Volume 3 Paper 22

The modelling of a galvanic cell series-connected with an external pure resistance and the source of an arbitrary external polarization.

A. A. Shagaev
Independent researcher, house 17-a apartment number 6, 660037 Kolomenskaya Str., Krasnoyarsk, Russia. E-Mail Address: or

Abstract

This paper is dedicated to the blessed memory of my parents, Maria and Arkadiy.

Based on the previously developed approach, the model (of the functioning of a galvanic battery under condition of the battery discharge on the external resistance and additional external polarization) is proposed. Design equations for discharge current, the emf of a galvanic battery (the potentials of separate electrodes), the corrosion currents of each electrode, the values of the emf of the external polarization of electrodes (which eliminate the corrosion of one separate electrode as well as that of all electrodes together) were obtained. The equations obtained allow the determination of factors that influence on the discharge current, the emf of a galvanic battery (the potentials of separate electrodes), the currents of corrosion of electrodes. A number of results (the existence of conditions of galvanic protection against corrosion) agree with the known electrochemical practice, other results (generalized equations for discharge current, corrosion currents, the emf of a galvanic battery (the potentials of electrodes), the emf of external polarization, protecting the electrodes against corrosion) have never been obtained before. The physical meaning of the measured electrode potential was shown in detail.

Keywords: galvanic element, modelling.

Introduction

The detailed studies of corrosion processes, which take place on metal (and any other) electrodes in various environments, are of great practical importance now. It serves the purposes of:

1. optimizing the characteristics of chemical power sources (CPS);
2. finding approaches to solving the problems concerning the corrosion protection of chemical and any other equipment;
3.  developing (on the basis of the electrochemical theory of catalysis) new kinds of catalysts for chemical-engineering processes;
4. solving the problems of the photoelectrochemical conversion of solar energy;
5. making new devices etc..

Obviously, it is impossible to find an approach to the solution of the problem without developing a theory adequately describing the behaviour of any electrode under conditions of various physical-chemical experiments (and in the course of the operating of the simplest galvanic circuit).

In our opinion, the well thought-out equivalent electric circuits in line with the well known chemical practice and thermodynamics can adequately model the behaviour of an arbitrary galvanic circuit under conditions of various physical-chemical experiments.

In the previous paper [1], the principles of the building of such circuits were set. The equivalent electric circuit, modelling corrosion processes in a metal-film-arbitrary oxidative medium was proposed in that paper also. This simple electric circuit obtained gives a theoretical approach to understanding the principles of the modelling of redox and ionic processes by means of equivalent electric circuits.

In this paper we tried to use the approach for the concrete model of a galvanic system consisting of 2 arbitrary electrodes series-connected with an external pure resistance and an external source of polarization (cathodic or anodic). At the same time it was taken into account that these two electrodes can corrode and be covered with an electroconductive (by means of the ions of a metal and electrons) film. Thus, we tried to lay the foundation of an approach to building equivalent electric circuits modelling the behaviour of an arbitrary galvanic element working under various conditions. The solution of the problem allows equations to be derived (for this model of a galvanic system) for the following values:

1.           the emf (emfel) of the element

2.           the internal resistance of the element (Rint)

3.           the working current (J6) of the element (when the element is series-connected with an external pure resistance (Rint) and an external source of arbitrary polarization )

4.           the currents (J6Ria and J6Ric) flowing between 2 similar electrodes, which are made from the anode or cathode materials.

5.           the corrosion currents of separate electrodes (the corrosion current of the anode (Ja) and the corrosion current of the cathode (Jc)) when the element is series-connected with an external pure resistance (Rint) and an external source of arbitrary polarization.

6.           the current of joint corrosion of the anode and cathode Jjc (see the details below)

7.           the current of total corrosion in the galvanic element Jcel

8.           the external polarization when Ja is minimal

9.           the external polarization when Jc is minimal

10.      the external polarization when current of the joint corrosion of the anode and cathode (Jjc) is minimal

11.      the external polarization when current of total corrosion in the galvanic element (Jcel) is minimal

All above-enumerated values are the functions of the parameters of the system. Thus, the application of the mathematic methods of the multifactor optimizing allows the determination of the following parameters of the system:

1.     the optimal composition of a battery (in accordance with the required parameters of the operation of a battery )

2.     the optimal conditions of the operation and storage of batteries

3.     the optimal conditions of the cathodic and anodic protection of metals against corrosion.

The author of the paper realizes that a number of models of functioning can be proposed even for the same galvanic system. The equivalent electric circuit of the system will change in accordance with these models. Consequently, the search of the optimum solution for this system will not always produce the correct result (if at all). The result of the search is determined by the adequacy of the model to its galvanic system.

The model of the functioning of a galvanic element, proposed by the author of the paper.

As a sample of modelling of a galvanic element functioning, the author of the paper proposes a model based on the following assumptions:

1.           the cathode (M2) (and anode (M1)) of the galvanic element can corrode in the electrolyte environment of the element;

2.           all elementary chemical (electrochemical) processes that occur in the system, are reversible reactions;

3.           there is only one oxidant (for the anode and cathode) in the system.

4.           the oxidant on the cathode forms only one anion (a2) in the course of corrosion and the operation of the galvanic element

5.           the oxidant forms only one anion (a1) in the course of the corrosion of the anode

6.           only one cation (c2) forms in the course of the corrosion of the cathode

7.           only one cation (c1) forms in the course of the corrosion and the operation of the galvanic element on the anode

8.           the product of cathode corrosion (c2a2) either dissolves in the solution or is deposited in the form of a film on the surface of the cathode

9.           the product of anode corrosion (c1a1) either dissolves in the solution or is deposited in the form of a film on the surface of the anode

10.      the electroconductivity of the film (c2a2) on the cathode is conditioned only by the transferring of cations (c2) and electrons

11.      the electroconductivity of the film (c1a1) on the anode is conditioned only by the transferring of cations (c1) and electrons

12.      in the course of the transferring to the solution, the cation (c2) forms only one product of its solvation (c2s)

13.      in the course of the transferring to the solution, the cation (c1) forms only one product of its solvation (c1s)

14.      in the course of the transferring to the solution, the anion (a2) forms only one product of its solvation (a2s)

15.      in the course of the transferring to the solution, the anion (a1) forms only one product of its solvation (a1s)

16.      solvated anions (a2s) and solvated cations (c2s) interact in the solution forming a neutral compound c2sa2s (for example an ion pair);

17.      solvated anions (a2s) and solvated cations (c1s) interact in the solution forming a neutral compound c1sa2s (for example an ion pair);

18.      solvated anions (a1s) and solvated cations (c2s) interact in the solution forming a neutral compound c2sa1s (for example an ion pair);

19.      solvated anions (a1s) and solvated cations (c1s) interact in the solution forming a neutral compound c1sa1s (for example an ion pair);

20.      during the storage and in the course of the operation of the galvanic element, anions (a2) and cations (c2) are not implanted into the film (c1a1)

21.      during the storage and in the course of the operation of the galvanic element, anions (a1) and cations (c1) are not implanted into the film (c2a2).

Guided by the aforesaid assumptions the author of the paper proposes the following equivalent electric circuit to describe this model of a galvanic element:

Fig. 1. the equivalent electric circuit proposed by the author of this paper (a text description is available in the Appendix for visually impaired readers).

where:

RM11, RM12 - the Ohmic resistances of the metal of the anode (M1), respectively from cathodic and anodic areas (of the corrosive microelement on the metal/film interface) to the electric contact of the external pure resistance, connected to the anode. (RM11 + RM12) = the Ohmic resistance of the metal of the anode in the corrosive microelement.

RM21, RM22 - the values that are similar to RM11 and RM12, but for cathode (M2)

Rc1, Rc2 - respectively, the sums of the cationic resistances of metal/film (RcMe/film) phase boundaries, the film (Rcfilm) and film/solution phase boundaries (Rcfilm/s), for the anode and cathode.

Re1, Re2 - respectively, the sums of the electronic resistances of metal/film (ReMe/film) phase boundaries, the film (Refilm) and film/solution phase boundaries (Refilm/s) for the anode and cathode.

Rs1ox, Rs2ox - respectively, the resistance of the transferring of the molecules (ions) of an oxidant to the film surface for anode and cathode.

Ras2 and Ras1 - the resistance of the transferring of the molecules of a solvent to the anodic and cathodic areas of corrosive elements on the surface of the film/solution phase boundary (for anode) to solvate the ions formed.

Rcs3 and Rcs4 - the resistance of the transferring of the molecules of a solvent to the anodic and cathodic areas of corrosive elements on the surface of the film/solution phase boundary (for cathode) to solvate the ions formed.

Rciajaj, Rciajci - the anionic and cationic resistance of the migration of delocalized cations (ci) and anions (aj) on the surface of a film (ciaj) on the electrode (if i=j=1, then the electrode is an anode; if i=j=2 then the electrode is a cathode). These are resistances from the areas where anions are formed or the areas where cations outcrops on the surface of the film/solution phase boundary to the areas, where they interact, which results in the forming of a crystal lattice.

Rcjsaisais, Rcjsaiscjs, Rscjsais, Rsaiscjs - the anionic and cationic resistance of the migration of solvated cations (cjs) and anions (ais) in the solution. These are resistances from the areas where anions are formed or the areas where cations outcrops on the surface of the film/solution phase boundary to the areas, where they interact in the solution, which results in the forming of a neutral substance cjsais (for example, an ion pair).

Rext - the external pure resistance

Ем1, Ем2 - the emf, corresponding to the direct reactions of the processes: M1  → c1 + e (for anode) and M2  →c2 + e (for cathode).

ER1, ER2 - the emf, corresponding to the direct reactions of the processes: Ox + е → a1 (for anode) and Ox + е → a2 (for cathode).

Efcc2a2, Efac1a1- the emf, corresponding to the direct reactions of the processes: c2 + а2 → c2a2 (on the cathode) and c1 + а1 → c1a1 (on the anode).

Esci, Esai - the emf, corresponding to the direct reactions of the processes of the solvation of cation ci and anion ai (i=1 for anode and i=2 for cathode).

Escisajs - the emf, corresponding to the direct reaction of the process of the formation of a neutral substance cisajs (in the solution) from solvated ions cis and ajs (i=1¸2; j=1¸2)

Ep - the emf (or voltage) of an external source of emf (voltage).

Each circuit of the equivalent electric circuit describes the definite process, taking place in the galvanic element described by this model. The definition of this process can be easily made on the basis of an algebraic sum of the vectors of the emf of this circuit. The detailed definitions of the mechanisms of processes, taking place in a number of circuits are shown in the table below and are necessary, as examples, for understanding the functioning of the circuit.

Table 1. the list of some circuits of the total circuit with the description of chemical processes taking place in them.

The author of the paper assumes that the positive circuit path-tracing direction (for the vectors of emfs and currents) is anticlockwise. If the direction of the vector of the emf is similar to the circuit path-tracing direction then this emf will be included in the expression for the second Kirchhoff's law for this circuit with the plus sign and this emf corresponds to the emf of the forward process. Otherwise, this emf is included in the expression for the second Kirchhoff's law for this circuit with the sign minus, so, the emf corresponds to the reverse process. The examples shown in the Table 1do not describe all currents (and all circuits) of the total circuit. Any reader of this paper, who wants to do it, can consider any other circuit of the total circuit and understand the physical meaning of the chemical processes taking place in this circuit.

The calculation of the equivalent circuit was made after some insignificant simplification shown below:

Fig.2. The result of the simplification of the author’s initial equivalent circuit.

where:

 The following currents were calculated for this circuit:

1. the current of corrosion of the anode, that is accompanied by the formation of the film on the anode - J1 (the circuit R1-R2-R3);
2. the current of corrosion of the anode, accompanied by the formation of the soluble products of corrosion in the solution - J2 (the circuit R1-R10-R14-R9-R3);
3. the current of corrosion of the cathode, accompanied by the formation of the film on the cathode - J3 (the circuit R7-R6-R8);
4. the current of corrosion of the cathode, accompanied by the formation of the soluble products of corrosion in the solution - J4 (the circuit R7-R11-R15-R12-R8);
5. the current of the joint corrosion of the anode and cathode - J5, flowing in the circuit R1-R10-R4-R11-R7-R8-R12-R5-R9-R3. In reality it is the current of self-discharge of the element;
6. the working current of the galvanic element - J6 (the circuit R1-R10-R4-R11-R7-R13);

The calculation of the currents in this equivalent electric circuit was made on the basis of the second Kirchhoff's law for these 6 aforesaid circuits.

The results of the calculation of parameters of the proposed equivalent circuit.

It was found:

1. there are 47 subcircuits which exert influence upon all electric currents, calculated in this paper

2. According to the principle of currents superposition, total current Ji (that flows through the fixed circuit i-branch) can be described as the sum     , where m is the number of currents that flow through the circuit i-branch, J1j is the j-current that flows through the circuit i-branch. 

        

In our case , where is the algebraic sum of the emfs contained in the closed subcircuit (where J1j flows). As regards the sign of Ef see the text below.

So, the total current Ji (that flows through the fixed circuit i-branch) is given by:

                (1)

where n is equal to the number of the subcircuits which exert influence upon the electric current Ji

sum(for f=1 to m of (Ef)) is the sum of the values of the emfs (Ef), which form such closed j-subcircuit, which has an influence upon the electric current Ji.

m is the number of the emfs which form the given closed subcircuit j. The value of the m depends on the value j, only.

K1j/K is the equivalent conductivity of the circuit, which is connected to the closed j-subcircuit.

    where K1j is the function of the sum (for f=1 to m of (Ef)). The list of values of K1j is represented in Table 2.

K is a constant value for all currents. Its value is derived in Appendix 1

The list of the 47 subcircuits that exert influence upon all electric currents, calculated in this paper, is shown in Table 2.

Table 2. The list of 47 subcircuits that exert influence upon the electric currents.

The analysis of the obtained expressions shows:

1. every sum of emfs is the emf of the well-defined chemical/electrochemical process. According to the principle of currents superposition, and the determination of the K1i/K, every factor K1i is the numerator of the equivalent conductivity of all subcircuits, connected with this closed j-subcircuit (subcircuit describing this chemical/electrochemical process). The values of K1i/K can be obtained as results of the solution of corresponding combined equations. However, some of factors K1i can be quickly calculated according to the following simplest rules, which were obtained by the paper author:
1. if the subcircuit (which describes the completed electrochemical/chemical process) by-passes (when we go round this subcircuit) the single resistances of the circuit (that are not connected with each other), then the factor K of the emf of this process is equal to the product of these resistances. If the factor of the emf (E3 + E2 - E4 - E6 + E14 - E12 - E11 + E8 + E7 + E15) is equal to R3 × R10 × R5 × R11 × R8 then this subcircuit by-passes single resistances R3, R10, R5, R11 and R8;
2. if the subcircuit (which describes the completed electrochemical/chemical process), by-passes (when we go round this subcircuit) one complex combination of resistances (e.g. a star connection of resistances), then the factor K of the emf of this process contains a function of these resistances as a subfactor. For example, the sum (R1 × R10 + R1 × R2 + R2 × R10), contained in the factor K1i of the emf (-E1 - E4 - E6 + E14 + E10 - E8 - E9 + E15), corresponds to the star connection of resistances R1-R2-R10 that were by-passed by the subcircuit (-E1 - E4 - E6 + E14 + E10 - E8 - E9 + E15);
2. the value of the K1i (and every resistance contained in the K1i) shows the influence of the value of the emf of the given process (that corresponds to the K1i) on the value of a current which is the function of this emf The value of every resistance (or group of resistances), contained in the K1i, can increase (or decrease) the influence of the emf of the given chemical/electrochemical process on the value of the current which is the function of this emf. For example, the growth of the cation resistance of the film, on the anode, R1 increases the influence of the emfs of the processes (whose factors K1i/K contain R1) on the current J6. These processes are accompanied by the reduction of the oxidant on the surface of the anode film. Thus, the processes of the reduction of an oxidant will predominate over the processes of the oxidation of a metal, on the anode. The other consequence of these processes is the growth of the electrode potential of the anode.

Note: It is necessary to take into consideration that the sign of the same emf of the process can vary in the equations of the different currents. The sign of every emf, included in the emf of the process, is positive if the vector of this emf has the same direction as the path-tracing direction of this subcircuit. For example, the currents J1 and J2 have opposite signs of the sum of emfs (E2 - E4 - E5 - E6). The direction of the vector of the emf E2 is the same as the direction of the current J1. So, the current J1 is the function of the emf (E2 - E4 - E5 - E6). Moreover, the value E2 is positive in the emfs (contained in the current J1), which contain the value E2. In the case with a current J2 the situation is opposite;

The working current (or the measured current) of a galvanic element (J6).

It was found, that the current J6 depends on the overall emfs of the subcircuits (chemical/electrochemical processes), that can run down on the external resistance R13 (the subcircuits containing the resistance R13):

                          (2)

where sum(for j=1 to m of (Ej)) is the sum of the values of the emfs -Ej, which describes a closed j-subcircuit (subcircuit of the chemical/electrochemical process), which contains resistance R13 (or emf E15).

The signs of the aforesaid sums must guarantee the positive sign of the vector of E15.

It is important to draw your attention to the fact that, the directions of both the current and emf (of the galvanic element) can be reversed when there is a certain ratio of the circuit resistances, but the emfs values are constant. An outside observer can interpret it as a polarity reversal of the galvanic element (the change of the sign of the electrode potential). This phenomenon can be observed in practice and this circuit shows the possibility of its existence.

The emf of a galvanic element (emfel).

On the basis of the expression for the working current of a galvanic element it is easy to determine the expression for the emf of the element.

In practice the emf of a power source is determined with the help of the opposition method when an external opposite polarization, that stops the current (flowing in the external circuit), is connected to the sockets of the electrodes of the battery. This polarization value is equal to the emf of the power source, but its direction is opposite to the emf of the power source. Therefore, for the determination of the emf of a chemical power source it is necessary to determine such polarization (E15) when the discharge current of the battery J6 is equal to 0. So, it is necessary to determine such E15 value when the numerator of J6 is equal to 0. It was found, that the emf (emfel) of the galvanic element (the potential of the single electrode) is equal to:

                                   (3)

The equation for Kemf is given in Appendix 1.

The value (E15 - sum (for j=1 to m of (Ej))) is similar to the sum (for j=1 to m of (Ej)), in the equation (2), but these sums differ by the sign and the final sum in the equation (3) doesn't contain the value of E15

Since the equation (3) contains the subcircuits, which can run down on the external resistance R13 only, it is obvious, that:

1. The value of the emfel doesn’t depend on the resistance of the measuring instrument (R13) as well as the value of the external polarization (E15) (see equation 3.). It is correct, because this value (emfel) is the function of the nature of the components of this system, but not the characteristics of a measurement system.
2. This expression for the emfel (the electrode potential) notably differ from the well-known expression for the emf of a galvanic element (the electrode potential), where the emf of only one reaction, which generates the current, was included. The role of other reactions, which generate the current, has not been taken into account before. Now it is possible to say with confidence, that the emf of a galvanic element (the electrode potential) is a function of all possible chemical (electrochemical) processes which can run down on an external resistance.
3. Every item of the emfel is equal to the sum of the difference of the potentials of the anode and cathode reactions of the fixed process + the potential of the ion-ion interaction (in the solution) of this process. This fact is observed, because the emf of the galvanic element is equal to the change of the Gibbs free energy of a chemical process divided by the nF value. Thus, the emf of a galvanic element (the electrode potential) is not equal to the ordinary difference of the electrode potentials of the anode and cathode (the difference of the electrode potentials of the working electrode and reference electrode). It is the well-known fact for every electrochemist and it often used in the course of the calculation of the emf of galvanic elements. Unfortunately, most electrochemists use the ordinary difference of the electrode potentials of the anode and cathode (the difference of the electrode potentials of the working electrode and reference electrode) instead of the real value of the emf (the electrode potential) and, so, ignore this fact very often. Such operation may be justified if the change of the Gibbs free energy of the ion-ion (and other) interaction, in the solution, is small; otherwise it is necessary to take this fact into account. Now we can see, that the using of this fact can be useful for the determination of the contributions of the different ion-ion (and other) processes to the emf of the galvanic element (the electrode potential) in the course of the measurements of the emf (electrode potential).
4. the potential of any i’s electrode (the emf of a galvanic element) depends on:
1. the predominant resistance (cationic or anionic) of films and interface boundaries on the anode (R1 and R3) and on the cathode (R7 and R8);
2. the sum value of the migration resistances of cations and anions on the film surface on the anode (R2) and cathode (R6);
3. the predominant resistance of the transfer of the molecules of the solvent to the place where cation goes out on the film/solution phase boundary (R10) or anion is formed (R9) for the anode (R11 or R12 correspondingly for the cathode);
4. the sum value of the migration resistances of solvated cations and anions (which were obtained on the same electrode) from the places where they form to the places, in the solution, where they interact (R14 for the anode and R15 for the cathode);
5. the sum value of the migration resistances of solvated cations and anions (which were obtained on the different electrodes) from the places where they form to the places, in the solution, where they interact (R4 or R5);
6. the values of the ratio of emfs (Ei) between each other;
5. the expression numerator for the electrode potential (the emf of a galvanic element) is practically similar to the expression of the numerator of the current J6. The difference between these values lies in the fact, that they can have opposite signs and the value E15 is certainly absent in the expression for the emf. This applies to all expressions for the emfs of this model.

The currents flowing through the system containing 2 similar electrodes (made out of the anode material - J6Ria and the cathode material - J6Ric) and an electrolyte (which contains an oxidant).  The process of the DC measurement of the electrolyte resistance.

J6Ria={2 × E15 × [[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10)] × [[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10)] × (R4 + R14) + R14 × (R1 + R2 + R3) × (2 × R4 + R14)] + R4 × (R1 × R14 + R2 × R14 + R3 × R14) × (R1 × R14 + R2 × R14 + R3 × R14)]/[KJ6Ria] } equation (4)

The equation for KJ6Ria is given in Appendix 1.

J6Ric={2 × E15 × (((R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)) × (((R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)) × (R5 + R15) + R15 × (R6 + R7 + R8) × (2 × R5 + R15)) + R5 × (R8 × R15 + R6 × R15 + R7 × R15) × (R8 × R15 + R6 × R15 + R7 × R15))/[ KJ6Ric]} equation (5)

The equation for KJ6Ric is given in Appendix 1.

The J6Ria current corresponds to the case, when E1=E7, E3=E9, E2=E8, E4=E10, E5=E11, E6=E12, E14=E13, R1=R8, R3=R7, R2=R6, R9=R11, R10=R12, R4=R5, R15=R14. The J6Ric current corresponds to the case, when E1=E7, E3=E9, E2=E8, E4=E10, E5=E11, E6=E12, E14=E13, R1=R8, R3=R7, R2=R6, R9=R11, R10=R12, R5=R4, R14=R15. The result that was obtained shows that these currents do not depend on any process in the system. They depend, only, on the equivalent resistances of the obtained system of electrodes. The values of these currents can be used for the calculation of the parameters of an electric circuit.

The internal resistance of a galvanic element (Rint).

The expression for the internal resistance of a galvanic element, which is described by means of the aforesaid model, can be obtained on the basis of the expressions of a working current value (J6), the emf of a galvanic element (emfel) and Ohm's law for the complete circuit:

                          (6)

Rint={[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [R4 × R5 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)]] + R14 × R15 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] + R14 × R15 × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9] + [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [[(R4 × R10 + R4 × R14 + R10 × R14) + R5 × R10] × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × (R3 + R9)] + R1 × (R4 + R5 + R14) × (R2 × R9 + R3 × R2 + R3 × R9) + R5 × R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × (R1 + R10) ]] + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [[(R4 × R11 + R4 × R15 + R15 × R11) + R5 × R11] × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × (R8 + R12)] + R7 × (R4 + R5 + R15) × (R6 × R12 + R6 × R8 + R8 × R12) + R5 × R15 × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × (R7 + R11)]] + [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)] × R15 × [R1 × [(R2 × R9 + R3 × R2 + R3 × R9) + R3 × (R10 + R14)] + R10 × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9]] + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10)] × R14 × [R7 × [(R6 × R12 + R6 × R8 + R8 × R12) + R8 × (R11 + R15)] + R11 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12]]}/{(R4 + R5) × [[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)]]] + R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)]] + R15 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)]]}        equation (7)

One can say with confidence, that this expression is the internal resistance of the galvanic couple, with potential emfel, such that the current through the resistance in the external circuit (R13 or Rext) is given by (emfel + E15)/(Rint + R13).

The currents of the formation of insoluble corrosion products on the anode (J1) and cathode (J3).

The expressions for corrosion currents J1 (for anode) and J3 (for cathode) are symmetrical (they are mirror images for each other), because this circuit is symmetrical:

                              (8)

where the sum for j=1 to m of (Ej) is equal to the emf of the process, contained in the Table 2, which includes the value of E2.

                                (9)

where the sum for j=1 to m of (Ej) is equal to the emf of the process, contained in the Table 2, which includes the value of E8.

The signs of the aforesaid sums must guarantee the positive sign of the vectors of E2 (in the case of J1) and E8 (in the case of J3).

The equations (8 and 9) show that the currents J1 and J3 depend on the emfs of the processes which run down on the resistance R13 (and do not run down on the resistance R13). So, we can say, that:

1.     the currents J1 and J3 are sums of the following currents:

1.1.                    currents that do not depend on the external polarization (E15):

1.1.1.  currents which depend on the emf of the processes of the formation of the films of corrosion products  (E1 + E2 + E3) (for the current J1) and (E7 + E8 + E9) (for the current J3);

1.1.2.  currents which depend on the emf of processes of the crystallization (from the solution) of the soluble corrosion products on the film surface (E2 - E4 - E5 - E6) (for the current J1) and (E8 - E10 - E11 - E12) (for the current J3);

1.1.3.  the electric currents depending on the processes whose emfs are equal to (E2 - E4 - E5 - E14 - E13 + E12) (for the current J1) and (E6 - E13 - E14 - E10 - E11 + E8) (for the current J3). These processes include the following stages: 

1.1.3.1.     the compounds c1sa2s and c2sa1s dissociate in the solution;

1.1.3.2.     in the case of the current J1, the formed ions c2s and a2s form the compound c2sa2s in the solution. In the case of the current J3, the formed ions c1s and a1s form the compound c1sa1s in the solution.;

1.1.3.3.  In the case of the current J1, the formed ions c1s and a1s decompose with the formation of the ions c1 and a1 and the molecules of the solvent. In the case of the current J3, the formed ions c2s and a2s decompose with the formation of the ions c2 and a2 and the molecules of the solvent. We call this process “desolvation (to desolvate)”. So I shall use this term below.

1.1.3.4.  in the case of the current J1, the corrosion product film grows on the anode. In the case of the current J3, the corrosion product film grows on the cathode.

1.1.4.  the electric current depending on the process whose emf are equal to (E2 - E4 - E5 - E13 - E14 - E10 - E11 + E8) (for the current J1and current J3). This process includes the following stages:

1.1.4.1.     the corrosion product films located on the cathode and anode dissociate in the solution. The ions c2s, a2s, c1s and a1s are formed at this stage;

1.1.4.2.     the ions c2s and a2s (c1s and a1s) interact in the solution. The ion pairs c2sa2s and c1sa1s are formed at this stage;

1.1.5.  the electric currents depending on the processes whose emfs are equal to (E2 - E4 - E5 - E13 - E14 - E10 - E11 - E7 - E9) (for the current J1) and (-E1 - E3 - E4 - E5 - E13 - E14 - E10 - E11 + E8) (for the current J3). These processes include the following stages:

1.1.5.1.     the compounds c1sa2s and c2sa1s dissociate in the solution. The ions c1s, a1s, c2s and a2s are formed at this stage;

1.1.5.2.     in the case of the current J1, formed ions c2s and a2s are desolvated then. In the case of the current J3, the formed ions c1s and a1s are desolvated then;

1.1.5.3.     in the case of the current J1, the desolvated ion c2 transferring through the film on the cathode surface is reduced then on the cathode; the desolvated ion a2 is oxidized on the film on the cathode surface. In the case of the current J3, the desolvated ion c1 transferred through the film on the anode surface is reduced then on the anode; the desolvated ion a1 is oxidized on the film on the anode surface;

1.1.5.4.     in the case of the current J1 (J3), the solvated ions c1s and a1s (c2s and a2s) are desolvated then on the surface of the film on the anode (cathode);

1.1.5.5.     the corrosion product film c1a1 (c2a2) grows on the anode (cathode);

1.2.                    the sum of currents, which depend on the external polarization (E15);

2.     in the case of the anode, it is clear, that when the sum of the resistances of the transfer of the molecules of the solvent to the place where a cation goes out on the film/solution phase boundary (and an anion is formed) (R9 + R10) is greater, than the sum value of the migration resistances of cations and anions on the film surface on the anode (R2), then the formation of the insoluble corrosion products film on the anode takes place. In a similar manner, when the sum of the resistances of the transfer of the molecules of the solvent to the place where cation goes out on the film/solution phase boundary (and anion is formed) (R11 + R12) is greater than the sum value of the migration resistances of cations and anions on the film surface on the cathode (R6), then the formation of the insoluble corrosion products film takes place on the cathode;

3.      the value of the current J1 (J3) depends on the currents (which depend on the external polarization), which flow through the branch of the circuit 1-2 (7-8) and that are accompanied by the formation of the film of products of corrosion on the anode (cathode) (it is a result of either corrosion or the solution crystallization).

4.      the current J1 equal to 0 (J3 is equal to 0) when E15 have certain value (see below).

The values of an external polarization (E15) when the current J1 (or J3) of the formation of insoluble products of corrosion for the anode (cathode) is equal to 0.

The expressions of the values of these emfs are symmetrical (they are mirror images for each other), because this circuit is symmetrical. The expressions of these values (E15J1=0 for the anode) and (E15J3=0 for the cathode) are illustrated below:

                  (10)

                            (11)

where the sum for j=1 to m of (Ej) - the sum of the items of the emfel, whose emfs (emfi) of processes include values of E2 or E8. The signs of the aforesaid sums must guarantee the negative sign of the vectors of E2 (in the case of E15J1) and E8 (in the case of E15J3).

The equations for Ke1 and Ke3 are given in Appendix 1.

These results represent the values of the external polarization when corrosion currents (that determine the growth of the films of corrosion products) of the anode and cathode are equal to 0. These values are one of the conditions of the protection against corrosion for the anode and cathode. The expressions, obtained, can be useful for the determination of the condition of the cathodic and anodic protection of metals against corrosion.

The currents (J2 and J4), which form the anode's (cathode's) soluble corrosion products in the solution.

The expressions for corrosion currents J2 (for anode) and J4 (for cathode) are symmetrical (they are mirror images for each other), because this circuit is symmetrical.  

                                           (12)

where the sum for j=1 to m of (Ej) is equal to the emf of the process, contained in the Table 2, which includes the value of E6.

                                 (13)

where the sum for j=1 to m of (Ej) is equal to the emf of the process, contained in the Table 2, which includes the value of E12.

The signs of the aforesaid sums must guarantee the positive sign of the vectors of E8 (in the case of J3) and E12 (in the case of J4). The results obtained show that:

1.1.         which do not depend on the external polarization (E15):

1.1.1.  the currents, which depend on the corrosion processes on the anode (whose emf is equal to (E1 + E3 + E4 + E5 + E6)) and cathode (whose emf is equal to (E7 + E9 + E10 + E11 + E12));

1.1.2.  the currents, which depend on the processes of the physical dilution of the corrosion products of the anode (cathode) in the solution (whose values of emf are equal to (-E2 + E4 + E5 + E6) and (-E8 + E10 + E11 + E12));

1.1.3.  the currents, which depend on the processes of the exchange of ions in the solution whose emf is equal to (E6 + E12 - E13 - E14);

1.1.4.  the electric currents depending on the processes whose emfs are equal to (E6 - E14 - E13 - E10 - E11 + E8) (for the current J2) and (E2 - E4 - E5 - E13 - E14 + E12) (for the current J4). These processes include the following stages:

1.1.4.1.   the compounds c1sa2s and c2sa1s dissociate in the solution;

1.1.4.2.   in the case of the current J2, the formed ions c1s and a1s form a compound c1sa1s in the solution. In the case of the current J4, the formed ions c2s and a2s form a compound c2sa2s in the solution.;

1.1.4.3.   in the case of the current J2, the formed ions c2s and a2s desolvate. In the case of the current J4, the formed ions c1s and a1s desolvate.

1.1.4.4.   in the case of the current J2, the corrosion product film grows on the cathode. In the case of the current J4, the corrosion product film grows on the anode.

1.1.5.  the electric currents depending on the processes whose emfs are equal to (E6 - E14 - E13 - E10 - E11 - E7 - E9) (for the current J2) and (-E1 - E3 - E4 - E5 - E13 - E14 + E12) (for the current J4). These processes include the following stages:

1.1.5.1.   the compounds c1sa2s and c2sa1s dissociate in the solution;

1.1.5.2.   in the case of the current J2, formed ions c2s and a2s are desolvated then on the surface of the film on the cathode. In the case of the current J4, formed ions c1s and a1s are desolvated then on the surface of the film on the anode;

1.1.5.3.   in the case of the current J2, the desolvated ion c2 transferred through the film on the cathode surface is reduced then on the cathode; the desolvated ion a2 is oxidized on the film on the cathode surface. In the case of the current J4, the desolvated ion c1 transferring through the film on the anode surface is reduced then on the anode; the desolvated ion a1 is oxidized on the film on the anode surface;

1.1.5.4.   in the case of the current J2, the compound c1sa1s forms in the solution. In the case of the current J4, the compound c2sa2s forms in the solution;

1.2.         the current, which depends on the external polarization (E15);

3.  only the currents that depend on the external polarization and which flow through subcircuits 3-4 or 5-6 exert influence on the currents J2 and J4. These currents are accompanied by the formation of soluble products of corrosion of the anode (cathode)  (this process take place due to corrosion or due to physical dilution of the film on the anode (cathode) or due to the process of the ion exchange in the solution);

5.  the current J2 equal to 0 (J4 is equal to 0) when E15 have a certain value (see below).

The values of an external polarization emfs when J2 is equal to 0 (E_15J2=0) and J4 is equal to 0 (E_15J4=0).

The expressions of these values are mirror images for each other, because this circuit is symmetrical.

    (14)

    (15)

where the sum for j=1 to m of (Ej) - the sum of the items of the emfel, whose emfs (emf_i) of processes include values of E6 or E12.

The signs of the aforesaid sums must guarantee the negative sign of the vectors of E6 (in the case of E15J3) and E12

The equations for Ke2 and Ke4 are given in Appendix 1.

The summed currents of corrosion of the anode (Ja = J1 + J2) and the cathode (Jc = J3 + J4).

The expressions of the values of the summed currents of corrosion Ja and Jc are symmetrical (they are mirror images for each other) because this circuit is symmetrical.

Ja=J1 + J3                                                    (16)

Jc=J2 + J4                                                     (17)

The results obtained show that:

1. only the currents, which flow through subcircuits 1-2, 3-4, 5-6, 7-8 and which correspond to the formation of the insoluble and soluble products of corrosion of the anode and cathode (which, also, is the result of corrosion, or the result of the processes of the ion exchange in the solution) can influence the values of the currents of the summed corrosion of the anode (J1 + J2) and cathode (J3 + J4);
2. we can determine a certain E15 value, when the current of summed corrosion of the anode (J1 + J2) is equal to 0 and a certain E15 value when the current of summed corrosion of the cathode (J3 + J4) are equal to 0 (see below).

The values of an external polarization (E15) when current Ja= J1 + J2 =0 (E15J120) and Jc=J3 + J4=0 (E15J340).

The expressions of these values (E15J120 and E15J340) are mirror images for each other, because the circuit is symmetrical.

E15J120=                                       (18)

                           (19)

where  is the sum of the items of the emfel, whose emfs (emf_i) of the processes include one (not two) of the values of E2, E6, E8, E12. For example, in the case of E15J120, this sum can’t include the emfs of the processes that contain the values E2 and E6 at the same time. The sign of the sum must guarantee the counter-clockwise direction of the vectors of E2, E6, E8, E12.

The equations for Ke12 and Ke34 are given in Appendix 1.

The results obtained show that it is possible to stop corrosion processes on the anode and cathode when certain values of the external polarization are applied. The aforesaid expressions have an important practical meaning. They either allow us to determine the numerical values of the external polarization protecting the electrodes (or a corrodible metal) and show such factors of the system that exert an influence on the value of this polarization. It is obvious, that the similar expressions may be obtained for all combinations of the currents (processes). This allows us to stop any chemical/electrochemical processes in this system with the help of the corresponding value of external polarization.

The current of the joint corrosion of the anode and cathode (Jjc).

This type of corrosion is the result of the interaction, in the solution, of cationic (anionic) products of corrosion of the anode with the anionic (cationic) products of corrosion of the cathode.

                        (20)

where sum for j=1 to m of (Ej) is equal to the emf of the process, contained in the Table 2, which includes the value of E13.

The signs of the aforesaid sums must guarantee the positive sign of the vector of E13.

The result obtained shows that:

1. only the currents, which flows through subcircuits 3-5 exert influence on the value of the current of joint corrosion of the anode and cathode;
2. we can determine a certain E15 value, when the total current of the joint corrosion of the anode and cathode (Jjc) is equal to 0 (see below).

The value of an external polarization (E15) when Jjc is equal to 0 (E15Jjc=0).

                     (21)

where sum for j=1 to m of (Ej) - the sum of the items of the emfel, whose emfs (emf_i) of processes include value of E13.

The sign of the sum must guarantee the positive sign of the vectors of E13.

The equation for Kejc is given in Appendix 1.

The current of the total corrosion in a galvanic element Jcel=J1 + J2 + J3 + J4 + Jjc

Jcel=J1 + J2 + J3 + J4 + Jjc                                     (22)

The result obtained shows that:

1.  the total corrosion of the anode and cathode takes place due to currents, which do not depends on the external polarization (E15);

2. the total corrosion of the anode and cathode takes place due to currents, which depends on the external polarization (E15) also;
3. we can determine certain E15 value, when the total corrosion current of the galvanic element (Jcel) is equal to 0 (see below).

The value of the emf of an external polarization (E15) when Jcel=0 (E15Jcel0)

                    (23)

where: the sign of the sum must guarantee the counter-clockwise direction of the vectors of E2, E6, E8, E12, E13.

The equation for Ke12345 is given in Appendix 1.

The result obtained shows that for this model there is such value of the external polarization (E15), when the sum of all corrosion currents of the anode and cathode in the galvanic element is equal to 0. This polarization allows carrying out the complete protection of a galvanic element from any forms of the self-discharge (that take place due to corrosion of separate electrodes, as well as the joint corrosion of the electrodes). This phenomenon can find the practical application for the:

1. realization of the ideal galvanic protection of metals against corrosion;
2. long time storage of galvanic elements before sale (or at their non-working periods).

Conclusions.

1. On the basis of the approach developed earlier [1] for the modelling of ionic processes, as an example, the model of the functioning (under conditions of discharge on an external resistance and when an additional external polarization is applied) of a galvanic element is proposed;
2. the fundamentally new design equations for the working current, the emf of a galvanic element (the potentials of separate electrodes), the currents of corrosion of each electrode, the values of an external polarization (which eliminates the corrosion of separate electrodes as well as that of all electrodes simultaneously) are obtained for this model;
3. the expressions obtained (paragraph 2) allow us to determine the factors, which exert influence on the working current, the emf of a galvanic element (the potentials of separate electrodes), the currents of corrosion of electrodes. Some of these results agree with the well-known electrochemical practice (the possibility of the galvanic protection of metals against corrosion). The other results are obtained for the first time in practice (general expressions for the working current, the corrosion currents of electrodes in a battery, the emf of a galvanic element, the potentials of separate electrodes, the external polarizations which eliminate the corrosion of separate as well as that of all electrodes, the internal resistance of the galvanic element). The physical meaning of the measured potential of an electrode (the emf of a galvanic element) is described in detail. All expressions obtained have a clear thermodynamic meaning.
4. thus, the possibility of the application of the earlier proposed approach [1] is demonstrated for the description, modelling and optimizing of the existent and radically new galvanic elements. This approach can be useful, also, for the description and understanding of the results, which have been obtained in the course of different electrochemical research activities.

Acknowledgments

Special thanks to Freydman Leonid Isaacovich, engineer-creator of the experimental-design office ART (for his advices and significant remarks concerning the material of this paper) and to Panov E. E. (for his important assistance in the translation of this paper)..

References

1.     “The new principles of the building of equivalent circuits for the modeling of metal-film-corrosive environment systems”, Shagaev A. A., Abolin O. E., Danilov V. G.. The Journal of Corrosion Science and Engineering, 2002, vol. 3, paper 14.

Appendix

Values of some terms

The value of some of the terms used in the text are given by the following expressions:

K={[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [[(R2 × R9 + R2 × R3 + R3 × R9) + R1 × R9] × [(R4 × R10 + R4 × R14 + R10 × R14) + R5 × R10] + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] × [(R4 + R5 + R14) × R13 + R4 × R5] + [(R4 × R10 + R4 × R14 + R10 × R14) + (R5 × R9 + R5 × R14 + R9 × R14) + (R4 × R9 + R5 × R10)] × R1 × R3 + (R4 + R5 + R14) × R1 × R2 × (R3 + R9) + R14 × [(R1 + R2 + R3) × (R4 × R13 + R4 × R5 + R5 × R13) + R5 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10]]] + R15 × [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)] × [R13 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] + (R1 × R10 + R1 × R2 + R2 × R10) × (R3 + R9) + R3 × R9 × (R1 + R10)] + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] × [(R6 × R12 + R6 × R8 + R8 × R12) × [(R4 × R11 + R4 × R15 + R11 × R15) + R5 × R11 + R7 × (R4 + R5 + R14 + R15)] + [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × (R8 + R12)] × R14 × R11 + [(R4 × R11 + R4 × R15 + R11 × R15) + R5 × R11] × R7 × (R8 + R12) + [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × (R7 + R11)] × R5 × R15] + R14 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] × [ R15 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] + (R1 + R2 + R3) × [(R4 × R11 + R4 × R15 + R11 × R15) + R5 × R11]] + R14 × R7 × (R1 + R2 + R3) × (R4 + R5 + R15) × [(R6 × R12 + R6 × R8 + R8 × R12) + R8 × R11] + R14 × R15 × [[(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] × [(R2 × R9 + R2 × R3 + R3 × R9) + R1 × R9 + R5 × (R1 + R2 + R3)] + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] × R7 × R8 + [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)] × R1 × R3 + R7 × R8 × (R1 + R2 + R3) × (R4 + R5)]} equation (24)

Kemf={[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × (R4 + R5) × [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] + R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)]] + R15 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)]]}   equation (25)

KJ6Ria={[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [[(R2 × R9 + R2 × R3 + R3 × R9) + R1 × R9] × [(R4 × R10 + R4 × R14 + R10 × R14) + R4 × R10] + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] × [2 × R13 × (R4 + R14) + R4^2] + [(R4 × R10 + R4 × R14 + R10 × R14) + (R4 × R9 + R4 × R14 + R9 × R14) + R4 × (R9 + R10)] × R1 × R3 + (2 × R4 + R14) × R1 × R2 × (R3 + R9) + R14 × R4 × [(R1 + R2 + R3) × (R4 + 2 × R13) + [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10]]] + R4 × R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)]^2 + [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] × [(R1 × R10 + R1 × R2 + R2 × R10) × [(R4 × R9 + R9 × R14) + R4 × R9 + 2 × R3 × (R4 + R14)] + [(R1 × R10 + R1 × R2 + R2 × R10) × (R3 + 2 × R9) + 2 × R3 × R9 × (R1 + R10)] × R14 + [2 × R4 × R9 + 2 × R4 × R14 + R9 × R14 + 2 × R14^2] × R1 × R3 + (2 × R4 + R14) × R3 × R9 × R10] + R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] × [R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] + (R1 + R2 + R3) × (2 × R4 × R9 + R4 × R14 + R9 × R14)] + R14 × R3 × (R1 + R2 + R3) × (R4 + R4 + R14) × [(R1 × R10 + R1 × R2 + R2 × R10) + R1 × R9] + R14^2 × [[(R2 × R9 + R2 × R3 + R3 × R9) + R1 × R9] × [(R2 × R9 + R2 × R3 + R3 × R9) + R1 × R9 + R4 × (R1 + R2 + R3)] + 2 × R1 × R3 × R4 × (R1 + R2 + R3)]}  equation (26)

KJ6Ric={[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] × [(R5 × R12 + R5 × R15 + R12 × R15) + R5 × R12] + [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)] × [2 × R13 × (R5 + R15) + R5^2] + [(R5 × R12 + R5 × R15 + R12 × R15) + (R5 × R11 + R5 × R15 + R11 × R15) + R5 × (R11 + R12)] × R8 × R7 + (2 × R5 + R15) × R8 × R6 × (R7 + R11) + R15 × R5 × [(R6 + R7 + R8) × (R5 + 2 × R13) + [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12]]] + R5 × R15 × [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)]^2 + [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)] × [(R6 × R12 + R6 × R8 + R8 × R12) × [(R5 × R11 + R11 × R15) + R5 × R11 + 2 × R7 × (R5 + R15)] + [(R6 × R12 + R6 × R8 + R8 × R12) × (R7 + 2 × R11) + 2 × R7 × R11 × (R8 + R12)] × R15 + [2 × R5 × R11 + 2 × R5 × R15 + R11 × R15 + 2 × R15^2] × R8 × R7 + (2 × R5 + R15) × R7 × R11 × R12] + R15 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] × [R15 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] + (R6 + R7 + R8) × (2 × R5 × R11 + R5 × R15 + R11 × R15)] + R15 × R7 × (R6 + R7 + R8) × (2 × R5 + R15) × [(R6 × R12 + R6 × R8 + R8 × R12) + R8 × R11] + R15^2 × [[(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11 + R5 × (R6 + R7 + R8)] + 2 × R8 × R7 × R5 × (R6 + R7 + R8)]} equation (27)

Ke1={[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [-R1 × [(R9 × R5 + R14 × R5 + R9 × R14) + R9 × R4] + R3 × [(R10 × R4 + R14 × R4 + R10 × R14) + R10 × R5]] + R15 × [+(- R1 × R9 + R3 × R10) × [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11)] + R14 × [-R1 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] + R3 × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11]]]} equation (28)

Ke3={[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [-R7 × [(R12 × R5 + R15 × R5 + R12 × R15) + R12 × R4] + R8 × [(R11 × R4 + R15 × R4 + R11 × R15) + R11 × R5]] + R14 × [(-R7 × R12 + R8 × R11) × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R2 × R3 + R3 × R9) + (R1 × R9 + R3 × R10)] + R15 × [-R7 × [(R2 × R9 + R2 × R3 + R3 × R9) + R1 × R9] + R8 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10]]]} equation (29)

Ke2={ [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [R5 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] - R4 × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9]] + R15 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] - R15 × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9]}  equation (30)

Ke4={[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [R5 × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] - R4 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12]] + R14 × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9] - R14 × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10]} equation (31)

Ke12={[(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [-(R1 × R10 + R1 × R2 + R2 × R10) × R5 + (R2 × R9 + R3 × R2 + R3 × R9) × R4 + (R4 × R10 + R4 × R14 + R10 × R14) × R3 - (R5 × R9 + R5 × R14 + R9 × R14) × R1] + R15 × [(R2 × R9 + R3 × R2 + R3 × R9) + R3 × (R10 + R14)] × [(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11] - R15 × [(R1 × R10 + R1 × R2 + R2 × R10) + R1 × (R9 + R14)] × [(R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12]} equation (32)

Ke34={ [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [R5 × (R6 × R11 + R6 × R7 + R7 × R11) - R4 × (R6 × R12 + R6 × R8 + R8 × R12) - R8 × (R4 × R11 + R4 × R15 + R11 × R15) + R7 × (R5 × R12 + R5 × R15 + R12 × R15)] + R14 × [(R6 × R11 + R6 × R7 + R7 × R11) + R7 × (R12 + R15)] × [(R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9] - R14 × [(R6 × R12 + R6 × R8 + R8 × R12) + R8 × (R11 + R15)] × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10]} equation (33)

Kejc={R4 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [ [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)]] + R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + R3 × R10] × [ [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)]] + R15 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [[(R6 × R11 + R6 × R7 + R7 × R11) + R8 × R11]]} equation (34)

Ke12345={[(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + (R1 × R9 + R3 × R10) + R14 × (R1 + R2 + R3)] × [R5 × [(R6 × R11 + R6 × R7 + R7 × R11) + R7 × (R12 + R15)] + R7 × R12 × R15] + [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + (R7 × R12 + R8 × R11) + R15 × (R6 + R7 + R8)] × [R5 × [(R1 × R10 + R1 × R2 + R2 × R10) + R1 × (R9 + R14)] + R1 × R9 × R14] + R14 × [(R1 × R10 + R1 × R2 + R2 × R10) + (R2 × R9 + R3 × R2 + R3 × R9) + R1 × R9] × [[(R6 × R11 + R6 × R7 + R7 × R11) + R7 × (R12 + R15)]] + R15 × [(R6 × R11 + R6 × R7 + R7 × R11) + (R6 × R12 + R6 × R8 + R8 × R12) + R7 × R12] × [[(R1 × R10 + R1 × R2 + R2 × R10) + R1 × (R9 + R14)]] + R14 × R15 × [R6 × (R1 × R10 + R1 × R2 + R2 × R10) + R2 × (R6 × R11 + R6 × R7 + R7 × R11)] + R4 × [(R1 × R10 + R1 × R2 + R2 × R10) + R1 × R9 + R14 × (R1 + R2)] × [(R6 × R11 + R6 × R7 + R7 × R11) + R7 × R12 + R15 × (R6 + R7)] - (R2 × R9 + R3 × R2 + R3 × R9) × [R8 × (R4 × R11 + R4 × R15 + R15 × R11) + R4 × (R6 × R12 + R6 × R8 + R8 × R12)] - R3 × (R10 × R4 + R14 × R4 + R10 × R14) × [(R6 × R12 + R6 × R8 + R8 × R12) + R8 × R11] - R3 × R15 × R8 × [(R10 × R4 + R14 × R4 + R10 × R14) + R11 × (R10 + R14)]} equation (35)

Text description of Figure 1

The figure shows the initial equivalent circuit used in this paper. There are following major nodes in the circuit:

node 0 corresponds to the bulk of the metal of the anode

node 1 corresponds to the place on the anode film where delocalized anion (a1) is generated

node 2 corresponds to the place, on the anode, where the delocalized cation (c1) of the anode metal goes out on the film surface on the anode

node 3 corresponds to the place, in the solution, where solvated anion (a1s) is generated

node 4 corresponds to the place, in the solution, where solvated cation (c1s) of the anode metal is formed

node 5 correspond to the place, in the solution, where solvated cation (c2s) of the cathode metal is formed

node 6 corresponds to the place, in the solution, where solvated anion (a2s) is generated

node 7 corresponds to the place, on the cathode, where the delocalized cation (c2) of the cathode metal goes out on the film surface on the cathode

node 8 corresponds to the place on the cathode film where delocalized anion (a2) is generated

node 9 corresponds to the bulk of the metal of the cathode.

Then the following circuit elements are connected in series:

RM11 + Re1 + ER1 + Rs1ox between node 0 and node 1 (this branch of the circuit describes the transfer of the electron and oxidant molecule towards each other, on the anode, and their interaction. The delocalized anion a1 is the result of this interaction)

RM12 + EM1 + Rc1 between node 0 and node 2 (this branch of the circuit describes, first of all, the formation of the cation c1 and electron in the volume of the metal of anode and, secondly, the migration of the cation c1 to the metal/film phase boundary and, thirdly, the migration of the electron to the place where it interacts with the molecule of the oxidant.)

Rc1a1a1 + Efac1a1 + Rc1a1c1 between node 1 and node 2 (this branch of the circuit describes the migration of the delocalized cation c1and anion a1 towards each other, on the surface of the film, and their interaction. The film substance c1a1 is the result of this interaction).

Esa1 + Ras1 between node 1 and node 3 (this branch of the circuit describes the transfer of the solvent molecule to the places where the anion a1 is formed on the anode film surface)

Esc1 + Ras2 between node 2 and node 4 (this branch of the circuit describes the transfer of the solvent molecule to the places where cation c1 goes out on the film surface on the anode)

Rc1sa1sa1s + Esc1sa1s + Rc1sa1sc1s between node 3 and node 4 (this branch of the circuit describes the migration of the solvated cation c1s and anion a1s towards each other, in the solution, and their interaction. The ionic pair c1sa1s is the result of this interaction)

Rsc1sa2s + Esc1sa2s + Rsa2sc1s between node 4 and node 6 (this branch of the circuit describes the migration of the solvated cation c1s and anion a2s towards each other, in the solution, and their interaction. The ionic pair c1sa2s is the result of this interaction)

Rsa1sc2s + Esc2sa1s + Rsc2sa1s between node 3 and node 5 (this branch of the circuit describes the migration of the solvated cation c2s and anion a1s towards each other, in the solution, and their interaction. The ionic pair c2sa1s is the result of this interaction)

Rc2sa2sc2s + Esc2sa2s + Rc2sa2sa2s between node 5 and node 6 (this branch of the circuit describes the migration of the solvated cation c2s and anion a2s towards each other, in the solution, and their interaction. The ionic pair c2sa2s is the result of this interaction)

Esc2 + Rcs3 between node 5 and node 7 (this branch of the circuit describes the transfer of the solvent molecule to the places where cation a2 goes out on the film surface on the cathode)

Esa2 + Rcs4 between node 6 and node 8 (this branch of the circuit describes the transfer of the solvent molecule to the places where anion a2 is formed on the film surface on the cathode)

Rc2a2a2 + Efcc2a2 + Rc2a2c2 between node 7 and node 8 (this branch of the circuit describes the migration of the delocalized cation c2 and anion a2 towards each other, on the surface of the film, and their interaction. The film substance c2a2 is the result of this interaction).

RM21 + Re2 + ER2 + Rs2ox between node 8 and node 9 (this branch of the circuit describes the transfer of the electron and oxidant molecule towards each other and their interaction. The delocalized anion a2 is the result of this interaction)

RM22 + EM2 + Rc2 between node 7 and node 9 (this branch of the circuit describes, first of all, the formation of the cation c2 and electron in the volume of the cathode metal and, secondly, the migration of the cation c2 to the metal/film phase boundary and, thirdly, the migration of the electron to the place where it interacts with molecule of oxidant.)

RM21 + Re2 + ER2 + Rs2ox between node 8 and node 9 (this branch of the circuit describes the transfer of the electron and oxidant molecule towards each other, on the cathode, and their interaction. The delocalized anion a2 is the result of this interaction)

Ep + Rext between node 0 and node 9 (this branch of the circuit describes an external circuit, that contains a pure resistance and the source of an external polarization)

The author of the paper took into account following currents flowing in this circuit:

current J1, flowing from node 0 to node 1, next to node 2 and finally to node 0 (this current describes the rate of the growth of the corrosion product (c1a1) film on the anode);

current J2, flowing from the node 0 to node 1, then to node 3, then to node 4, then to node 2 and finally to node 0 (this current describes the rate of the formation of the soluble corrosion product (c1sa1s) of the anode in the solution);

current J3, flowing from node 9 to node 8, next to node 7 and finally to node 9 (this current describes the rate of the growth of the corrosion product (c2a2) film on the cathode);

current J4, flowing from the node 9 to node 8, then to node 6, then to node 5, then to node 7 and finally to node 9 (this current describes the rate of the formation of the soluble corrosion product (c2sa2s) of the cathode in the solution);

current J5, flowing from node 0 to node 1, then to node 3, then to node 5, then to node 7, then to node 9, then to node 8, then to node 6, then to node 4, then to node 2 and finally to node 0 (this current describes the rates of the formation of the soluble corrosion products c1sa2s and c2sa1s in the solution. It is the current flowing between anode and cathode when an external circuit is open, also);

current J6, flowing from the node 0 to node 9, then to node 8, then to node 6, then to node 4, then to node 2 and finally to node 0. It is the current flowing through the external resistance R13. We can measure this current by means of the usual VA measurement.

End of description of Figure 1