonlinear algebraic equation for the hydronium concentration
that can be solved numerically. The result is the equilibrium
concentration of [H+] in free solution, which in combination
with eq 12 gives the equilibrium concentration of all ionic
species. These equilibrium concentrations are applied as fixed
Dirichlet boundary conditions at x2 and x5 (Figure 1) under the
assumption that these positions are far enough from both the
driving electrodes at x1 and x6 and the membrane interfaces at
x3 and x4 so as not to experience any significant change in the
concentrations during the time of interest. In practice, widths
L23 = x3 − x2 and L45 = x5 − x4 of the fully resolved parts of the
microchannel are taken to be around 15 times the membrane
width Lmem = x4 − x3, while widths L12 = x2−x1 and L56 = x6 −
x5 of the remaining parts of the microchannel are on the order
of centimeters, i.e., tens of thousands of times larger than Lmem.
However, as we shall see, most of the voltage drop occurs in the
computational region shortly after the voltage is applied.
Matching of Electric Field. The electrostatic boundary
condition is derived by assuming uniform conductivity from the
driving electrodes at x1 and x6 to positions x2 and x5. This
agrees with the assumption of negligible concentration
variations. Furthermore, the overpotentials driving the
electrode reactions are assumed to be at most O(1 V), which
is negligible compared to the typical applied voltages of O(100
V) and above. Thus, in accordance with Ohm’s law,
i = σE
L12
ϕ|x6 − ϕ|x5
L56
=
=
∂ϕ
∂x
x2
(14a)
∂ϕ
∂x
x5
(14b)
in which the electrode potentials are ϕ|x1 = ΔV and ϕ|x6 = 0.
Initial Condition with Membrane. The initial condition is
found numerically as the steady-state solution to the governing
equations (eqs 1 and 8) with the above-mentioned boundary
conditions. In practice, to find the steady-state solution, an
iterative procedure is employed in which the membrane
concentration cmem is increased from a small fraction of its
final value and where the initial guess for each iteration is the
solution from the previous step. In this manner, the initial
condition for the unsteady calculations is ensured to be
physically equilibrated and numerically consistent. The initial
condition found in this manner produces a Donnan potential
consistent with theory.55
Numerical Scheme. We develop our own numerical
scheme to solve the governing equations (eqs 1 and 8). The
numerical integration is challenging in particular because the
system is characterized by large separations in time and length
scales, ranging from the small charge-relaxation time (∼0.1 ns)
to the diffusion time (∼10 s) and from the small Debye length
(∼1 nm) to the microchannel length (∼1 cm). Our numerical
scheme uses a nonuniform grid that fully resolves the smallest
features, such as the EDL and ESC boundary layers, and
models the large microchannels through effective boundary
conditions. Additionally, the scheme is based on an implicit
time integration, and the time step is adjusted appropriately
throughout the simulation such that the transient dynamics is
properly resolved. The scheme conserves mass at the level of
discretization and is second-order accurate in both space and
time. It is based on a so-called delta-form iteration to converge
(13)
where i, σ, and E = −∂xϕ are the current density, conductivity,
and electric field. The electric field is uniform from x1 to x2 and
from x5 to x6. Additionally, since at x2 and x5 the current is
ohmic, the continuity of the current density results in mixed
Robin boundary conditions for the potential at x2 and x5,
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estimates from the literature.58 Again, the model shows little
sensitivity to the exact value of kb,buf. For the ubiquitous water
splitting, the rate constant in free solution is known as kfree
b,w =
1.4 × 108 mM−1 s−1.58 However, as explained, there is a large
body of literature showing that the rate constant of water
splitting is enhanced by orders of magnitude in concentration
polarization.9,13,15,18,25 We therefore use kb,w = 109 mM−1 s−1 as
the nominal value and present the impact of variation of this
constant over orders of magnitude. We emphasize that the
simple uniform increase in kb,w employed here mimics the more
detailed models quite naturally since the actual value of kb,w
becomes important only in the concentration depletion zone.
The membrane factor αmem accounts for the impacts of
porosity, tortuosity, and constrictivity on the ionic transport in
the membrane. It is difficult to determine a priori and it was not
measured in the experiments. Thus, in accordance with the
literature,28,44,59 we use αmem = 0.2.
Comparison to Experimental Data: Spatiotemporal
pH Fronts and Current. Figure 2 shows model prediction of
on the solution for each time step. The governing equations are
solved in a dimensionless form, which can be seen in the
Supporting Information. Moreover, details of the delta-form
discretization are also found in the Supporting Information.
The scheme is implemented and solved in Matlab.
■
RESULTS AND DISCUSSION
We model the experimental setup used in Mai et al.14 and note
that, even though the data from this experiment is novel, it is
also limited and there is uncertainty in the experimental value
of some of the parameters in the model. A list of the parameters
and their values is given in Table 2. Thus, we do not aim for a
conclusive determination of the model parameters by rigorously
fitting to the experimental data. Rather, we will emphasize the
fact that the model predicts qualitatively the same novel
spatiotemporal pH dynamics as seen in the experiment.
Additionally, the model provides detailed insight into the
dynamics and the electrochemical structure of the system,
which are not observable in experiments. Moreover, the model
illustrates to which degree the assumption of chemical
equilibrium holds. Some of the reasoning behind the values
in Table 2 chosen for the model is as follows.
Regarding the salt and buffer concentrations, Mai et al.14
used a commercial buffer specified as 150 mM sodium chloride
(NaCl) with 10 mM phosphate (NaH2PO4, Na2HPO4, and
Na3PO4). For the phosphate component, we ascribe 10 mM to
sodium and adjust cbuf until the pH value at equilibrium in the
model matches that in the experiment, which results in cbuf = 6
mM. We then follow the stated values and use can = 150 mM
and ccat = 160 mM, with the latter being the sum of
contributions from the sodium chloride and phosphate
components. We note that for these concentrations the
Debye length is λD ≈ 0.8 nm outside the membrane.
The total concentration of surface groups in the membrane
cmem is adjusted by hand until optimal agreement is found
between model and experiment, leading to cmem = 300 mM.
This is a reasonable value considering that Mai et al.14 used an
acrylamide monomer and bis(acrylamide) cross-linker mixture
with a concentration of around 6 × 103 mM to polymerize the
membrane. That concentration sets the upper limit for the
concentration of surface groups in the membrane. Typically,
only a small fraction of the amine groups on the acrylamide
monomer hydrolyze into carboxylic groups, and the ratio of
∼1/20 here is consistent with typical numbers in the literature.
We also note that, for this membrane concentration, the Debye
length is λD ≈ 0.5 nm inside the membrane, which our model
fully resolves.
The nature of the chemical groups inside the membrane is
unknown. Complementary experiments would be needed for
an independent determination of pKmem. We use pKmem = −2
since this gives robust agreement between the model and data.
This low value of pKmem minimizes current-induced membrane
discharge effects and renders the membrane fully ionized. A
more rigorously determined value of pKmem would be based on
additional experimental data that is currently not available.
Rate constants can be difficult to find in the literature. For
the surface protonation/deprotonation reaction in the membrane, we note that Hoogerheide et al.57 measured a backward
reaction rate for silanol groups on silica of 7 × 108 mM−1 s−1,
which indicates that surface reactions can proceed fast. On the
basis of this, we settle for kb,mem = 107 mM−1 s−1 and note that
the model shows little sensitivity to the actual value of kb,mem.
For the buffer reaction, we use kb,buf = 107 M−1 s−1 based on
Figure 2. pH profiles across the membrane at different times. (a)
Experimental data. (b) Model prediction. The left edge of the
membrane is at x = 0, and its width is 50 μm. The inset shows the full
range of pH predicted by the model.
pH fronts around an ion-selective membrane in qualitative
agreement with the experimental data. Both the experiment and
model show pH variations up to several units in magnitude
emanating from the membrane over the course of tens of
seconds. These pH modulations occur despite the relatively
large phosphate buffer concentration. The immediate conclusion for biomolecule separation and analysis is clear, namely,
that the pH around ion-selective membranes cannot be
assumed to be uniform.
There are several potential causes of the quantitative
discrepancy between the model and the data in Figure 2. For
one, there is the experimental uncertainty, which manifests
itself in various ways: (i) The accuracy of the ratiometric
fluorescence microscopy is highest at a pH of around 7 and
saturates below pH = 6 and above pH = 9. Titration
experiments in bulk solution show the ratiometric approach
to be relatively insensitive to pH below 6, and the dye was
depleted below its detection limit in the depletion zone. pH
values outside the range between 6 and 9 from experimental
measurements should be viewed as being uncertain, and as seen
in Figure 2, the pH moves significantly outside these limits. (ii)
The system may not have been completely equilibrated before
the start of the experiment, since at t = 0 (blue curve in Figure
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surprising. The other model curves in Figure 3(a) pertain to
increasing values of kb,w and will be addressed below.
Measurements of the current provide a complementary
observable for comparison between the model and data. Figure
3(b) plots the current as a function of time. The time
resolution of the experimental current measurements is
somewhat coarse with a measurement only every 30 s. In
contrast, as seen in Figure 3(b) and its inset, the model predicts
variations in the current on time scales down to 0.1 s. Given
these few experimental data points in the time interval captured
by the model, we also plot in Figure 3(b) the experimental
asymptote in the limit of long times (the experimental current
settles at this level after approximately 120 s). From Figure
3(b) we note that the model and data agree with respect to the
magnitude of the current and its decreasing trend in time
toward an asymptotic value.
Concentration Polarization. It is well known that
concentration polarization occurs around ion-selective membranes when subjected to an electric field.53 In this
phenomenon, concentration decreases on one side of the
membrane and increases on the other. The concentration
difference across the membrane grows both in time and with
the strength of the applied electric field. For a suddenly applied
strong electrical forcing, either at or above the limiting current,
Sand’s time,
2) the pH to the left of the membrane does not equal that to
the right of the membrane, but is around 0.5 unit higher. From
this effect alone, we could expect deviations between the model
and experiment of up to 0.5 pH unit.
Before moving on to a more quantitative analysis, we address
some uncertainties in the model. As seen in Figure 2, the
agreement in pH between the model and experiment is less
successful to the left of the membrane, where the propagation
of the pH front is too fast in the model prediction. To the right
of the membrane, the model predicts very low pH values.
These discrepancies could be due to the omission of the two
phosphate buffer reactions
H3PO4 ⇌ H 2PO4 − + H+, pK = 2.1
(15a)
HPO4 2 − ⇌ PO4 3 − + H+, pK = 12.7.
(15b)
Given that the pK value of the phosphate reaction that we do
account for is 7.2, the neglect of eq 15 is less accurate for low
and high pH values and is especially critical for pH below 4 and
above 10. The reactions in eq 15 act to buffer, or slow down
and minimize, the speed and magnitude of the pH fronts, and
as is clear in Figure 2, this effect would improve the agreement
between the model and data. The reason that the reactions in
eq 15 are not included in the model in its current
implementation is primarily that the additional dependent
variables would make the problem less numerically tractable.
However, even with these limitations, the model reasonably
captures the main trends of the data in Figure 2.
We proceed with a more quantitative comparison between
the model and experiment. As mentioned above, for both the
experiment and the model, we have the greatest confidence in
intermediate pH values of around 7. It therefore makes sense to
use this pH value for a more quantitative comparison, and we
introduce xpH = 7.0 as the position to the right of the membrane
(x > x4) where the pH equals 7. Figure 3(a) plots xpH = 7.0 as a
τsand =
2
π ⎡ zFc0 ⎤
D⎢
⎥
4 ⎣ i ⎦
(16)
is the characteristic time for the concentration to reach
approximately zero on the depletion side of the membrane.
We assume z = 1, D = 1.5 × 10−9 m2 s−1, and c0 = 150 mM,
while the inset in Figure 3(b) gives the current density, i = 5 ×
103 A m−2. With these parameters, τsand ≈ 0.01 s, which is an
order of magnitude less than the 0.1 s observed by the kink in
the curve in the inset of Figure 3(b). However, this discrepancy
is expected since the expression for Sand’s time in eq 16 is
based on an ideally selective membrane. As the ratio of the
electrolyte ionic strength to the membrane concentration is
quite high, here I/cmem ≈ 150/300 = 1/2, the membrane is far
from ideal. Thus, a longer time is expected for salt depletion to
develop, and this is in accordance with the observed
discrepancy.
The concentration polarization phenomenon can be seen in
Figure 4(a) and (b), which shows the concentration of cations
and anions at different times. These salt ion concentrations
reach values down to 10−4−10−3 mM in the depletion zone to
the left of the membrane (x < 0 μm), and they increase slightly
to the right of the membrane (x > 50 μm). The transitions in
the concentration fields across the EDL boundary layers (at x =
0 and 50 μm) appear to be discontinuous on the micrometer
scale, but the inset in Figure 4(a) illustrates that it is in fact
smooth and fully resolved by the model.
While the salt ions behave as expected throughout the
concentration polarization process, there is a different and
richer behavior of the chemically reactive ions. Figure 4(c)−(f)
indicates that the dynamics of the buffer and water ions are
strongly influenced by chemistry. At the initial times, t ≲ 0.1 s
(red curves in Figure 4), before enhanced water splitting sets in,
the buffer ions follow classical concentration−polarization
depletion/enrichment behavior. While the same is true for
the hydroxyl ion, interestingly, the hydronium depletes to the
right of the membrane and enriches to the left of the
Figure 3. Comparison between the experiment (red circles) and the
model employing different values of the water-splitting backward
reaction rate constant, kb,w/(mM−1 s−1) = 109 (blue curve), 1010 (cyan
curve), 1011 (orange curve), and 1012 (purple curve). Kw is kept
constant. (a) Position where the pH equals 7 versus time. (b) Current
density versus time. The experimental asymptote is indicated (red
dashed line).
function of time and shows for kb,w = 109 mM−1 s−1 good
agreement between the model and data, both of which exhibit a
linear growth in time with similar slope, or speed of
propagation of the pH front. The model prediction of
xpH = 7.0 is displaced slightly below the experimental one, but
given the uncertainties discussed above, this is not too
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Figure 4. Concentration versus position of all of the ionic species at different times. The inset in panel (a) illustrates that the model fully resolves the
rapid but smooth and continuous transition across the EDL boundary layers at the microchannel−membrane interfaces.
system to be carried by that ion. The same is true regarding the
hydroxyl ion in the part of the system to the left of the
membrane. Moreover, the transference number shows that only
around 1% of the current is carried by the water ions.
Extended Space Charge. We turn our focus to the
concentration-depletion zone next to the membrane on its left
side. This is also where the extended space charge, or ESC,
layer forms. Figure 6 shows the pH, electric field, ionic strength,
membrane, opposite to the behavior of all of the other ions in
the system. Then, at intermediate times 0.1 s ≲ t ≲ 10 s (green
and black curves in Figure 4), the effect of water splitting is
seen by propagating fronts of hydronium: an enrichment front
going to the right and a depletion front going to the left. The
change in the hydronium concentration drives the buffer
reaction in eq 4a toward the protonation of the buffer to the
right of x = 0 and vice versa to the left of x = 0. Finally, at
longer times, t ≳ 10s (magenta curves in Figure 4), the
propagating hydronium front moving right makes it through
the membrane and continues in the microchannel to the right.
Of course, these hydronium fronts are the same as the pH
fronts in Figure 3, but instead of observing only the pH, we
have, through the model, gained additional insight into the
dynamics of all of the ions in the system.
As discussed above, water splitting leads to an increased flux
of water ions out of the depletion zone. As seen in Figure 4(f),
the concentration of water ions reaches significant values up to
at least 1 mM. Considering their uniquely high mobility, the
water ions start to carry a considerable fraction of the current. A
measure of the fraction of current carried by an ion is given by
the transference number,
ti =
ziFji
i
(17)
The transference numbers of the hydronium and hydroxyl
ions are plotted in Figure 5. In accordance with the above
observations, the front of hydronium propagating to the right
causes an increasing fraction of the current in this part of the
Figure 6. Dynamics of various fields in the ESC region: (a) pH, (b)
electric field E, (c) ionic strength I, and (d) normalized free charge
density ρe/F.
and free charge density in the ESC layer. In fact, as seen in
Figure 6(d), the ESC layer can be defined as the appearance of
a considerable amount of free charge density, which extends
much farther away from the membrane than the EDL. In this
case, at t = 40 s, the ESC has an extension of approximately 15
μm, which is more than 104 times the EDL width of 0.8 nm.
The large separation of the free charge from the membrane
surface makes the ESC layer hydrodynamically unstable, which
can lead to strong local advection and overlimiting
current.26,27,29 However, we neglect such hydrodynamic effects
here.
Figure 5. Transference number versus position at different times: (a)
Hydroxyl transference number tOH−. (b) Hydronium transference
number tH+.
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There are additional interesting phenomena in the ESC layer
that are worth noting, especially in the context of biochemical
separation and analysis. The state and transport properties of
biomolecules primarily depend on the pH, local electric field,
and ionic strength. Thus, the observations made here can
provide more information on detailed modeling, such as
molecular dynamics simulations,19 regarding the value of these
fields in the critical microscopic region where biomolecules
enter the membrane. As seen in Figure 6(a), the pH is almost
uniform and close to its original value in the ESC. However, the
pH is much larger to the left of the ESC and much lower inside
the membrane. Figure 6(b) displays the huge electric field in
the ESC, in this case up to at least 107 V m−1. This field
strength has to be contrasted to the average field between the
electrodes of 2500 V m−1, whereby it is clear that almost the
entire applied voltage drops across the microscopic ESC region.
These field strengths make the linearized Poisson−Boltzmann
equation inapplicable. It is also interesting to compare these
field strengths to those across biological membranes. The
voltage drop across a cell (∼60 mV) or a muscle cell (∼80 mV)
occurs over a 3- to 4-nm-thick membrane.60 That gives a field
on the order of 107 V m−1, which is comparable to those
observed here. Finally, Figure 6(c) shows that the ionic
strength reaches very low values in the ESC, in this case, down
to 10−2 mM from its equilibrium value of approximately 150
mM, i.e., a drop by a factor of more than 104.
Nonequilibrium Chemistry and Sensitivity to the
Water-Splitting Reaction Rate Constant. Chemical equilibrium is often assumed when dealing with the transport of
ions.16,33−38 By this assumption, it is often possible to reduce
the number of dependent variables and eliminate numerically
troublesome fast chemical time scales. In the present case, that
assumption leads to the relations in eq 12. The assumption of
chemical equilibrium is based on the chemical time scale τchem
≈ 1/(kbc0) being much faster than the transport time scales of
diffusion τdiff ≈ L2/D and electromigration τem ≈ LVT/(zDE).
However, when considering hydronium in the ESC region, kb =
109 mM−1 s−1 (the rate enhanced by a factor of 10), c0 =
O(10−4 mM), E = O(107 V m−1), and L = O(1 μm) whereby
τchem = O(10−5 s) and τem = O(10−7 s). Thus, by this simple
estimate, transport by electromigration is faster than chemical
reaction in the ESC region, and chemical equilibrium is
doubtful.
We can use our model to further investigate the assumption
of chemical equilibrium. We quantify the deviation from
equilibrium in the water-splitting and buffer reactions by the
departure from zero in a rearranged version of the relations in
eq 12,
[OH−][H+]
−1=0
Kw
(18a)
[B2 −][H+]
−1=0
[HB−]Kbuf
(18b)
Figure 7. Deviation from chemical equilibrium versus position at
different times: (a) water-splitting reaction and (b) buffer reaction.
magnitude. When unity is subtracted, the result is close to −1
(Figure 7). The deviation for the buffer ions extends further
than that of the water ions due to the slower chemical time
scale for the buffer reaction. The positive values of the deviation
for the water ions (Figure 7(a)) are likely related to the
coupling of H+ to the buffer reaction.
To investigate the consequences of assuming chemical
equilibrium of the water-splitting reaction, formally corresponding to the limit kb,w → ∞, we consider in our model three
additional values of kb,w, which are successively increased by an
order of magnitude, while keeping Kw constant. It was shown in
Figure 3(a) how these increases in kb,w lead to an overprediction of the speed of the propagating pH front. This
clearly illustrates that the assumption of chemical equilibrium is
erroneous when studying dynamical phenomena in concentration polarization, like a propagating pH front as done here.
Additionally, this is also a strong illustration of the power of the
present study in accounting for both space and time, thereby
gaining insight not obtainable when considering either of these
two dimensions separately, as done in previous studies.
Nevertheless, our model shows that the assumption of
chemical equilibrium indeed can lead to good approximations
of the global pH distribution in special cases such as the steady
state.28,30,31,38 Figure 7 shows the evolution of the pH at two
different positions in the system, inside the membrane (x = 25
μm) and outside the membrane (x = 60 μm), for the four
different values of kb,w. While the transition from high to low
pH proceeds faster with increasing kb,w, the steady-state pH
profile appears relatively insensitive to kb,w.
That is, the left-hand sides in eq 18 quantify the deviation from
chemical equilibrium. As shown in Figure 7, the left-hand-sides
of eq 18 are not zero in the ESC region, and chemical
equilibrium does not hold there. Moreover, the deviations
approach minus unity, which can be explained in the following
way. In and near the ESC layer, there is a local depletion of H+
and OH− and a stronger local depletion of B2− relative to HB−.
Thus, [OH−][H+] and [B2−][H+]/[HB−] drop toward zero
Figure 8. pH as a function of time at two different positions for
increasing values of the water-splitting backward reaction rate
constant, kb,w/(mM−1 s−1) = 109 (blue curve), 1010 (cyan curve),
1011 (orange curve), and 1012 (purple curve): (a) inside the membrane
at x = 25 μm and (b) outside the membrane at x = 60 μm.
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warfare agents.14,20,21,50 Since biomolecules are often very
sensitive to pH, accounting for variations in pH with a model of
the present type is important for interpreting observations.
Furthermore, the model can be used to design and optimize
microfluidic-membrane systems. We re-emphasize that this
model and the dramatic shifts observed for pH warrant
attention with respect to the possibility of similar phenomena
impacting broad-ranging devices where microchannels intersect
with nanofeatures.
Finally, there is potential in the present type of study to
elucidate many of the still unanswered questions regarding
transport in concentration polarization. The spatiotemporal
analysis uniquely allows the testing of aspects not possible
through purely spatial or temporal approaches. For instance,
the present study shows that the much debated assumption of
chemical equilibrium is erroneous for dynamical phenomena in
concentration polarization. Nonetheless, it is furthermore
shown that chemical equilibrium can lead to a good
approximation of the global pH distribution under special
circumstances such as steady-state conditions reached after long
times. Also, the present model gives insight that allows us to
develop simple scaling arguments. For instance, as shown in eq
19, we find that the hydronium ion flux scales with the length of
the extended space charge region, the rate constant, and the
equilibrium constant.
Our model also allows us to gain insight into the mechanisms
driving the water splitting in the ESC region. Figure 9 shows
Figure 9. Absolute magnitude of two dominating terms in the
equation of mass conservation for hydronium: (a) divergence of the
electromigration flux and (b) source term due to water splitting.
the absolute magnitude of two of the terms in the governing
equation for mass conservation of the hydronium ion, the
divergence of the electromigration flux ∂ x j He m+ =
∂x(−αDH+[H+]∂xϕ/VT), and the water-splitting source term r5
= kb,wKw(1 − [OH−][H+]/Kw). The remaining terms in the
governing equation are not shown. As can be seen in Figure 9,
the model nevertheless shows that the primary balance in the
ESC region is between those two terms, ∂xjH+ ≈ r5, where we
have assumed jH+ ≈ jem
H+ . This is in contrast to the traditional
nonreacting models that assume a balance between diffusion
and electromigration in the ESC. At the same time, Figure 7(a)
shows that the water-splitting term can be approximated by
unity, 1 − [OH−][H+]/Kw ≈ 1, whereby the balance simplifies
to ∂xjH+ ≈ kb,wKw. We integrate this balance from the left to the
right edge of the ESC, while employing, as shown in Figure
5(b), that jH+ is approximately zero at the left edge of the ESC.
Thus, the hydronium flux injection at the right edge of the ESC
becomes
jH+ |x = 0 ≈ k b,wK wL ESC
■
ASSOCIATED CONTENT
S Supporting Information
*
Details of the nondimensionalization and numerical discretization. This material is available free of charge via the Internet at
http://pubs.acs.org/.
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail: slrempe@sandia.gov.
*E-mail: alimani@stanford.edu.
(19)
Present Address
where LESC is the width of the ESC region. For t = 10 s, LESC ≈
15 μm, and by eq 19, jH+|x=0 = 1.5 × 10−4 mM m s−1. Then, with
a current of i = 103 A m−2, this flux corresponds via eq 17 to a
transference number of tH+ ≈ 0.015, in good agreement with the
full simulation results in Figure 5. The approximation in eq 19
is very useful since the scaling of the length of the ESC is
already known from the existing literature,52 whereby eq 19
provides insightful a priori estimates of the hydronium ion flux.
(D.M.R.) Department of Chemistry, University of South
Florida, Tampa, FL 33620, United States.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
This project received support from the Defense Threat
Reduction Agency-Joint Science and Technology Office for
Chemical and Biological Defense (IAA number
DTRA10027IA-3167). This work was also supported in part
by Sandia’s LDRD program and the National Science
Foundation under grant no. PHYS-1066293 and the hospitality
of the Aspen Center for Physics. Sandia National Laboratories
is a multiprogram laboratory managed and operated by Sandia
Corporation, a wholly owned subsidiary of Lockheed Martin
Corporation, for the U.S. Department of Energy’s National
Nuclear Security Administration under contract DE-AC0494AL8500.
■
CONCLUSIONS
We have developed a 1D model capable of accurately
integrating the unsteady Poisson−Nernst−Planck equations.
The model is verified by comparison against experimental data
by Mai et al.14 of pH measurements around a nanoporous
membrane in a microfluidic channel subjected to an external
electric field. A major novelty of the present work is the
consideration of both space and time for the study of pH
dynamics in concentration polarization. This aspect is highlighted by the model being able to predict the propagating pH
fronts observed in the experiment.
Both model and experiment clearly emphasize that pH
cannot be assumed to be uniform or constant during
concentration polarization. This is an important point since
this type of microfluidic-membrane system has gained
considerable popularity as a means of biochemical processing
and analysis in applications such as the detection of biological
■
REFERENCES
(1) Jong, J. d.; Lammertink, R. G. H.; Wessling, M. Lab Chip 2006, 6,
1125−1139.
(2) Khandurina, J.; Jacobson, S. C.; Waters, L. C.; Foote, R. S.;
Ramsey, J. M. Anal. Chem. 1999, 71, 1815−1819.
(3) Song, S.; Singh, A. K.; Kirby, B. J. Anal. Chem. 2004, 76, 4589−
4592.
7911
dx.doi.org/10.1021/la5014297 | Langmuir 2014, 30, 7902−7912
Langmuir
Article
(4) Wang, Y.-C.; Stevens, A. L.; Han, J. Anal. Chem. 2005, 77, 4293−
4299.
(5) Foote, R. S.; Khandurina, J.; Jacobson, S. C.; Ramsey, J. M. Anal.
Chem. 2005, 77, 57−63.
(6) Strathmann, H. Desalination 2010, 264, 268−288.
(7) Suss, M. E.; Mani, A.; Zangle, T. A.; Santiago, J. G. Sens. Actuators,
A 2011, 165, 310−315.
(8) Probstein, R. F.; Hicks, R. E. Science 1993, 260, 498−503.
(9) Krol, J. J.; Wessling, M.; Strathmann, H. J. Membr. Sci. 1999, 162,
145−154.
(10) Mani, A.; Zangle, T. A.; Santiago, J. G. Langmuir 2009, 25,
3898−3908.
(11) Mani, A.; Bazant, M. Z. Phys. Rev. E 2011, 84, 061504.
(12) Chang, H.-C.; Yossifon, G.; Demekhin, E. A. Annu. Rev. Fluid
Mech. 2012, 44, 401−426.
(13) Nikonenko, V. V.; Pismenskaya, N. D.; Belova, E. I.; Sistat, P.;
Huguet, P.; Pourcelly, G.; Larchet, C. Adv. Colloid Interface Sci. 2010,
160, 101−123.
(14) Mai, J.; Miller, H.; Hatch, A. V. ACS Nano 2012, 6, 10206−
10215.
(15) Simons, R. Electrochim. Acta 1984, 29, 151−158.
(16) Ramírez, P.; Mafé, S.; Alcaraz, A.; Cervera, J. J. Phys. Chem. B
2003, 107, 13178−13187.
(17) Nikonenko, V.; Pis’menskaya, N.; Volodina, E. Russ. J.
Electrochem. 2005, 41, 1205−1210.
(18) Zabolotskii, V.; Bugakov, V.; Sharafan, M.; Chermit, R. Russ. J.
Electrochem. 2012, 48, 650−659.
(19) Leung, K.; Rempe, S. B.; Lorenz, C. D. Phys. Rev. Lett. 2006, 96,
095504.
(20) Hatch, A. V.; Herr, A. E.; Throckmorton, D. J.; Brennan, J. S.;
Singh, A. K. Anal. Chem. 2006, 78, 4976−4984 00141.
(21) Meagher, R. J.; Hatch, A. V.; Renzi, R. F.; Singh, A. K. Lab Chip
2008, 8, 2046−2053.
(22) Nikonenko, V. V.; Kovalenko, A. V.; Urtenov, M. K.;
Pismenskaya, N. D.; Han, J.; Sistat, P.; Pourcelly, G. Desalination
2014, 342, 85−106.
(23) Grossman, G. J. Phys. Chem. 1976, 80, 1616−1625.
(24) Simons, R. Desalination 1979, 28, 41−42.
(25) Zabolotskii, V. I.; Shel’deshov, N. V.; Gnusin, N. P. Russ. Chem.
Rev. 1988, 57, 801−808.
(26) Rubinstein, I.; Zaltzman, B. Phys. Rev. E 2000, 62, 2238.
(27) Druzgalski, C. L.; Andersen, M. B.; Mani, A. Phys. Fluids 2013,
25, 110804.
(28) Andersen, M. B.; Soestbergen, M. v.; Mani, A.; Bruus, H.;
Biesheuvel, P. M.; Bazant, M. Z. Phys. Rev. Lett. 2012, 109, 108301.
(29) Davidson, S. M.; Andersen, M. B.; Mani, A. Phys. Rev. Lett. 2014,
112, 128302.
(30) Lu, J.; Wang, Y.-X.; Zhu, J. Electrochim. Acta 2010, 55, 2673−
2686.
(31) Yeh, L.-H.; Zhang, M.; Qian, S. Anal. Chem. 2013, 85, 7527−
7534.
(32) Nielsen, C. P.; Bruus, H. Phys. Rev. E 2014, 89, 042405.
(33) Nikonenko, V.; Lebedev, K.; Manzanares, J. A.; Pourcelly, G.
Electrochim. Acta 2003, 48, 3639−3650.
(34) Volgin, V.; Davydov, A. J. Membr. Sci. 2005, 259, 110−121.
(35) Danielsson, C.-O.; Dahlkild, A.; Velin, A.; Behm, M. Electrochim.
Acta 2009, 54, 2983−2991.
(36) Tanaka, Y. J. Membr. Sci. 2010, 350, 347−360.
(37) Roelofs, S. H.; Soestbergen, M. v.; Odijk, M.; Eijkel, J. C. T.;
Berg, A. v. d. Ionics 2014, 1−8.
(38) Dykstra, J. E.; Biesheuvel, P. M.; Bruning, H.; Heijne, A. T.Phys.
Rev. E 2014, submitted for publication.
(39) Andersen, M.; Frey, J.; Pennathur, S.; Bruus, H. J. Colloid
Interface Sci. 2011, 353, 301−310.
(40) Simons, R. Electrochim. Acta 1985, 30, 275−282.
(41) Choi, J.-H.; Moon, S.-H. J. Colloid Interface Sci. 2003, 265, 93−
100.
(42) Cheng, L.-J.; Chang, H.-C. Biomicrofluidics 2011, 5, 046502.
(43) Stodollick, J.; Femmer, R.; Gloede, M.; Melin, T.; Wessling, M.
J. Membr. Sci. 2014, 453, 275−281.
(44) Dhopeshwarkar, R.; Crooks, R. M.; Hlushkou, D.; Tallarek, U.
Anal. Chem. 2008, 80, 1039−1048.
(45) Hlushkou, D.; Dhopeshwarkar, R.; Crooks, R. M.; Tallarek, U.
Lab Chip 2008, 8, 1153−1162.
(46) Hlushkou, D.; Perdue, R. K.; Dhopeshwarkar, R.; Crooks, R. M.;
Tallarek, U. Lab Chip 2009, 9, 1903.
(47) Martin, C. R.; Nishizawa, M.; Jirage, K.; Kang, M.; Lee, S. B.
Adv. Mater. 2001, 13, 1351−1362.
(48) Chun, K.-Y.; Stroeve, P. Langmuir 2002, 18, 4653−4658.
(49) Ali, M.; Ramírez, P.; Mafé, S.; Neumann, R.; Ensinger, W. ACS
Nano 2009, 3, 603−608.
(50) Sommer, G. J.; Mai, J.; Singh, A. K.; Hatch, A. V. Anal. Chem.
2011, 83, 3120−3125.
(51) Malek, P.; Ortiz, J.; Richards, B.; Schäfer, A. J. Membr. Sci. 2013,
435, 99−109.
(52) Yariv, E. Phys. Rev. E 2009, 80, 051201.
(53) Probstein, R. Physicochemical Hydrodynamics: An Introduction;
John Wiley & Sons: New York, 2005.
(54) Bard, A.; Faulkner, L. Electrochemical Methods: Fundamentals and
Applications; Wiley: New York, 1980; Vol. 2.
(55) Galama, A. H.; Post, J. W.; Cohen Stuart, M. A.; Biesheuvel, P.
M. J. Membr. Sci. 2013, 442, 131−139.
(56) Haynes, W. M., Ed. CRC Handbook of Chemistry and Physics;
CRC Press/Taylor and Francis: Boca Raton, FL, 2010.
(57) Hoogerheide, D. P.; Garaj, S.; Golovchenko, J. A. Phys. Rev. Lett.
2009, 102, 256804.
(58) Eigen, M. Angew. Chem., Int. Ed. 1964, 3, 1−19.
(59) Elattar, A.; Elmidaoui, A.; Pismenskaia, N.; Gavach, C.;
Pourcelly, G. J. Membr. Sci. 1998, 143, 249−261.
(60) Hille, B. Ion Channels of Excitable Membranes; Sinauer:
Sunderland, MA, 2001; Vol. 507.
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dx.doi.org/10.1021/la5014297 | Langmuir 2014, 30, 7902−7912