This page consists of 3 sections:

•Charge Transfer Reaction

•Chemical Reaction

•Species

1. Adding a Charge-Transfer Reaction

Click on the first empty line headlined Charge-Transfer Reaction. Enter the name of the species involved in the charge transfer process in the appearing dialog box

 
and select  the type of the CT-reaction provided in the Combo-Box.

Syntax:

oxidized species + n e- = reduced species                (kf and kb computed from Butler-Volmer or Marcus equation (the latter option applies only to n= 1))

oxidized species + n e- => reduced species        (kb=0, kf computed from Butler-Volmer or Marcus equation the latter option applies only to n= 1)

oxidized species + n e- <= reduced species        (kf=0, kb computed from Butler-Volmer or Marcus equation the latter option applies only to n= 1)

 
Note, that Marcus-kinetics is allowed only for charge-transfer reactions with n= 1. All charge-transfer reactions are reset to Butler-Volmer-kinetics when adding a reaction with n>1.

Charge transfer reactions are always formulated to proceed as a reduction from the left to the right. Whether a charge transfer step does really proceed as reduction or oxidation is determined by the relation between initial and standard potential and the direction of the potential scan entered on the Property Page: Simulation Parameters and Chemical Reactions.

 
The entire string entered into the left text field is considered the name of the oxidized species. Analogously, the string on the right-hand site is considered the name of the reduced species. The names for the oxidized and reduced species can be freely chosen but the input must not violate the following rules:

•the species name of oxidized and reduced form must not be equal or empty

•any character (including space) can be used in a species name but the first character must not be X, Y or Z

•the number of electrons must be an integer ranging from 1 to 9
 

Checkbox: Enable Adsorption

Activate the Enable Adsorption checkbox if adsorption reactions need to be taken into account for the species involved in the underlying charge-transfer reaction. The adsorption parameters can be entered on the Property Page: Surface Reactions (see below) as soon as the input on the Property Page: Chemical Reactions has been completed. Note that E° and Keq must have been specified for each entered reaction and the required analytical concentrations must have been entered. Otherwise, access to the Property Page: Surface Reactions remains disabled.

CT-Reactions involving adsorbed species are written in red as shown in the following picture:

View video clip to see an example.

Linking the heterogeneous rate constant of a CT reaction to the surface coverage of an adsorbed species. A heterogeneous rate constant, ks, written in red has been linked to the surface coverage of an adsorbed species. Click here for more details.
 

Checkbox: Enable termolecular CT-Reaction
Ticking this checkbox transforms the dialog box to the following form

This enables the user to enter a termolecular charge transfer reaction (two species + electrode) of the following form

Ox + P + e = Red + Q

where P or Q might be an empty string as well. If both, P and Q are empty the CT-Reaction is treated as a "normal" bimolecular charge transfer.

 
Termolecular CT-Reactions such as the Concerted Proton-Electron-Transfer (CPET)

Ox + H3O+ = HRed + H2O

have attracted increasing interest during the last couple of years.
 

In order to accomplish that the heterogeneous rate constant, ks, has the usual unit of cm/s it is assumed in DigiElch that the concentration of species P and Q is dimensionless. For this purpose the "real-world concentrations" of P and Q must be divided by 1 mol/l.    

2. Adding a Homogeneous Chemical Reaction

Click on the first empty line headlined Chemical Reaction and enter the name of the involved species in the appearing dialog box:

Second-order chemical reactions comprising  up to four species can be modeled:
 

reversible reactions (Keq = kf/kb):

Species1 + Species2 = Species3 + Species4

irreversible reactions (kb=0, independetly of Keq):

Species1 + Species2 => Species3 + Species4

 
The input must be compliant with the following rules:

•if the name of a species starts with X, Y or Z it is considered a buffer/excess component. The concentration of such a species does not change in the course of the simulation

•excess components or empty strings can be entered only for Species 2 and/or Species 4

•(pseudo-) first-order reaction: Species 2 and Species 4 are empty strings (or excess components)

•second-order reaction: no species name or only the name of Species 2 or 4 is empty or that of an excess component
 

Note that numbers are not considered stoichiometric numbers. The notation “2A” is therefore not an abbreviation for "A + A" but simply interpreted as a name of a species. For this reason the reaction “A+ + A- = 2A” (where “2” is a stoichiometric number) must be entered as “A+ + A- = A + A”

3. Editing or Removing a Reaction Equation

Reaction equations can be edited/removed simply by clicking on these reaction equation. The appearing dialog box provides the following options:

•Cancel
does nothing

•Copy Reaction
the reaction equation is copied if the dialog box is closed by OK.

•Remove Reaction
the reaction will be removed after asking for a confirmation.

•OK
Accept modifications (if any).
 

4. Meaning of the Chemical/Electrochemical Parameters

•Heterogeneous Reactions

oEo (V), ks (cm/s)
Standard potential, Eo, and the heterogeneous rate constant, ks, have their usual physical meaning. The forward and backward rate constants actually used in the simulation are computed from Eo, α and ks by means of the Butler-Volmer equation or Marcus-Equation. The latter depends on whether the button α or the button λ (eV) has been selected.

oButton α and λ (eV)
When selecting "α" all charge transfer reactions are treated to obey the Butler-Volmer-equation while Marcus-Kinetics is assumed when selecting " λ (eV)". The latter option is restricted to one-electron charge-transfer steps.

•Homogeneous Reactions

oKeq, kf, kb
Each homogeneous chemical reaction must be characterized by equilibrium constant, Keq, and rate constant, kf ,associated with the "forward reaction" i.e. the reaction proceeding from the left to the right in the underlying reaction equation. The value of the backward reaction, kb, is computed by DigiElch and cannot be entered. The unit of kf  is s-1 if a first-order chemical reaction has been entered and it is l*mol-1s-1 in the case of a second-order chemical reaction.

odepends on whether a first- or second-order chemical reaction has been entered.
 

Note that DigiElch is able to recognize thermodynamically superfluous reactions (TSR). For example, in the case of the mechanism shown above the equilibrium constant of the second-order cross reaction (third chemical reaction) and the standard potential of the second charge transfer reaction is unambiguously characterized by the remaining equilibrium constants. In order to prevent the user from entering a Keq- or E°-value that violates thermodynamics, these input fields are blocked after having completed the input for the other reaction equations. If, for instance, the user wants to enter Keq for the second-order cross reaction the input field for one of the remaining equilibrium constants must be emptied. If the mouse cursor has left the emptied field the input associated with Keq of the second-order cross reaction becomes active. After entering the Keq-value the input field of the reaction which are TSR's now will be blocked. Blocked ("read only") input fields are displayed on a grey background.
 

5. Meaning of the Species Parameters

•D (cm²/s)
Diffusion coefficient of the species associated with this input

•Canal (mol/l), Cinit (mol/l)
Canal denotes the analytical concentrations of the individual species in mol/l. In the presence of chemical reactions, the analytical concentrations may be different from the equilibrium values, Cinit, (the initial concentrations from which the simulation is actually started). See also more below where the Pre-Equilibrium option on the Property Page: Simulation Parameters is described.

•Boundary

oBRB
"blocked right boundary", i.e. no concentration gradient at the right-hand boundary

oORB
"open right boundary", i.e. concentration values at the right-hand boundary remain constantly at Cinit in the course of the simulation.

If electrode geometry is Spherical (Hg) the user has the choice to specify whether a particular species is forming an amalgam (i.e. diffusing into the mercury drop) or not.

This Page can be entered only if
 

•there is at least one CT-reaction for which the check box Enable Adsorption has been activated as shown above.

•a value for E° (V) and Keq must have been entered for each charge-transfer and chemical reaction, respectively.

•the required analytical concentrations, Canal (mol/l), of initially present species must have been entered.

•the diffusion modus must not be Semi-Infinite 2D
 

If only a single CT-reaction of the form Ox + e = Red has been entered on the Property Page: Chemical Reactions, the Property Page: Surface Reactions looks as follows:

Note that all relevant parameters are zero by default. Consequently, the activation of the Enable Adsorption check box has no effect on the simulated CV if the parameters on the Property Page: Surface Reactions have not been modified.
 

You cannot directly add a new CT-Reaction of Adsorbed Species or Adsorption Reaction on this page. This must be done by entering a new Charge-Transfer-Reaction on the Property Page: Chemical Reactions and activating the Enable Adsorption checkbox for this reaction.

1. Defining a Reaction of Adsorbed Species

Unlike chemical reactions on the Property Page: Chemical Reactions, the definition of reactions between adsorbed and desorbed species is currently restricted to two predefined types:
 

•Type 1: Red1* + Ox2 = Ox1* + Red2

•Type 2: A* + P = Q + S

 
where adsorbed species are marked by a star.

Click on the mouse button on the first empty line headlined Reaction of Adsorbed Species and select the reaction type from the appearing combo box.

Then use the combo boxes in the appearing dialog box to define the species involved in this reaction

Note that any species required for formulating such a reaction must have been defined on the Property Page: Chemical Reactions before. This should be accomplished by entering (on the Property Page: Chemical Reactions) an analogously defined reaction equation for the desorbed species even if the reaction between the desorbed species is negligibly slow (kf ~ 0).

2. Meaning of Parameters in CT-Reactions of Adsorbed Species

The charge-transfer parameters in this section refer to the direct reduction of the adsorbed species. The latter are marked by an "*".
 

•Eo* (V), ks* (1/s)
Standard potential, Eo*, and the heterogeneous rate constant, ks*, have their usual physical meaning. The forward and backward rate constants actually used in the simulation are computed from Eo*, a and ks* by means of the Butler-Volmer equation or Marcus-Equation. The latter depends on whether the button α* or the button λ (eV)* has been selected on the Property Page: Chemical Reactions.

•α* and λ (eV)*
The symbol α*"appears if Butler-Volmer-kinetics has been selected on the Property Page: Chemical Reactions . " λ (eV)*" appears in the case of Marcus-kinetics.
 

3. Meaning of Parameters in Adsorption Reaction

•K*, kf*
Each adsorption reaction is characterized by equilibrium constant, K*, and rate constant, kf* ,associated with the "forward reaction" i.e. the reaction proceeding from the left to the right in the underlying reaction equation. The value of the backward reaction, kb*, is obtained as kb* = K*/kf*. The relationship between kf*, kb* and the surface coverage, Γ, has been formulated such as to lead to the Frumkin isotherm under equilibrium conditions. In the case of reaction Ox = Ox* it is as follows:
 

•α*
The self-interaction parameter in the Frumkin-isotherm.
 

4. Meaning of Parameters in Reaction of Adsorbed Species

•Keq, kf, kb
Each reaction is characterized by equilibrium constant, Keq, and rate constant, kf ,associated with the "forward reaction". The value of the backward reaction, kb, is computed by DigiElch and cannot be entered.

5. Meaning of Parameters in Adsorbed Species

•Γ max
The maximum surface coverage in mol/cm².
 

•Θ
The relative surface coverage Γ/Γmax computed from the entered values of K* and Γ max assuming that the adsorption reactions do not change the equilibrium concentrations, Cinit

View video clip to see an example.

A value of Cdl (F) written in magenta indicates that a time/potential dependent double layer capacity has been entered by the user. The polynomial coefficients describing the dependence of the double layer capacity as function of the electrode potential can be edited by clicking with the right mouse button while the cursor is localized over the input field associated with Cdl (F).

1. Scan Parameters:

•Scan segment, Estart (V), Eend (V), v (V/s)
DigiElch provides a highly flexible definition of the potential scan used in the cyclic voltammetric experiment. The overall scan can be composed of up to 20 scan segments characterized by starting potential, Estart (V), end-potential, Eend (V) and scan rate, v (V/s). In the case of cyclic voltammetry a scan segment is identical with a half cycle (i.e. forward or backward scan) where the scan rate in each individual scan segment may be different.
 
Examples 1: (classical CV, 2 symmetrical scan segments are executed)

Estart - values plotted on a grey background are automatically filled in. They are "read only" and cannot be edited/modified. This ensures a "smooth scan" where the starting potential of a scan is always equal to the end potential of the previous scan.
 

•Check Box: use the same value of v(V/s) in each scan segment
The possibility that an individual scan rate can be defined for each scan segment leads inevitably to more inconvenience, if, for instance, the scan rate in a CV composed of 20 scan segments with equal scan rate must be changed. For this reason, this option can be disabled if only "classical" CVs are simulated. In the latter case, any modification of the scan rate will be taken over for all other scan segments if the checkbox use the same value of v(V/s) in each scan segment is activated.

•Check Box: apply background correction
This checkbox is visible only if both Ru (Ohm) and Cd (F) are different from zero. Click here for further details.

•Potential steps (V)
By default all simulations are executed with potential steps of 5mV. This value optimizes speed and accuracy of most simulations.

Removing scan segments:
A scan segment (and all subsequent scan segments) can be removed by entering an empty string into the associated Eend - input filed. For instance, if a user wants that only the first scan segment is executed in Example 2, an empty string must be entered into the input field "Eend (V)" associated with the scan index 2.

2. Pre-Equilibrium

•Disabled
the simulation starts using the analytical concentrations, i.e. Cinit = Canal for each species.

•Enabled
all equilibrium constants (chemical and electrochemical ones) entered on the Property Page: Chemical Reactions are used to compute the equilibrium distribution of all species from the analytical concentrations. The equilibrium values are used as initial concentration Cinit when starting the simulation. This option may lead to unexpected results as the quiet time before starting an experiment is usually not large enough to accomplish that the equilibrium distribution predicted by the Nernst equation is obtained in the entire diffusion layer.

•Smart
The relation between Estart (V) and the standard potentials, Eo (V), decides how the analytical concentrations are assigned to the oxidized or reduced form of a redox couple. The initial concentrations are computed then in a second step taking only the chemical reactions into consideration. This option is optimal for all practical purposes unless the user is going to simulate a CV using Estart (V) not quite different from Eo (V).

3. Diffusion

•Semi-Infinite 1D
diffusion is treated in one space direction either by making use of symmetry properties of the electrode or by neglecting contributions from other space directions. The user should be aware that this approach may lead to a considerable error in the current density when approximating a "small" planar (two-dimensional) electrode in terms of an infinitely large (effectively one-dimensional) electrode.

•Finite 1D
simulation is executed for a thin layer electrode. When selecting this mode, a dialog field appears that enables the user to enter the Thickness of the thin layer electrode while and the right-hand boundary condition can be selected on the Property Page: Chemical Reactions.

•Semi-Infinite 2D
diffusion is simulated in two space directions. This option is much more time consuming because diffusion occurring parallel to the electrode surface is not neglected in 2D-simulations. Watch video clip to see an example.

4. Geometry

Depending on the selected option for Diffusion the following geometry options are available
 

When selecting Spherical (Hg) the user has the choice to specify whether a particular species is going to form an amalgam (i.e. diffusing into the mercury drop) or not.

5. Experimental Conditions:

•Ru (Ohm), Cdl (F)
Uncompensated ohmic resistance of the solvent and double layer capacity of the electrode which are to be included into the simulation. Cdl (F) is assumed to have a constant value independently of the potential - a simplification  which does hardly apply to a real electrochemical measurements. Hence, when fitting experimental data the effect of both parameters should be preferably eliminated by using IR-compensation and subtracting the background currents. If the dependence of Cdl (F) on the electrode potential is known and expressible by a polynomial the coefficient of the polynomial approximation can be entered in a dialog box that comes up when clicking with the right mouse button while the cursor is localized over the input field of Cdl (F) :
 

 
Cdl (F) is the constant value of the double layer capacity left over in the limiting case where the dependence of the double layer capacity on the electrode potential expressed by C1, C2, C3 and C4 remains negligibly small. After closing this dialog box only the value of Cdl (F) is displayed in the list of Simulation Parameters but the value of Cdl (F) is written in magenta if one of the coefficients C1, C2, C3, C4 is different from zero.
 

•Temp (K)
The temperature entered in this field is exclusively used for computing the effect of the Nernst-factor. Since DigiElch has no special knowledge about the temperature dependence of thermodynamic and kinetic parameters involved in a particular reaction scheme, it should be obvious, that such information cannot be obtained by executing a series of simulations referring to different temperatures !
 

6. Model Parameters

•Noise level (%)
In order to mimic experimental data files a user-defined level of noise can be added to simulated current curves. This may be useful to investigate how the noise level affects the accuracy of parameters estimated by the fitting routine. The percentage refers to the maximum absolute current.

•Gauss-Newton-Iterations
This input field is active only in the presence of second-order chemical reactions where the underlying equations become non-linear and must be solved iteratively.

•Expansion Factor x-grid (perpendicular to electrode), Truncation error (%)
DigiElch provides two different simulation tools:

oThe first one is a particular implementation of the box method which is the only simulation technique reported in the literature that yields exponential converges for the flux error towards zero when refining the grid expansion factor. The importance of the Expansion factor and the Relative truncation error for the accuracy of the simulated flux has been discussed in detail in a series of papers listed on the ElchSoft Homepage. The user is strongly advised to deal with these papers when working with DigiElch. Using the default setting for the Relative truncation error ensures that the flux error becomes independent of the selected grid expansion factor to the greatest possible extend.
Provided the grid expansion factor is not larger than 0.5 the correctness of the simulated flux can be guaranteed for pure diffusion systems and kinetic-diffusion systems comprising only first-order chemical reactions. However, the default parameter setting work also well for the majority of systems comprising complex second-order reactions provided the diffusion coefficients of the individual species do not differ by several orders of magnitude.

oThe second simulation technique implemented into DigiElch is an adaptive grid simulator. The starting grid is identical to that described above for the box method. However, the adaptive grid simulator does not result in exponential convergence for the simulated flux error. The error level originating from using an exponentially expanding space grid is therefore usually much larger than the truncation error. Consequently, the accuracy of the simulated flux can be guaranteed only when using a sufficiently small Local FEM Error (see more below). Also note that the adaptive grid simulator is not fully implemented. The following simulations cannot be executed with this tool:

▪simulation of two-dimensional diffusion systems

▪simulations involving amalgam forming species
 

•Xmax / SQRT(Dt)
It is a well-known fact that the concentration of a species does not change in an electrochemical experiment if the distance from the electrode is farther that 6(Dt)½ where D is the diffusion coefficient and t denotes the duration of the entire experiment. Hence the default value of Xmax / SQRT(Dt) is 6 but it can be modified if necessary.
 

7. 2D-Simulations

•Expansion factor y-grid (parallel to electrode) , Boxes for electrode
The associated input fields are active only when selecting the diffusion mode Semi-Infinite 2D where the space grid must also be specified for the second space direction. In principle this space grid is characterized again by grid expansion factor and Truncation error. Since the latter becomes smaller the more grid points/boxes are used for modeling the electrode surface, it was more lucid to enter here the number of Boxes for electrode as a second parameter. Only five boxes are used by default. This number seems to be very small but it is easy to verify that there is hardly any situation where the accuracy of the simulation is significantly improved when increasing this number.
 

8. FEM- (Adaptive Grid-) Simulation

•Local FEM Error
The value entered in the associated input field is operative only when simulating the current curve by means of the adaptive grid simulator (based on the Finite Element Method). If the input for Local FEM Error is 0 the FEM-Simulator is forced to switch off adaptive grid refinement and to work on the fixed expanding space grid specified by the parameters described above. Adaptive grid refinement will be enabled if Local FEM Error > 0. The smaller the value the higher the accuracy and the computational costs but the simulation may fail if the value for the accepted local error level is unattainably small. Also, the effect of the local error level on the accuracy of the simulated flux can be hardly predicted in advance and must be figured out by "trial and error". (Compared to the box method, a considerably higher number of grid points is usually required by the FEM-Simulator to get a comparable level of accuracy for the simulated flux).
 

9. Level of Multi-Core CPU-Support

•Level 0:
no multi-core cpu support. As in previous versions of DigiElch, all simulations will be executed in a single task and do not become faster when working on a dual core or quad core processor.

•Level 1:
multi-core cpu support for data fitting. The computation time (for a fitting project comprising a large number of parameters) is decreased by a factor of 2 or 4 when working on a dual core or quad core processor.

•Level 2:
multi-core cpu support for the 2D-(micro-electrode) simulation algorithm. The computation time for a 2D-simulation without inclusion of IR-drop and double layer charging can be decreased by a factor of 2 when working on a dual core or quad core processor.

•Level 3:
multi-core cpu support for the matrix inversions executed in the 2D-simulation algorithm. The computation time for a 2D-simulation can be decreased by a factor of 2 when working on a dual core or quad core processor. The positive effect on computation time is increased the bigger the size of the matrixes (i.e. the smaller the size of the electrode and the more species are involved in the mechanism) . It works for both 2D-simulations with and without inclusion of IR-drop and double layer charging.

•Level 4:
multi-core cpu support for both the 2D-(micro-electrode) simulation algorithm and all executed matrix inversions. The computation time for a 2D-simulation without inclusion of IR-drop and double layer charging can be decreased by a factor of 4 when working on a quad core processor.

 
Note that the higher the level of multi-core cpu support the higher the computational overhead required for synchronizing all running tasks.  Levels > 1 will therefore lead to a positive effect only if sufficiently "large" systems are simulated where the single-task computation time is significantly greater than the time required for synchronizing tasks. For this reason, multi-core cpu support is not provided for 1D-simulations where a positive effect could be only expected for mechanisms comprising an unrealistically high number of species.

10. Simulation Name

There is a single-line edit control on the bottom of the Property Page: Simulation Parameters that can be used for giving the simulation a more meaningful name as that automatically generated by DigiElch. The name referring to the active simulation is indicated in the frame window of the simulation document after closing the CV-Properties Dialog. When exporting a simulation the default file name is either “simulation name.use” or “simulation name.txt ”. For this reason, the simulation name must not contain ‘\’ or any other character leading to problems in file names.