Study of Metrological Properties of Voltammetric Electrodes in the Time Domain

Metrological properties of voltammetric electrodes, in the situation where on their surface an electrochemical reaction of oxidizing/reduction takes place, were analyzed in this chapter. The properties of electrodes on which a reaction controlled by ion transport process takes place were taken into consideration. Also, it was analyzed how the electrode ’ s shape and the voltage polarizing the electrode influence this electrode ’ s metrological properties. The result of the analysis conducted is that in case of a reaction controlled by charge exchange process, such a voltammetric electrode functions like a converter type 0. Its metrological properties in the time domain are defined solely by sensitivity. However, if on the surface of the electrode there is a reaction controlled by ion transport process, the electrode will function like a converter type I. Its metrological properties in the time domain are defined by the sensitivity and time constant. Numeric simulations were conducted in order to determine the influence of the electrode ’ s shape and the polarizing voltage on metrological properties of the electrode. The results show that both the sensitivity and the time constant of the electrode can be influenced by choice of an electrode ’ s shape and the shape of the polarizing voltage. It results from the above analysis presented that metrological properties of a voltammetric electrode are described solely by sensitivity. This parameter is characteristic to an electrode both in the steady state and in the transient state. In such electrochemical reactions, the electrode does not present any delays or dynamic errors. Its sensitivity is determined by parameters defining marked ions, area of the electrode, electrochemical reaction rate on the surface of the electrode, and the voltage polarizing the electrode.


Introduction
Voltammetric measurements are one of the most frequently conducted measurements in order to determine ion concentration in water [1][2][3][4][5][6][7]. Their commonness is connected most of all to its simplicity and relatively high accuracy. There are many different types of voltammetric methods [7][8][9][10][11][12][13][14][15]. These methods differ from each other mainly in the voltage shape polarizing the voltammetric electrode, and in result, they also differ in accuracy of measurements conducted.
These methods are successfully applied in electrochemical measurements in stationary conditions when the marked ion concentration in the volume of analyzed solution is constant in time. At the same time, much more voltammetric measurements are conducted in situ, where the concentration of marked ions can change during the marking process [7,10]. Some questions raise concerning the accuracy of conducted measurements, metrological properties of voltammetric electrodes, and methods of their improvement [12][13][14][15]. Hence, some work is undertaken in order to define metrological properties of voltammetric electrodes and the influence of the electrode's shape and the shape of polarizing voltage on these electrodes.

Metrological properties of voltammetric electrodes
Generally, metrological properties of voltammetric electrodes as measuring converters can be divided into static and dynamic ones.
Static properties are the characteristic of voltammetric electrodes which are in the steady state, i.e., in the state in which the concentration of marked ions does not change in the volume of the analyzed solution nor on the surface of the electrode. Dynamic properties are the characteristics of the electrode in the transient state, when these concentrations change while the voltammetric measurements are being conducted.
In order to simplify the analysis of the voltammetric electrode metrological properties following assumptions have been accepted: • the input signal is the marked ion concentration C 0 i ðtÞ in the analyzed electrolyte volume, • the output signal is the current i i ðtÞ of the electrochemical reaction on the voltammetric electrode's surface, • the time of charge exchange between the ions in the analyzed electrolyte and the voltammetric electrode equals 0.
Generally, the electrochemical reaction of oxidizing/reduction on the voltammetric electrode's surface is divided into several stages. The first stage is about delivery of depolarizer's ions from the volume of the electrolyte into the vicinity of the electrode's surface. The second stage of the electrochemical reaction is to transport, to or from, an electron or electrons through the depolarizer's ion. The third stage is to transport away the products from the reaction to the volume of analyzed solution. Stage four of the reaction is when the products of the electrochemical reaction can still react with other ions in the electrolyte after being transported away.

Electrochemical reaction controlled by a process of charge exchange on the surface of the voltammetric electrode
When on the surface of the voltammetric electrode an electrochemical reaction controlled by a process of charge exchange takes place, then the marked ion concentration in oxidizing/reduction form on the electrode surface equals the concentration of the same ion form in the analyzed solution volume [8]: The value of the output signal of a voltammetric electrode, which is the current on this electrode, is defined by the Butler-Volmer equation [8]: where the values of the reaction rate coefficients k i, ox and k i, red are defined by the following relations [8]: Keeping in mind that an oxidizing/reduction reaction may take place on the surface of the voltammetric electrode, the relation (3) may be denoted as follows: It is immediately clear that the voltammetric electrode functions exactly like a converter type 0. Hence, its metrological properties are defined solely by the sensitivity coefficient denoted as follows: by taking into consideration relations (6) and (7) we get: It results from the above analysis presented that metrological properties of a voltammetric electrode are described solely by sensitivity. This parameter is characteristic to an electrode both in the steady state and in the transient state. In such electrochemical reactions, the electrode does not present any delays or dynamic errors. Its sensitivity is determined by parameters defining marked ions, area of the electrode, electrochemical reaction rate on the surface of the electrode, and the voltage polarizing the electrode.

Electrochemical reaction controlled by a process of ion transport to the surface of the voltammetric electrode
In the case when an electrochemical reaction controlled by a process of ion transport takes place on the surface of the electrode, its flux to the surface is defined by this relation [8]: Distribution of ion concentration ∇D i ¼ 0 in the solution volume as a function of time t is defined by the flux divergence. Hence [8]: Keeping in mind the relation (11) and assuming that ∇D i ¼ 0 we get as a result: In real terms, voltammetric measurements are conducted with stationary electrodes in presence of excess of concentrated basic electrolyte, which allows to simplify the relation (13) to: It is clear that in such a case the ion transport to or from the surface of a voltammetric electrode is determined solely by the diffusion.

Metrological properties for a general case
In general cases, without making any assumptions about reversibility or irreversibility of electrochemical reactions happening on the surface of the voltammetric electrode, one may present the relation (14) using the finite difference method Which, after transformation, leads to: Keeping in mind the relation (3) and transforming it we get: Substituting the relation (16) with relations (17) and (18) we get: Keeping in mind all the assumptions, we can transform the above relation to: It results from the presented analysis that in this case the voltammetric electrode functions like converter type I. Its metrological properties are defined by sensitivity and the time constant. Static properties of the electrode are defined by sensitivity, and its dynamic properties are characterized by sensitivity and the time constant. Values of these parameters define the following relations: N T, i, red ðtÞ ¼ In the case of this type of electrochemical reactions, an electrode will present a dynamic error whose values are determined above all by the time constant and the nature of changes in the marked ion concentration. One can see in the presented relations that the parameters defining metrological properties of the voltammetric electrode are determined by the parameters defining marked ions, areas of the electrodes, a thickness of the diffusion layer, a rate of the electrochemical reaction happening on the surface of the electrode, and thereby voltage polarizing the electrode.

4.2.
The influence of the voltammetric electrode's shape on its metrological properties in the time domain Not only flat electrodes are being used in voltammetric measurements, but spherical and cylindrical as well. That is why we also analyzed the influence of the voltammetric electrode on its metrological properties.

Spherical voltammetric electrode
In the case of spherical voltammetric electrode, the relation of ion transport to/from the electrode's surface give in relation (14) appears as follows: Assuming that the marked ion concentration on the surface of the voltammetric electrode is not determined by θ and ϕ, so and relation of the ion transport given in relation (25) is simplified as follows: and it results in: We can present the relation using the finite difference method: which after transformation leads to the relation: and it results in: Substituting the relation (34) with relations (17) and (18) we get: which can be denoted as: The above relation proves that the spherical voltammetric electrode functions as converter type I, both for the oxidizing reaction and for the reduction reaction. Its metrological properties are defined by sensitivity and the time constant. Static properties of the electrode are defined by sensitivity, and its dynamic properties are defined by sensitivity and the time constant. Values of these parameters are defined by the following relations: Presented relations show that a change of the shape of voltammetric electrode does not cause a change of the measuring converter's type. A spherical electrode functions as converter type I. This electrode, in this type of electrochemical reactions, will produce a dynamic error whose value is determined above all by the time constant and by the nature of changes of the marked ion concentration. The relations also show that the parameters defining metrological properties of this voltammetric electrode are determined by the parameters defining marked ions, an electrode radius, a thickness of the diffusion layer, a rate of electrochemical reaction taking place on the surface of the electrode, and thereby by the voltage polarizing the electrode.

Cylindrical voltammetric electrode
The relation of ion transport to/from the surface of the cylindrical voltammetric electrode shown by relation (14) is: Assuming that the concentration of marked ions on the surface of the cylindrical electrode is determined by ϕ, i.e.,: we get: Defining this relation with finite difference method we get: which finally leads to: and it results in: Substituting relation (46) with relations (17) and (18) we get: Keeping in mind the assumptions, the above relation may be transformed as follows: In the above relation, it is clear that a cylindrical voltammetric electrode functions as a measuring converter type I both for the oxidizing reaction and for the reduction reaction. Its metrological properties are defined by sensititvity and the time constant. Static properties of the electrode are defined by sensitivity, and its dynamic properties are characterized by sensitivity and the time constant. Values of the parameters are described as follows: In presented relations, one may notice that the change of the shape of a voltammetric electrode does not change its type as a measuring converter. A cylindrical electrode functions also as a converter type I. This electrode in case of this type of electrochemical reaction will create a dynamic error whose value is determined above all by the time constant and the nature of changes in marked ion concentration. In presented relations, one may notice that parameters defining metrological properties of the voltammetric electrode are determined by parameters of marked ions, an electrode radius, a thickness of the diffusion layer, a rate of the electrochemical reaction taking place on the surface of the electrode, and thereby by voltage polarizing the electrode.

The influence if polarizing voltage on metrological properties of a voltammetric electrode
In order to enforce a certain course of an electrochemical reaction on the surface of a voltammetric electrode, it should be polarized with a proper voltage. In measuring practice, there are different voltammetric methods applied. Most frequently used method is direct current voltammetry. In this method, an electrode used is polarized by voltage with a value changing linearly. The advantage of such a solution is the simplicity of the measuring system. At the same time, its disadvantage is relatively low accuracy. It is a solution connected with relatively great influence of capacitive current.
In order to eliminate the influence of the volume of the double layer on accuracy of voltammetric markings, there are different types of alternating currents voltammetry applied. In such cases, the voltage polarizing a voltammetric electrode has two components: variable and static. A static component is identical with one in the direct current voltammetry, and a variable component may be for example sinusoidal voltage, square wave voltage, or triangle wave voltage.

The influence of polarizing voltage with a value changing linearly
In the method of direct current voltammetry, a voltammetric electrode used for measuring is polarized with a voltage described as follows: In such a case, coefficients of the rate of an electrochemical reaction which are defined by relations (4) and (5) will take on a form: and their derivatives are, respectively: Substituting relations (21) and (22) with relations (56) and (23) and relation (24) with relation (57) we get, respectively: and: In relations above, one can see that a flat voltammetric electrode polarized with a voltage as in relation (53) functions as a converter type I. Parameters defining its metrological properties are determined by the rate of changes in polarizing voltage. Hence, their values can be alternated by an appropriate choice of the rate of changes in this voltage.

The influence of polarizing voltage with a sinusoidal variable component
In the method of alternating current sinusoidal voltammetry, the voltammetric electrode used for measuring is polarized with a voltage described this way: In this case, coefficients of the rate of an electrochemical reaction described by the relations (4) and (5) are denoted: and their derivatives are, respectively: Because the variable component of polarizing voltage causes changes of ion concentration in the volume of the analyzed solution, also on the surface of the electrode, accordingly to the relation: then the relation (14) denoted on the surface of the voltammetric electrode polarized by voltage with a static and sinusoidal variable component is: Denoting this relation with the use of finite difference method we have: which results in: Substituting the relation (70) with relations (17) and (18) we get: Keeping in mind the assumptions taken, we can transform the above relation into: It is clear from the above relation that both for the oxidizing reaction and for the reduction reaction, a flat voltammetric electrode polarized by the voltage denoted as in relation (62) functions as a converter type I with properties defined as follows: It is clear that both the sensitivity of the electrode as well as its time constant are determined by the rate of changes of the static component of polarizing voltage and by the amplitude and sinusoidal frequency of the variable component. Hence, the parameters describing metrological properties of the electrode can be influenced by an appropriate choice of polarizing voltage parameters.

The influence of the triangle waveform variable component
In the method of triangular waveform AC voltammetry, a voltammetric electrode used for measuring is polarized by voltage denoted as follows: with a variable component of the polarizing voltage which can be denoted as: Triangular bipolar waveform component U t ðtÞ can be denoted with an expansion in the Fourier series: In this case coefficients of the rate of electrochemical reactions denoted by the relations (4) and (5) are as follows: and their derivatives are, respectively: Because the variable component of the polarizing voltage causes changes in the ion concentration of the analyzed solution, also on the surface of the electrode, accordingly to the relation: which leads to: then the relation (14) denoted for the surface of the voltammetric electrode polarized by the voltage with a static component and a triangular waveform variable component is: sin ωt À 1 9 sin 3ωt þ 1 25 sin 5ωt À … ∂C 0 i ðtÞ ∂t þωC 0 i ðtÞ cos ωt À Denoting this relation by using finite difference method, we get: sin ωt À 1 9 sin 3ωt þ 1 25 sin 5ωt À … ∂C 0 i ðtÞ ∂t þωC 0 i ðtÞ cos ωt À which results in: Substituting the relation (88) with relations (17) and (18), we get the relation: Keeping in mind the assuptions taken, the above relation may be transformed into: It is clear from the above relation that both for the oxidizing reaction and for the reduction reaction, a flat voltammetric electrode polarized by voltage as in relation (77) functions as a converter type I with properties defined as follows: cos ωt À 1 3 cos 3ωt þ 1 5 cos 5ωt À … Hence, the parameters describing metrological properties of an electrode can be influenced by an appropriate choice of polarizing voltage parameters.

Numerical simulations and discussion
Numeric simulations were conducted in order to determine how the electrode's shape and the shape of the voltage influencing the voltammetric electrode metrological properties.
The influence of the voltammetric electrode's shape on its metrological properties was analyzed assuming that the electrode is flat, spherical, and cylindrical and polarized only by linearly increasing voltage.
Also, the influence of the shape of the voltage polarizing the electrode on its metrological properties was analyzed assuming that the electrode is flat and polarized by the linearly increasing voltage, linearly increasing voltage with sinusoidal variable component and linearly increasing voltage with a triangular waveform variable component. It was assumed in the simulations that there is an oxidizing reaction of marked ions on the surface of the electrode and the values defining the marked ions, voltammetric electrode's shapes and polarizing voltages are: Numerical simulations prove that the shape of the electrode and its geometrical dimensions influence its sensitivity. When the electrode functions like a converter type 0, the geometrical values influence the sensitivity solely through the surface area of the electrode. However, if the electrode functions like a converter type I, the electrode's radius influences greatly the sensitivity. The sensitivity of the electrode is the highest when the electrode functions like a converter type 0, which is shown in Figure 1.
The highest sensitivity is the characteristic of a cylindrical electrode when there are determined: rate of changes in the voltage polarizing an electrode, a rate of oxidizing reaction, an ion diffusion coefficient, and equal geometrical dimensions. It was also proved that spherical and cylindrical electrodes sensitivity is determined by their radius. Reducing geometrical dimensions of a flat voltammetric electrode leads to reduction of its sensitivity. And reducing the radius of spherical and cylindrical electrode leads to an increase of their sensitivity.
Numeric simulation results show that the shape of the electrode influences its time constant. The lowest time constant is a characteristic of a cylindrical electrode when there are determined: rate of changes in the voltage polarizing an electrode, an ion diffusion coefficient, and equal geometrical dimensions of electrodes, which is shown in Figure 2.
The results of calculations show that the shape of the voltage polarizing a flat voltammetric electrode influences its metrological properties. It has been proved that a flat electrode polarized by voltage with a triangular waveform variable component has the highest sensitivity when there are determined: an ion diffusion coefficient, steady rate of the reaction, equal parameters of the polarizing voltage, which is shown in Figure 3.