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.
Now: let s consider again the interplay of kinetics and diffusion limitations. Kinetic limitations are represented by the rate constant exchange current density 0 0 b α b 1 α red ox 0 k (or equivalently by the j = nfk c c ). Diffusion limitations are represented by the diffusion constant D and by the diffusion layer thickness δ. D We can define a diffusion rate in the following way: kdiff =. The corresponding δ diffusion-limited current is The two rates, 0 k and b b nfdcox jdiff = nfkdiffcox =. δ k diff, determine the interplay between kinetics and mass transport. Reactions for which 0 k >> k are called reversible reactions. diff Reactions for which 0 k << k are called irreversible. In the chapter on cyclic diff voltammetry we will consider this distinction in more detail. The rates down to 10-14 0 k can vary over wide ranges (from 10 1 cm/s for facile reactions 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, 0 k 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 cm 2 /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 k diff. There are two principal ways to achieve that: Hydrodynamic devices forced convection. They help to confine concentration variations to a thin region near the electrode surface. 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: 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) 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 COTTRELLequation, L b ( ) = I t nfa c Hg ox Assume that the surface of the drop is an ideal sphere (this is of course a simplification!), i.e. constant mass flux A 2 Hg = 4π r0, where 0 D π t r is the radius of the drop. We assume a m Hg [mg s -1 ] of liquid Hg in the capillary. This mass flux determines the time-dependence of the drop radius, r 0 3m t 4 πρ Hg Hg = 1/ 3
where ρhg is the density of Hg. An effective diffusion coefficient has to be used in the Cottrell equation, D eff 7 = D. Using all these definitions in the Cottrell- 3 equation, the so-called ILKOVIC-equation is obtained, b ( ) = 708 I t nd c m t 1/ 2 2/ 3 1/ 6 L ox Hg where I L is in amperes, D in cm 2 /s, m Hg 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: Cd 2+ + 2 e - + Hg Cd(Hg) (reduction of Cd ions) Curve B: 2H + + 2 e - H 2 (g) (hydrogen evolution) Diffusion current I L
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 m Hg ). 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 ω. 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.
-1 At angular velocities ω < 60 s 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: molar flux: t-variation: c v J = vc D c ox t ox v = c + D c b ox ox ox convection + diffusion The results from linear sweep voltammetry indicate, that δ is now controlled by convection and not by diffusion! Stationary operation is possible.
c Consider the steady state limiting current density, i.e. consider the case ox = 0 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: δ = 1.61D ν ω 1/ 3 1/ 6 1/ 2 where ν is the kinematic viscosity [cm 2 s -1 ]. The mass transport limited current density is, thus, given as a function of the rotation speed as j L b cox = nfd δ = 0.602nFc b ox D ν ω 2/ 3 1/ 6 1/ 2 A plot of j vs. 1/ 2 L ω 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. slope nfc D ν b 2/ 3 1/ 6 = 0.602 ox.
Some notes on using the rotating disc electrode: The angular velocity ω 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 vl Re =, ν 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 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 = ω r and l = r is the radius of the disc. The critical Reynolds- Number is Re crit =10 5. The criterion for laminar flow is, thus ωr r ω A Re = = << 10 ν πν disc 5 Kinematic viscosities for dilute aqueous solutions are typically in the range ν 10 cm s 2 2-1 = and we obtain the criterion ω A << 2-1 disc << 3 10 3 cm s. Consider for example the following disc areas and corresponding upper limits on angular velocity: A A disc disc = << 2 3-1 1 cm ω << 3 10 s, = 1 mm ω << 3 10 s 2 5-1 Smaller electrode surface area operation at larger angular velicities possible. On the other hand, the angular velocity controls the thickness of the diffusion layer (see expression for δ in Levich theory), which controls the limiting current density due to diffusion. The larger ω is, the smaller is thinner is δ and the is j L. Therefore, ω should not be too small. The typical frequency range for operating an RDE with 0.3 cm radius is 10 s << ω << 3 10 s -1 3-1
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: Operation in the diffusion limited regime (at large η ): From a plot of j L vs. 1/ 2 ω it is possible to determine the diffusion coefficient of reactant species in the solution or determine the concentration of reactant. Operation in the kinetic or mixed regime (small to medium η ): The total current density j is determined by the following relation, involving an activation controlled current density density j diff jac and the diffusion-limited current 1 1 1 1 1 = + = + j j j j Bω ac diff ac 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 1/ 2 limited current has been inserted. It is evident, that a plot of 1 j over ω 1/ 2 (at a fixed potential) will result in a straight line. The intercept with the ordinate, i.e. the ω limit will give j ac, as shown in the following plot.
Summary: Three types of effects that you have to be able to distinguish Potential E Time scales, rates of processes 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!