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CHEM465/865, 2004-3, Lecture 20, 27 th 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) δ. Sometimes the diffusion layer is referred to as the “Nernst layer”.

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Page 1: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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”.

Page 2: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 3: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 4: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 5: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 6: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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

====

Page 7: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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

Page 8: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 9: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 10: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 11: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 12: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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 −−−−==== .

Page 13: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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<< << ⋅<< << ⋅<< << ⋅<< << ⋅

Page 14: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 15: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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.

Page 16: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering
Page 17: Hydrodynamic Electrodes and Microelectrodesaroudgar/Tutorials/OLD/Electrochemistry-09-2004/Lecture_19... · Hydrodynamic Electrodes and Microelectrodes So far we have been considering

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!