Chemistry Reference
In-Depth Information
where
u
(
x
) is the fluid velocity,
Δ
p
is the pressure difference across the channel, and
l
is
the length of the channel. We take ψ(
x
) to be the equilibrium distribution, which is
justified as long as the applied electric field gradients are too weak to significantly distort
the double layer, i.e. smaller than
k
B
T
κ.
20
It is also conventional to apply the no-slip
boundary condition at the channel surfaces.
In the absence of an applied pressure gradient and taking the electrical mobility of
the ions to be the bulk value, the solutions to Equations 1.1 and 1.2 can be used to calculate
the total conductance of a channel. This was the approach used by Levine to calculate the
ionic conductance in a narrow channel with charged walls.
21
However in order to
accurately describe our experimental conductance data, it was necessary to replace the
constant surface potential boundary condition that had been commonly used. We found
that a constant effective surface charge density, σ, described the data extremely well and
could be imposed on our transport model using Gauss' Law, i.e.
σ=±
εε
0
k
B
T
e
d
ψ
dx
x
=±
h
/2
(1.3)
Our ionic transport model described the experimental data very well, as can be seen from
the theoretical fits in Figure 1.2(b). The model contains only a single fit parameter,
namely σ, which was found to agree well with published values for silica surfaces obtained
by chemical titration experiments.
22
The ion transport model also provides insight into the very different behaviour
that was observed in the high and the low salt regimes. At high
n
, the number of ions in the
double layer is overwhelmed by the number in the bulk fluid. The conductance of a
nanochannel at high
n
therefore increases with
n
just as the conductivity of bulk solution.
At low
n
, by contrast, the counter-ions in the double layer dominate. Their number is fixed
by the requirement of overall charge neutrality, and so the conductance of the nanochannel
becomes governed by the charge density at the surfaces. The crossover between high-salt
and low-salt behaviour occurs when | σ|≈
enh
for monovalent salt. It is important to note
that this does not correspond to double layer overlap. The data in Figure 1.2(b) clearly
show, for example, that a 380 nm high channel is in the low-salt conductance plateau at
n
= 10
4
M
, where the Debye length is only 30 nm.
Solid-state nanopores and nanotubes are systems in which ion transport in the low
salt regime is particularly relevant. Due to their small diameter (<10 nm typically), the
onset of the conductance plateau in a nanopore occurs at salt concentrations as high as
hundreds of millimolar. In addition, nanopore experiments typically involve the insertion
of an individual DNA molecule, which is itself a highly charged object. The backbone of
double-stranded DNA carries two electronic charges for every 3.4 Å of length. DNA
insertion into a solid-state nanopore therefore entrains a high concentration of mobile
counter-ions into the pore, which actually increases the measured conductance for salt
concentrations below ~0.4 mM.
23-25
The electrically driven transport of ions in nanochannels reveals an interesting
parallel with integrated circuits. The dependence of channel conductance on the surface
charge is analogous to the conductance modulations in a field effect transistor (FET) that
can be induced by the charge on the gate. It is therefore possible to “gate” the conductance
of a nanofluidic channel by chemically modifying its surface charge density, as we have
shown in Figure 1.2(c). The conductance of an
h
= 87 nm channel in the low-salt regime
was clearly reduced by treatments with octadecyltrichlorosilane (OTS), whose attachment
to silica neutralizes the surface. Other groups have employed this phenomenon as a sensing
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