11.2 Nanochannel Templates Obtained with Top-Down Synthesis Methods
369
Solution of the mass transport problem is often given for two limiting cases:
(i) Short-time approximation. Here, the initial reactant concentration is the same
as the bulk one. Neither the diffusion field can extend up to the pore ending
when the polarization starts, nor the nanochannel can be shortened significantly
due to the growth of the deposit within. In this case, a Cottrell-type current
transient can be obtained:
i (t) = zFr
2 c
∗
Dπ
t
1/2
(11.1)
(ii) At long times, the diffusion field is steady state, and the total reactant flux is the
same at the tip of the growing nanowire and at the pore mouth. If the diffusion
fields of the neighbouring nanopores in the bulk solution do not overlap and
no convection is applied, one can obtain for the steady-state current of a single
nanopore that
i =
zFr
2 Dπ c
∗
L + π r/4
(11.2)
If the overlap of the diffusion fields outside the pores is to be considered and
the pore arrangement is random (i.e., track-etched membranes are used), the Avrami
theorem can be used to calculate the mean effective area that provides the material
supply of a single pore [37]. This means that the effective “collection surface area”
of a single pore can be calculated with the help of an exponential weighting factor
instead of a linear one, which is due to the randomness of the pore arrangement.
The models for calculation use a single diffusion coefficient. It is to be noted that
in accord with the measurements, the diffusion coefficient may vary with the pore
diameter [20], and it is not identical to that measured in the bulk solution either.
The results of a series of complex calculations of the filling process of regularly
arranged nanopores have been published [38]. The thickness of the diffusion layer
in the free solution was a variable determined by the hydrodynamic conditions,
and this was taken into account through a geometric factor depending also on the
porosity of the template. The kinetics of the electrode reaction was considered with
a single Tafel-type exponential function. Chronoamperometric functions and pore
filling times are given for various deposition potentials and reactant concentrations. In
the second part of the work [39], the inhomogeneity of the pore filling was addressed.
It was shown that the initial inhomogeneity of the pore lengths leads to a selfenhancing differentiation of the nanowire growth. The widening of the nanowire
length distribution curve proved to the higher, the larger the deposition rate (i.e., the
more negative the electrode potential).
369
Solution of the mass transport problem is often given for two limiting cases:
(i) Short-time approximation. Here, the initial reactant concentration is the same
as the bulk one. Neither the diffusion field can extend up to the pore ending
when the polarization starts, nor the nanochannel can be shortened significantly
due to the growth of the deposit within. In this case, a Cottrell-type current
transient can be obtained:
i (t) = zFr
2 c
∗
Dπ
t
1/2
(11.1)
(ii) At long times, the diffusion field is steady state, and the total reactant flux is the
same at the tip of the growing nanowire and at the pore mouth. If the diffusion
fields of the neighbouring nanopores in the bulk solution do not overlap and
no convection is applied, one can obtain for the steady-state current of a single
nanopore that
i =
zFr
2 Dπ c
∗
L + π r/4
(11.2)
If the overlap of the diffusion fields outside the pores is to be considered and
the pore arrangement is random (i.e., track-etched membranes are used), the Avrami
theorem can be used to calculate the mean effective area that provides the material
supply of a single pore [37]. This means that the effective “collection surface area”
of a single pore can be calculated with the help of an exponential weighting factor
instead of a linear one, which is due to the randomness of the pore arrangement.
The models for calculation use a single diffusion coefficient. It is to be noted that
in accord with the measurements, the diffusion coefficient may vary with the pore
diameter [20], and it is not identical to that measured in the bulk solution either.
The results of a series of complex calculations of the filling process of regularly
arranged nanopores have been published [38]. The thickness of the diffusion layer
in the free solution was a variable determined by the hydrodynamic conditions,
and this was taken into account through a geometric factor depending also on the
porosity of the template. The kinetics of the electrode reaction was considered with
a single Tafel-type exponential function. Chronoamperometric functions and pore
filling times are given for various deposition potentials and reactant concentrations. In
the second part of the work [39], the inhomogeneity of the pore filling was addressed.
It was shown that the initial inhomogeneity of the pore lengths leads to a selfenhancing differentiation of the nanowire growth. The widening of the nanowire
length distribution curve proved to the higher, the larger the deposition rate (i.e., the
more negative the electrode potential).