hydrodynamic electrodes and...
TRANSCRIPT
CHEM465/865, 2004-3, Lecture 20, 27th Sep., 2004
Hydrodynamic Electrodes and Microelectrodes
So far we have been considering processes at planar electrodes. We have focused
on the interplay of diffusion and kinetics (i.e. charge transfer as described for
instance by the different formulations of the Butler-Volmer equation). In most
cases, diffusion is the most significant transport limitation. Diffusion limitations
arise inevitably, since any reaction consumes reactant molecules. This
consumption depletes reactant (the so-called electroactive species) in the vicinity
of the electrode, which leads to a non-uniform distribution (see the previous
notes).
______________________________________________________________________
Note: In principle, we would have to consider the accumulation of product species
in the vicinity of the electrode as well. This would not change the basic
phenomenology, i.e. the interplay between kinetics and transport would remain
the same. But it would make the mathematical formalism considerably more
complicated. In order to simplify things, we, thus, focus entirely on the reactant
distribution, as the species being consumed.
______________________________________________________________________
In this part, we are considering a semiinfinite system: The planar electrode is
assumed to have a huge surface area and the solution is considered to be an
infinite reservoir of reactant. This simple system has only one characteristic
length scale: the thickness of the diffusion layer (or mean free path) δδδδ. Sometimes
the diffusion layer is referred to as the “Nernst layer”.
Now: let’s consider again the interplay of kinetics and diffusion limitations.
Kinetic limitations are represented by the rate constant 0k (or equivalently by the
exchange current density b b
red ox
0 0 1j nFk c cα αα αα αα α−−−−==== ). Diffusion limitations are
represented by the diffusion constant D and by the diffusion layer thickness δδδδ .
We can define a diffusion rate in the following way: diff
Dk
δδδδ==== . The corresponding
diffusion-limited current is b
b oxdiff diff ox
nFDcj nFk c
δδδδ= == == == = .
The two rates, 0k and diffk , determine the interplay between kinetics and
mass transport. Reactions for which diff
0k k>>>>>>>> are called reversible reactions.
Reactions for which diff
0k k<<<<<<<< are called irreversible. In the chapter on cyclic
voltammetry we will consider this distinction in more detail.
The rates 0k can vary over wide ranges (from 101 cm/s for facile reactions
down to 10-14 cm/s for rather slow reactions). They are determined by the
electronic structure of the metal (the Fermi level), the structure of the solution and
the LUMO (lowest unoccupied molecular orbital) for reduction or the HOMO
(highest occupied molecular orbital) for oxidation of the species in solution
undergoing reduction or oxidation. In general, 0k depends on the type of metal,
its surface structure, the type of electrolyte and the redox species.
The values of diffusion coefficients in aqueous solution are usually in the
range of 10-5 cm2/s, with only a small range of variation. This parameter cannot be
controlled in an experiment.
In this part we will learn ways to control the interplay between kinetics and
mass transport. Which options exist to influence this interplay? As you know
already, the rate of electron transfer in an electrochemical system is controlled by
the electrode potential E. This is the most important variable in an electrochemical
experiment. We have studied its effects already in detail.
What are the other options of experimental control? Subsequently,
experimental techniques will be discussed, which allow to control the rates of
mass transport, i.e. allow to control diffk .
There are two principal ways to achieve that:
Ø Hydrodynamic devices – forced convection. They help to confine
concentration variations to a thin region near the electrode surface.
Ø Control the electrode geometry, i.e. use (ultra-)microelectrodes.
Overall, these measures raise the rates of mass transport. Fast rates of mass
transport make it possible to study the kinetics of fast electron transfer reactions.
Hydrodynamic Devices
These devices use convection to enhance and control the rate of mass transport
to the electrode surface. Detectable currents are increased and the sensitivity of
voltammetric measurements is enhanced.
Two approaches are possible:
Ø The electrode is held in a fixed position and solution is flowed over the
electrode surface by an applied force, usually an applied pressure gradient
(e.g. wall-jet electrode)
Ø The electrode is designed to move which acts to mix the solution via
convection.
In order to be able to perform a quantitative analysis of the electrode processes,
the introduced convection must be predictable. The flow of the solution must be
laminar rather than turbulent in order to lead to well-defined, reproducible results.
The figure below compares turbulent and laminar flow.
Dropping Mercury Electrode (DME)
Historically, this is the first used hydrodynamic technique. The electrochemical
cell with potentiostat, working electrode (mercury), counter electrode and
reference electrode is shown below.
mercury drop electrode surface
A large reservoir of mercury is connected to a capillary. In the capillary,
mercury flows under the influence of gravitation. The drop at the opening of the
capillary grows in time until it reaches a critical size. At some point, the mercury
drop from detaches from the tip and falls down. The surface of the working
electrode, which is the surface of the drop, is thus refreshed in regular cycles. A
big advantage of this electrode is, that the continuous refreshing minimizes
problems of electrode poisoning.
Clearly, the measured current at will be a function of the surface area of the
drop. It increases continuously with drop size. When the drop falls of, the current
drops rapidly. The following picture shows the cyclic current at a DME as a
function of time (in this plot: for a fixed potential).
Evidently, the surface area is an important property of this electrode. We,
thus, have to consider current and NOT current densities at this electrode.
Consider the limiting current due to diffusion as determined by COTTRELL-
equation,
(((( )))) b
L Hg ox
DI t nFA c
tππππ====
Assume that the surface of the drop is an ideal sphere (this is of course a
simplification!), i.e. Hg
2
04A rππππ==== , where
0r is the radius of the drop. We assume a
constant mass flux Hgm [mg s-1] of liquid Hg in the capillary. This mass flux
determines the time-dependence of the drop radius,
Hg
Hg
1/ 3
0
3
4
m tr
πρπρπρπρ
====
where Hgρρρρ is the density of Hg. An effective diffusion coefficient has to be used in
the Cottrell equation, eff
7
3D D==== . Using all these definitions in the Cottrell-
equation, the so-called ILKOVIC-equation is obtained,
(((( )))) b
L ox Hg
1/ 2 2/ 3 1/ 6708I t nD c m t====
where LI is in amperes, D in cm2/s, Hgm in mg/s, and t in s.
The dropping mercury electrode can be used for voltammetric
measurements, i.e. measure the current as a function of applied electrode
potential E. For historic reasons, this method is called polarography. It was first
introduced by Heyrovsky in the 1920’s. The following picture shows a linear scan
polarograph (obtained by linear sweep voltammetry) for two reactions:
Curve A: Cd2+ + 2 e- + Hg � Cd(Hg) (reduction of Cd ions)
Curve B: 2H+ + 2 e- � H2(g) (hydrogen evolution)
Diffusion current
IL
Note: The spikes visible in this plot represent the cycles of the drop lifetime. The
drop lifetime is a constant (determined by the height of the mercury column in the
reservoir and by the mass flow rate Hgm ). At large cathodic overpotentials, the
current reaches a plateau. The height of this plateau is determined by the Ilkovic
equation, as specified above. In principle, this electrode is operated at steady
state.
Rotating Disc Electrode (RDE)
The rotating disc electrode is the most widely used is the first used hydrodynamic
electrode. A disc electrode is embedded into the bottom face of an insulating rod
(e.g. Teflon). The rod rotates at a constant angular velocity ωωωω . The rotation drags
solution to the electrode surface, resulting in a vortex, as shown below. Due to
this drag and the steady laminar flow to the electrode surface, solution is
continuously replaced. This electrode can be operated at steady state.
At angular velocities -1 s60ωωωω <<<< the flow profile will be laminar. Moreover, the
electrode disc is considered small compared to the surface area of the insulating
rod. This provides uniform conditions at the electrode surface. The figure below
shows potential sweep voltammograms (for a cathodic reaction, i.e. negative
current, negative electrode potential) measured at an RDE at various frequencies.
Obviously, the current in the mass transfer limited region (large |E|) is
independent of time, but it is controlled by the rotation speed.
In order to understand this behaviour, we have to return to the concept of the
diffusion layer.
What controls the thickness of the diffusion layer?
The transport equations are given by the following modified form of Fick’s
equations:
ox ox
boxox ox
molar flux:
t-variation:
convection + diffusion
J vc D c
cv c D c
t
= − ∇= − ∇= − ∇= − ∇
∂∂∂∂ = − ∇ + ∆= − ∇ + ∆= − ∇ + ∆= − ∇ + ∆∂∂∂∂
vvvv
vvvv
The results from linear sweep voltammetry indicate, that δδδδ is now controlled by
convection and not by diffusion! Stationary operation is possible.
Consider the steady state limiting current density, i.e. consider the case ox 0c
t
∂∂∂∂ ====∂∂∂∂
The theory for this electrode was developed by Levich: Levich theory!
Details of this theoretical solution will be skipped here. They can be found for
instance in the book: Electrochemistry – Principles, Methods and Applications,
C.M.A. Brett, A.M.O. Brett, Oxford University Press, Oxford, 1993, section 5.9). The
thickness of the diffusion layer is given by:
1/ 3 1/ 6 1/ 21.61Dδ ν ωδ ν ωδ ν ωδ ν ω −−−−====
where ν ν ν ν is the kinematic viscosity [cm2 s-1]. The mass transport limited current
density is, thus, given as a function of the rotation speed as
b
oxL
b
ox
2/ 3 1/ 6 1/ 20.602
cj nFD
nFc D
δδδδν ων ων ων ω−−−−
====
====
A plot of Lj vs. 1/ 2ωωωω will, thus result in a straight line.
The slope of this straight line is determined by the bulk concentration of reactant,
by the diffusion coefficient and by the kinematic viscosity, i.e.
b
ox
2/ 3 1/ 60.602slope nFc D νννν −−−−==== .
Some notes on using the rotating disc electrode:
Ø The angular velocity ωωωω should be small enough so that the flow profile will
be laminar. The so-called Reynolds number Re, a characteristic of the flow
profile, has to be smaller than a critical value. The Reynolds number is
defined by
vRe
l
νννν==== ,
where v is a characteristic velocity of the fluid relative to the solid surface
of the electrode, l is a characteristic length of the electrode, and ν ν ν ν is the
kinematic viscosity (a measure of the inner friction within the fluid). For an
RDE, v is the linear velocity at the outer edge of the disc electrode, given
by v rωωωω= ⋅= ⋅= ⋅= ⋅ and l r==== is the radius of the disc. The critical Reynolds-
Number is Recrit =105. The criterion for laminar flow is, thus
disc 5Re 10
Ar r ωωωωωωωων πνν πνν πνν πν
⋅⋅⋅⋅= = <<= = <<= = <<= = <<
Kinematic viscosities for dilute aqueous solutions are typically in the range
2 -1cm s210νννν −−−−==== and we obtain the criterion
2 -1
disc cm s33 10Aωωωω << ⋅<< ⋅<< ⋅<< ⋅ .
Consider for example the following disc areas and corresponding upper
limits on angular velocity:
2 -1
disc
2 -1
disc
cm s
mm s
3
5
1 3 10 ,
1 3 10
A
A
ωωωωωωωω
==== ⇒⇒⇒⇒ << ⋅<< ⋅<< ⋅<< ⋅
==== ⇒⇒⇒⇒ << ⋅<< ⋅<< ⋅<< ⋅
Smaller electrode surface area – operation at larger angular velicities
possible.
Ø On the other hand, the angular velocity controls the thickness of the
diffusion layer (see expression for δδδδ in Levich theory), which controls the
limiting current density due to diffusion. The larger ωωωω is, the smaller is
thinner is δ δ δ δ and the is Lj . Therefore, ωωωω should not be too small. The typical
frequency range for operating an RDE with 0.3 cm radius is
-1 -1 s s310 3 10ωωωω<< << ⋅<< << ⋅<< << ⋅<< << ⋅
Ø The electrode and rod need to be perfect cylinders to avoid wobbling
around the axis. This is not a problem for mm sized electrodes. It could be
a problem for smaller electrodes.
Capabilities of RDEs:
Ø Operation in the diffusion limited regime (at large ηηηη ): From a plot of Lj
vs. 1/ 2ωωωω it is possible to determine the diffusion coefficient of reactant
species in the solution or determine the concentration of reactant.
Ø Operation in the kinetic or mixed regime (small to medium ηηηη ): The total
current density j is determined by the following relation, involving an
activation controlled current density acj and the diffusion-limited current
density diffj
ac diff ac
1/ 2
1 1 1 1 1
j j j j Bωωωω= + = += + = += + = += + = +
This can be thought of as a series equivalent circuit of mass transport and
activation barrier that electroactive species have to overcome in order to
react. On the right hand side, the frequency dependence of the diffusion
limited current has been inserted. It is evident, that a plot of 1
j over
1/ 2ωωωω −−−−
(at a fixed potential) will result in a straight line. The intercept with the
ordinate, i.e. the ωωωω → ∞→ ∞→ ∞→ ∞ limit will give acj , as shown in the following plot.
Summary:
Three types of effects that you have to be able to distinguish
Ø Potential E
Ø Time scales, rates of processes
Ø Geometry, length scales!
The latter two are interrelated!
Voltammetry: electrochemistry techniques based on current
measurement as a function of applied electrode potential – what
do you need for that? Electrochemical cell!