the electrochemistry module model library manual · solved with comsol multiphysics 4.4 1 | current...

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Electrochemistry ModuleModel Library Manual

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Solved with COMSOL Multiphysics 4.4

Cu r r e n t D i s t r i b u t i o n i n a Ch l o r - A l k a l i Memb r an e C e l l

Introduction

The chlor-alkali membrane process is one of the largest processes in industrial electrolysis with production of roughly 40 million metric tons of both chlorine and caustic soda per year (Ref. 1). Chlorine’s largest use is in the production of vinyl chloride monomer, which in turn is used for the production of poly vinyl chloride (PVC). Among the applications of PVC are as electrical insulator in cables and as a material for pipes, carpets, raincoats, and many other products. The production of chlorine implies a simultaneous production of caustic soda (alkali), which is widely used in the chemical industry for alkalization and neutralization of acidic streams. Caustic soda is also used in alkaline batteries.

The traditional process for manufacturing chlorine and caustic soda is the mercury-cell process. This technology has been partly replaced by the diaphragm process, and in later years the membrane process has been the dominating process in retrofits and for new plants. The purpose of the diaphragm or membrane is to separate the products chlorine and caustic soda, which otherwise would react to produce hypochlorite and hydrochloric acid. Chlorine and caustic soda are produced at the anode and cathode, respectively. Figure 1 shows a diagram of the process.

Current density in membrane-cell technology has increased dramatically during the last decade as the membranes themselves have improved. This results in lower investment costs for greater production. However, the increase in current density implies an increase in power consumption if nothing is done to dampen the voltage increase. Advances in cell design by increased internal convection, decreased ohmic losses, and better membranes have allowed for large increases in current density with small increases in cell voltage. One of the important parameters in the design of modern membrane cells is the current-density distribution on the electrode surfaces. It is important, from the viewpoint of catalyst lifetime and minimization of losses, that the current density on the electrode frontal surfaces is as uniform as possible.

This example describes the current-density distribution in a realistic structure for the anodes and cathodes in a membrane cell. This discussion limits the model to one unit cell of the entire cell. This unit cell appears on the right side in Figure 1.

1 | C U R R E N T D I S T R I B U T I O N I N A C H L O R - A L K A L I M E M B R A N E C E L L


2 | C U R

Figure 1: Drawing of the unit cell.

The anode and cathode ribs are separated by the membrane, which is a cation-selective membrane. It is forced to adapt its shape to fit within the inter-electrode distance. The membrane prevents mixing between brine and chlorine on the anode side with the caustic soda and hydrogen on the cathode side.

A detailed description of the process of chlor-alkali electrolysis is available in Ref. 1.

Model Definition

This example models the current and potential distribution in a unit cell in the membrane cell sketched in Figure 1. This model is a secondary current-distribution model (see Ref. 2), which implies that you take into account the dependence of the electron transfer on the local potential, and that you assume constant composition in the subdomains. The electron transfer reactions at the anode and cathode surfaces are:

R E N T D I S T R I B U T I O N I N A C H L O R - A L K A L I M E M B R A N E C E L L


(1)

The domain in the model is half of the unit cell shown in Figure 1, as explained in Figure 2.

Figure 2: Model geometry.

The chemical reactions show that there is gas evolution in both the anodic and cathodic compartments, creating a vigorous internal convection in the respective compartments. This makes it possible to simplify the model by neglecting the concentration gradients in the anolyte and catholyte. The simplification implies that the transport of ionic current inside the cell takes place exclusively through migration, that is, the electric field induces a flux of ions. For this reason you do not need to model the complex problem of internal free convection of the two-phase flow in order to get an estimation of the current-density distribution in the cell (see also the Theory for the Current Distribution interfaces in the Electrochemistry Module Users Guide).

Model the current conduction in the membrane and two electrolyte chambers by using the Secondary Current Distribution interface, with different values for the electrolyte conductivity in each domain.

2Cl- Cl2(g) 2e-+ at the anode

2H2O 2e-+ H2(g) 2OH-

+ at the cathode

Catholyte

Anolyte

Anode

Cathode

Membrane



4 | C U R

For the cathodic reaction, use a Butler-Volmer expression to describe the relation between current density and potential in the electrode. Set the electric potential of the cathode to zero (ground).

The anode reaction is very fast, and small changes in potential provide large changes in current density. This implies that you can assume a constant potential (primary condition) at the anode’s surface. In this case this gives an error in potential of approximately 20 mV at that surface (Ref. 3). The potential condition for the electrolyte potential at the anode, l,a, assuming zero polarization losses, can be written as:

(2)

where Eeq,a is the equilibrium potential of the anode electrode reaction, and s,a the electrode (metal phase) potential at the anode.

The cell voltage, Ecell, can be written as the difference in metal potential between the anode and cathode:

(3)

Since the cathode is grounded (s,a=0) we get:

(4)

Define the cell polarization voltage as the deviation from the cell potential at equilibrium, so that the cell potential also may be written as

(5)

setting equilibrium potential of the cathode hydrogen reaction, Eeq,c, to zero, and inserting Equation 5 into Equation 2 we get

(6)

Use Equation 6 as boundary condition for the electrolyte potential at the anode.

Results and Discussion

Figure 3 shows the potential in the anode and cathode compartments as well as in the membrane electrolyte. From this plot note that the largest ohmic losses arise in the membrane, as expected from its low conductivity. The arrow plot of the

l a s a Eeq a–=

Ecell s a s c–=

Ecell s a=

Ecell s a Eeq a Eeq c– Epol+= =

l a Epol=



current-density vector shows how the distribution is more uniform on the cathode than on the anode surface.

Figure 3: Electrolyte potential.

Figure 4: Electrolyte current density norm.



6 | C U R

Figure 4 shows the modulus of the current-density vector, which illustrates “hot spots” in the electrolyte where the current density is large. These hot spots correspond to parts of the electrodes where the current density is high, and where hence catalyst can be lost due to accelerated wear.

References

1. H.S. Burney, “Past Present and Future of the Chlor-Alkali Industry, Chlor-Alkali and Chlorate Technology: R.B. Macmullin Memorial Symposium, Proc Electrochemical Society, vol. 99–21, 1999.

2. J.S. Newman, Electrochemical Systems, 2nd ed., Prentice Hall, 1991.

3. P. Bosander, P. Byrne, E. Fontes, and O. Parhammar, Current Distribution on a Membrane Cell Anode, Chlor-Alkali and Chlorate Technology: R.B. Macmullin Memorial Symposium, Proc Electrochemical Society, vol. 99-21, 1999.

Model Library path: Electrochemistry_Module/Electrochemical_Engineering/chlor_alkali

Modeling Instructions

From the File menu, choose New.

N E W

1 In the New window, click the Model Wizard button.

M O D E L W I Z A R D

1 In the Model Wizard window, click the 2D button.

2 In the Select physics tree, select Electrochemistry>Secondary Current Distribution

(siec).

3 Click the Add button.

4 Click the Study button.

5 In the tree, select Preset Studies>Stationary.

6 Click the Done button.



G L O B A L D E F I N I T I O N S

Start by loading the model parameters from a text file.

Parameters1 On the Home toolbar, click Parameters.

2 In the Parameters settings window, locate the Parameters section.

3 Click Load from File.

4 Browse to the model’s Model Library folder and double-click the file chlor_alkali_parameters.txt.

G E O M E T R Y 1

Import the model geometry from a file.

Import 1 (imp1)1 On the Home toolbar, click Import.

2 In the Import settings window, locate the Import section.

3 Click the Browse button.

4 Browse to the model’s Model Library folder and double-click the file chlor_alkali_geom.mphbin.

5 Click the Import button.

6 Click the Zoom Extents button on the Graphics toolbar.

M A T E R I A L S

Define the electrolyte conductivity in the three different domains by adding separate materials for the catholyte, the membrane and the anolyte.

Material 1 (mat1)1 In the Model Builder window, under Component 1 (comp1) right-click Materials and

choose New Material.

2 In the Material settings window, locate the Geometric Entity Selection section.

3 Click Clear Selection.

4 Select Domain 3 only.

5 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group

Electrolyte conductivity sigmal K_c S/m Electrolyte conductivity



8 | C U R

Material 2 (mat2)1 In the Model Builder window, right-click Materials and choose New Material.


3 In the Material settings window, locate the Material Contents section.

4 In the table, enter the following settings:

Material 3 (mat3)1 Right-click Materials and choose New Material.




S E C O N D A R Y C U R R E N T D I S T R I B U T I O N ( S I E C )

Set up the cathode current density by using an Electrolyte-Electrode Boundary Interface node, define the exchange current density in the Electrode Reaction child node.

Electrolyte-Electrode Boundary Interface 11 On the Physics toolbar, click Boundaries and choose Electrolyte-Electrode Boundary

Interface.

2 Select Boundaries 10–12 and 18 only.

Electrode Reaction 11 In the Model Builder window, expand the Electrolyte-Electrode Boundary Interface 1

node, then click Electrode Reaction 1.

2 In the Electrode Reaction settings window, locate the Model Inputs section.

3 In the T edit field, type T.

4 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Butler-Volmer.

5 In the i0 edit field, type i0_c.


Electrolyte conductivity sigmal K_m S/m Electrolyte conductivity


Electrolyte conductivity sigmal K_a S/m Electrolyte conductivity



Electrolyte Potential 1Define the potential on the anode by using an Electrolyte Potential node.

1 On the Physics toolbar, click Boundaries and choose Electrolyte Potential.

2 Select Boundaries 3, 8, 9, and 17 only.

3 In the Electrolyte Potential settings window, locate the Electrolyte Potential section.

4 In the l,bnd edit field, type E_pol.

Initial Values 1Provide an initial value for the electrolyte potential to reduce the computational time for this model.

1 In the Model Builder window, under Component 1 (comp1)>Secondary Current

Distribution (siec) click Initial Values 1.

2 In the Initial Values settings window, locate the Initial Values section.

3 In the phil edit field, type E_pol.

M E S H 1

Create a triangular mesh with a higher resolution at the electrode surfaces and on the membrane boundaries.

Free Triangular 1In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Free Triangular.

Size 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free

Triangular 1 and choose Size.

2 In the Size settings window, locate the Geometric Entity Selection section.

3 From the Geometric entity level list, choose Boundary.

Select the anode, cathode and membrane boundaries. The easiest way to do this is by using a selection box around the central part of the geometry.

4 Click the Select Box button on the Graphics toolbar.

5 Select Boundaries 3–5, 8–12, 14, 15, and 17–20 only.

6 Locate the Element Size section. From the Predefined list, choose Extremely fine.



10 | C U

Free Triangular 11 In the Model Builder window, right-click Mesh 1 and choose Build All.

The finalized mesh should now look like this:

S T U D Y 1

The problem is now ready for solving.

1 On the Home toolbar, click Compute.

R E S U L T S

Electrolyte Potential (siec)Modify the default plot by adding an arrow plot.

1 In the Model Builder window, under Results right-click Electrolyte Potential (siec) and choose Arrow Surface.

2 In the Arrow Surface settings window, locate the Expression section.

3 Click Electrolyte current density vector (siec.Ilx,...,siec.Ily) in the upper-right corner of the section. Locate the Arrow Positioning section. In the Points edit field, type 20.

4 In the Points edit field, type 20.

5 Locate the Coloring and Style section. From the Color list, choose White.


R R E N T D I S T R I B U T I O N I N A C H L O R - A L K A L I M E M B R A N E C E L L


Create a surface plot of the electrolyte current density norm to visualize the "hot spots" on the electrode surfaces.

2D Plot Group 21 On the Home toolbar, click Add Plot Group and choose 2D Plot Group.

2 In the Model Builder window, under Results right-click 2D Plot Group 2 and choose Surface.

3 In the Surface settings window, locate the Expression section.

4 Click Electrolyte current density norm (siec.NormIl) in the upper-right corner of the section. On the 2D plot group toolbar, click Plot.



12 | C U
R R E N T D I S T R I B U T I O N I N A C H L O R - A L K A L I M E M B R A N E C E L L


C y c l i c V o l t amme t r y a t a Ma c r o e l e c t r o d e i n 1D

Introduction

Cyclic voltammetry is a common analytical technique for investigating electrochemical systems. In this method, the potential difference between a working electrode and a reference electrode is swept linearly in time from a start potential to a vertex potential, and back again (see Figure 1). The resulting current at the working electrode is recorded and is plotted against the applied electrode potential in a voltammogram.

Figure 1: Potential of the working electrode during one voltammetric cycle. The potential is cycled between the vertex potentials 0.4 V and –0.4 V. The scan rate is 1 mV/s.

Voltammetry is a valuable technique because information about both the electrochemical reactivity and the transport properties of a system can be extracted simultaneously. For quantitative interpretation of voltammetry, however, we must use numerical methods to solve the physical equations that describe voltammetry. Then, unknown physical quantities in the system can be inferred by ‘fitting’ to experimental data.

1 | C Y C L I C V O L T A M M E T R Y A T A M A C R O E L E C T R O D E I N 1 D


2 | C Y C

This example demonstrates the use of a common approximation in which a large electrode (macroelectrode) is assumed to have uniform transport behavior across its surface, so only physics occurring normal to the surface need to be considered. By simplifying the model to 1D, an efficient time-dependent analysis is possible.

In this model, a Parametric Sweep is used to compare voltammetry recorded at different voltammetric scan rates.

Model Definition

The model contains a single 1D domain of length L, which is the maximum extent of the diffusion layer over the duration of the voltammetry experiment. A conservative setting for L is set to greatly exceed the mean diffusion layer thickness:

(1)

Here, D is the diffusion coefficient of the reactant and tmax is the duration of the cyclic voltammogram.

D O M A I N E Q U AT I O N S

We assume the presence of a large quantity of supporting electrolyte. This is inert salt that is added in electroanalytical experiments to increase the conductivity of the electrolyte without otherwise interfering with the reaction chemistry. Under these conditions, the resistance of the solution is sufficiently low that the electric field is negligible, and we can assume l0.

The Electroanalysis interface implements chemical transport equations for the reactant and product species of the redox couple subject to this assumption. The domain equation is the diffusion equation (also known as Fick’s 2nd law) to describe the chemical transport of the electroactive species A and B:

(2)

B O U N DA RY E Q U AT I O N S

At the bulk boundary (xL), we assume a uniform concentration equal to the bulk concentration for the reactant. The product has zero concentration here, as in bulk.

At the electrode boundary (x0), the reactant species A oxidizes (loses one electron) to form the product B. By convention, electrochemical reactions are written in the reductive direction:

L 6 Dtmax=

cit

------- Di ci =

L I C V O L T A M M E T R Y A T A M A C R O E L E C T R O D E I N 1 D


(3)

The stoichiometric coefficient is –1 for B, the “reactant” in the reductive direction, and +1 for A, the “product” in the reductive direction. This formulation is consistent even in examples such as this model where at certain applied potentials, the reaction proceeds favorably to convert A to B. The number of electrons transferred, n, equals one.

The current density for this reaction is given by the electroanalytical Butler-Volmer equation for an oxidation:

(4)

in which k0 is the heterogeneous rate constant of the reaction, c is the cathodic transfer coefficient, and is the overpotential at the working electrode. This overpotential is the difference between the applied potential and the equilibrium potential (formal reduction potential) of the redox couple of species A and B.

According to Faraday’s laws of electrolysis, the flux of the reactant and product species are proportional to the current density drawn:

(5)

This is expressed in the Electrode Surface boundary condition.

The applied triangular waveform for the cyclic voltammetry study is specified in the Electrode Surface boundary condition according to two vertex potentials—forming a potential window between –0.4 V and +0.4 V, either side of the equilibrium reduction potential—and a voltammetric scan rate, v (SI unit: V/s), which is the rate at which the applied potential is changed.

In the 1D approximation, the total current is related to the current density simply by multiplying by the electrode area A:

(6)

C Y C L I C VO LTA M M E T RY S T U DY

In the cyclic voltammetry experiment, the potential applied to the working electrode surface is varied linearly as a function of time. A Parametric Sweep is used to compare the voltammetry recorded at different scan rates.

B e– A+

iloc nFk0 cAexpn c– F

RT---------------------------- cBexp

cF–

RT----------------- –

=

n– NiiilocnF

-------------=

Iel ilocA=



4 | C Y C


The shape of the cyclic voltammogram (Figure 2) shows the relation between electrode kinetics and chemical species transport (diffusion).

Figure 2: Cyclic voltammetry recorded at a macroelectrode.

Initially, at reducing potentials, the oxidation reaction is not driven and negligible current is drawn. As the potential moves towards the reduction potential of the redox couple, the oxidation reaction is accelerated and the current increases. Once the oxidation reaction has consumed the reactant at the electrode surface, the current becomes limited by the rate of transport of A towards the working electrode. Therefore, a peak current is observed, and at higher potentials, the voltammetric current falls off at a potential-independent rate; this region is termed “diffusion-controlled” or “transport-controlled”.

On sweeping back towards more reducing potentials, the reconversion of the product B into the original reactant A gives a negative (cathodic, reductive) current. Depletion of the reacting species B causes a negative peak current and reconversion thereafter proceeds at a diffusion-controlled rate.

The magnitude of the current on the forward peak, Ipf, is a common diagnostic variable in voltammetry. For fast electrode kinetics and at a macroelectrode under the 1D approximation, its value is given theoretically by the Randles–Ševcík equation (see



Ref. 1 and Ref. 2 for a detailed discussion and derivation):

(7)

where A is the electrode area, c is the bulk concentration of the reactant, and D is the diffusion coefficient of the reactant.

The square-root relationship between peak current and scan rate is characteristic of macroelectrode cyclic voltammetry under the above conditions.

References

1. R.G. Compton and C.E. Banks, Understanding Voltammetry, 2nd Edition, London, 2011.

2. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Second Edition, Hoboken, 2001.

Model Library path: Electrochemistry_Module/Tutorial_Models/cyclic_voltammetry_1d



N E W




2 In the Select physics tree, select Electrochemistry>Electroanalysis (elan).


This model will solve for the two concentrations of a redox couple, change the default concentration variable names to cA and cB.

Ipf 0,446nFAc nFRT--------Dv=



6 | C Y C

4 In the Concentrations table, enter the following settings:


6 In the tree, select Preset Studies>Cyclic Voltammetry.



Add the model parameters from a text file.




4 Browse to the model’s Model Library folder and double-click the file cyclic_voltammetry_1d_parameters.txt.

G E O M E T R Y 1

Build the model geometry as a single interval, where the left boundary will later be defined as the electrode surface, and the right boundary will be the boundary towards the bulk.

Interval 11 Right-click Component 1>Geometry 1 and choose Interval.

Set the interval length to L. The value of L is defined in the Parameters node. Note that L will vary with the Voltammetric scan rate parameter v, which is also defined in the Parameters node.

2 In the Interval settings window, locate the Interval section.

3 In the Right endpoint edit field, type L.

cA

cB



4 In the Model Builder window, right-click Geometry 1 and choose Build All Objects.

The completed geometry should now look like this:

E L E C T R O A N A L Y S I S

Diffusion 1Start defining the physics by setting the diffusion coefficients for the two species of the redox couple.

1 In the Model Builder window, under Component 1>Electroanalysis click Diffusion 1.

2 In the Diffusion settings window, locate the Diffusion section.

3 In the DcA edit field, type DA.

4 In the DcB edit field, type DB.

Concentration 1Set the boundary to the right to bulk concentration values.

1 On the Physics toolbar, click Boundaries and choose Concentration.

2 Select Boundary 2 only.

3 In the Concentration settings window, locate the Concentration section.

4 Select the Species cA check box.

5 In the c0,cA edit field, type c_bulk.



8 | C Y C

6 Select the Species cB check box.

Electrode Surface 1Set up the electrode surface and the cyclic voltammetry settings on the left boundary.

1 On the Physics toolbar, click Boundaries and choose Electrode Surface.


3 In the Electrode Surface settings window, locate the Boundary Condition section.

4 From the Boundary condition list, choose Cyclic voltammetry.

5 In the Linear sweep rate edit field, type v.

The voltage will be cycled between the vertex potentials. When the start potential is not specified, the sweep will start at Vertex potential 2.

6 In the Vertex potential 1 edit field, type E_vertex1.

7 In the Vertex potential 2 edit field, type E_vertex2.

Electrode Reaction 1Specify the electrode reaction as an Electroanalytical Butler-Volmer reaction, which is concentration dependent as defined by the stoichiometric coefficients.

1 In the Model Builder window, expand the Electrode Surface 1 node, then click Electrode Reaction 1.

2 In the Electrode Reaction settings window, locate the Electrode Kinetics section.

3 In the k0 edit field, type k0.

4 Locate the Stoichiometric Coefficients section. In the cA edit field, type 1.

5 In the cB edit field, type -1.

Initial Values 1Specify the initial concentration values. This will set concentration values when the simulation starts at t=0.

1 In the Model Builder window, under Component 1>Electroanalysis click Initial Values

1.


3 In the cA edit field, type c_bulk.

S T U D Y 1

Solve the problem for various sweep rates.



Parametric Sweep1 On the Study toolbar, click Extension Steps and choose Parametric Sweep.

2 In the Parametric Sweep settings window, locate the Study Settings section.

3 Click Add.



R E S U L T S

Cyclic Voltammograms (elan)A number of plots are created by default. The first default plot shows the voltammograms created by the Cyclic Voltammetry feature in the Electrode Surface node.

1 In the 1D Plot Group settings window, click to expand the Legend section.

2 From the Position list, choose Upper left.

3 On the 1D plot group toolbar, click Plot.

Parameter names Parameter value list

v (10^range(-3,1,0))[V/s]



10 | C Y
C L I C V O L T A M M E T R Y A T A M A C R O E L E C T R O D E I N 1 D


Mode l i n g t h e D i f f u s e Doub l e L a y e r

Introduction

In the very vicinity of an electrode surface (in the range of up to a few nanometers), in the diffuse double layer, the assumption of electroneutrality is not valid due to charge separation. Typically the diffuse double layer may be of interest when modeling very thin layers of electrolyte, for instance in electrochemical capacitors and in atmospheric corrosion problems.

To model the behavior of the diffuse double layer, one needs to solve for the Nernst-Planck equations for all of the ions, in combination with the Poisson’s equation for the potential. The combination of these equations is frequently referred to as the Poisson-Nernst-Planck (PNP) equations.

This example shows how to couple the Nernst-Planck equations, solved using the Transport of Diluted Species physics interface, to the Poisson’s equation, solved using the Electrostatics physics interface.

A problem that arises when modeling the PNP equations is that of how to handle the boundary condition for the potential equation. In this example an assumption of a Stern layer with a constant capacity is used to derive surface charge boundary conditions for Poisson’s equation.

The model reproduces the results of Bazant and Chu (see Ref. 1 and Ref. 2).

Model Definition

The model geometry is in 1D (a single interval between 0 and L) and consists of one single domain, representing the electrolyte phase, including the diffuse double layer.

D O M A I N E Q U A T I O N S

The concentrations, ci (mol/m3, i=+,-), of two ions of opposite charge (+1/-1) are solved for in the electrolyte phase. The fluxes (Ni, mol/(m2·s)) of these are described by the Nernst-Planck equation

(1)

with Di (m2/s) being the diffusion coefficient, um,i (s·mol/kg) the mobility, F (C/

mol) Faraday’s constant, and (V) the potential.

Ni Di ci– um i ziFci –=

1 | M O D E L I N G T H E D I F F U S E D O U B L E L A Y E R


2 | M O D

Assuming no heterogenous reactions in the electrolyte, the governing equations for the two species become:

(2)

For the potential, Poisson’s equation states

(3)

where is the permittivity (F/m) and the charge density (C/m), depending on the ion concentrations according to:

(4)

B O U N D A R Y C O N D I T I O N S

The boundaries reside in the reaction plane of the electrodes on each side. The same electrode reaction, in which the positive ion, S+,participates, takes place on both electrodes.

(5)

The reaction rate r (mol/(m2·s)) is

(6)

where Ka and Kc (m/s) are the anodic and cathodic rate constants, cM the metal species activity (mol/m3, constant) and a and c the anodic and cathodic transfer coefficients. (V) is the difference in potential between the metal phase, (V), and the reaction plane:

(7)

The electrode reaction renders an inward flux for the positive ion according to

(8)

on both boundaries. For the negative ion, a zero flux condition is used.

(9)

Assuming the reaction plane to be placed at the boundary between the inner (compact) and diffuse double layer, and with the assumption of a Stern compact layer

Ni 0=

– =

F c+ c-– =

S+ e-+ S(s)

r KacMaF

RT----------------- exp K–

cc+

cF–

RT-------------------- exp=

M

M –=

n N+– r=

n N-– 0=

E L I N G T H E D I F F U S E D O U B L E L A Y E R


of a constant thickness, S (m), one can derive the following Robin type of boundary condition for the potential:

(10)

This condition reduces to a Dirichlet voltage condition for S = 0, that is, in the absence of a Stern layer. For the case of a non-zero stern layer thickness, the condition can be reformulated as a surface charge condition

(11)

C E L L P O T E N T I A L E Q U A T I O N

The problem formulated above can now be solved for given voltages of in the metal electrode phase for each side. Typically one grounds one electrode and specifies the cell voltage as V so that

(12)

However, to solve for a given cell current density, icell (A/m2), with V not known a priori, an additional global equation, solving for V, is used, fulfilling the condition:

(13)

G L O B A L C O N C E N T R A T I O N C O N S T R A I N T F O R T H E N E G A T I V E I O N

When solving this system for a stationary solution, the negative ion concentration needs an additional “boot-strap” to render a stable, unique, solution. This is done by adding the following global constraint to the equation system:

(14)

where c0 is the initial ion concentration (mol/m3), equal for both ions.

The constraint assures that the total number of negative ions is preserved during the iterative solver process. (For time-dependent simulations the constraint can be omitted.)

S n + M=

n – S---------–=

M

M x 0=0=

M x L=V=

icell Fr x L==

c0L c- xd0

L

=



4 | M O D

D I M E N S I O N L E S S N U M B E R S A N D P A R A M E T E R V A L U E S

A number of dimensionless numbers can be derived that govern the behavior of the cell. The problem is solved using a parametric study for a dimensionless parameter D = (0.001, 0.01, 0.1), defined as

(15)

where D is the Debye length.

The current of the cell is defined via the dimensionless number j=0.9,

(16)

where iD is the Nernstian limiting current density.

The cathodic reaction rate constant is defined using the dimensionless number kc10,

(17)

The rate of the anodic reaction term is governed by the dimensionless number kr10,

(18)

and the Stern layer thickness is set using the dimensionless number =0.1,

(19)


The following dimensionless variables are used when presenting the results:

D D L=

DRT

2F2c0

----------------=

j icell iD=

iD 4FD+c0 L=

kcKcL4D+-----------=

krKrLcM4D+c0------------------=

SD-------=



(20)

Figure 1 shows the dimensionless concentration, . The concentration gradients are steepest close to the electrodes.

Figure 1: Dimensionless concentration profile.

Figure 2 shows the dimensionless charge density profile. Charge separation occurs close to the electrodes. For higher values of D, the region of charge separation, the diffuse double layer, stretches further into the domain. This is expected since higher D values effectively mean a shorter domain length.

x xL----=

cc+ c-+

2c0---------------------=

c+ c-–

2c0--------------------=

FRT--------=

c



6 | M O D

Figure 2: Dimensionless charge density profile.

Figure 3 shows the potential profile. For higher values of D the voltage over the cell decreases. This is an expected result since a shorter domain length shortens the potential losses due to ion transport.



Figure 3: Dimensionless potential profile.

Notes About the COMSOL Implementation

The element order is set to 2 for the Transport of Diluted Species to match the default element order 2 of the Electrostatics physics.

References

1. M. Bazant, K. Chu, and B. Bayly. “Current-Voltage Relations for Electrochemical Thin Films”, SIAM Journal of Applied Math, vol. 65 (2005), no. 5, pp.1463–1484.

2. K. Chu, and M. Bazant, “Electrochemical Thin Films at and Above the Classical Limiting Current”, SIAM Journal of Applied Math, vol.65 (2005), no.5, pp.1485–1505.

Model Library path: Electrochemistry_Module/Tutorial_Models/diffuse_double_layer



8 | M O D



N E W




2 In the Select physics tree, select AC/DC>Electrostatics (es).


4 In the Electric potential edit field, type phi.

5 In the Select physics tree, select Chemical Species Transport>Transport of Diluted

Species (chds).


7 Click Add Concentration.


9 In the Select physics tree, select Mathematics>ODE and DAE Interfaces>Global ODEs and

DAEs (ge).



12 In the tree, select Preset Studies for Selected Physics>Stationary.



Start by loading some parameters from a text file.




4 Browse to the model’s Model Library folder and double-click the file diffuse_double_layer_parameters.txt.

cp

cm



G E O M E T R Y 1

Build the geometry as a single interval.

Interval 11 In the Model Builder window, under Component 1 right-click Geometry 1 and choose

Interval.



4 Click the Build All Objects button.


D E F I N I T I O N S

Proceed by adding some variable expressions. (Some of the expressions use variables that have not yet been defined and are hence marked in orange color, this is expected.)

Variables 11 In the Model Builder window, under Component 1 right-click Definitions and choose

Variables.

2 In the Variables settings window, locate the Geometric Entity Selection section.


4 Select Boundaries 1 and 2 only.

5 Locate the Variables section. Click Load from File.

6 Browse to the model’s Model Library folder and double-click the file diffuse_double_layer_variables.txt.

Variables 21 In the Model Builder window, right-click Definitions and choose Variables.




5 Locate the Variables section. In the table, enter the following settings:

Variables 31 Right-click Definitions and choose Variables.

Name Expression Unit Description

phiM 0[V] V Metal phase potential (ground)



10 | M O





Integration 1Add integration coupling operators to be used when setting up the global voltage equation and the global constraint.

1 On the Definitions toolbar, click Component Couplings and choose Integration.

2 In the Integration settings window, locate the Source Selection section.



Integration 21 On the Definitions toolbar, click Component Couplings and choose Integration.


E L E C T R O S T A T I C S

Charge Conservation 1Now start setting up the physics, begin with the Electrostatics physics (Poisson's equation).

1 In the Model Builder window, under Component 1>Electrostatics click Charge

Conservation 1.

2 In the Charge Conservation settings window, locate the Electric Field section.

3 From the r list, choose User defined.

Space Charge Density 11 On the Physics toolbar, click Domains and choose Space Charge Density.


3 In the Space Charge Density settings window, locate the Space Charge Density section.

4 In the v edit field, type F_const*Z*(cp-cm).


phiM V Metal phase potential (cell voltage)

D E L I N G T H E D I F F U S E D O U B L E L A Y E R


Surface Charge Density 11 On the Physics toolbar, click Boundaries and choose Surface Charge Density.


3 In the Surface Charge Density settings window, locate the Surface Charge Density section.

4 In the s edit field, type rho_s.

TR A N S P O R T O F D I L U T E D S P E C I E S

Now set up the model for the transport of the ions. Enable migration and change the element order to 2 to match the default element order of the Electrostatics physics.

1 In the Model Builder window, under Component 1 click Transport of Diluted Species.

2 In the Transport of Diluted Species settings window, locate the Transport Mechanisms section.

3 Clear the Convection check box.

4 Select the Migration in electric field check box.

5 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu.

6 Click to expand the Discretization section. From the Concentration list, choose Quadratic.

Diffusion and Migration1 In the Model Builder window, under Component 1>Transport of Diluted Species click

Diffusion and Migration.

2 In the Diffusion and Migration settings window, locate the Model Inputs section.

3 In the V edit field, type phi.


5 Locate the Diffusion section. In the Dcp edit field, type Dp.

6 In the Dcm edit field, type Dm.

7 Locate the Migration in Electric Field section. In the zcp edit field, type Z.

8 In the zcm edit field, type -Z.

Flux 11 On the Physics toolbar, click Boundaries and choose Flux.


3 In the Flux settings window, locate the Inward Flux section.



12 | M O

4 Select the Species cp check box.

5 In the N0,cp edit field, type r.

Initial Values 11 In the Model Builder window, under Component 1>Transport of Diluted Species click

Initial Values 1.


3 In the cp edit field, type cref.

4 In the cm edit field, type cref.

5 In the Model Builder window’s toolbar, click the Show button and select Advanced

Physics Options in the menu.

Global Constraint 1Add a global constraint to boot-strap the average concentration of negative ions to the initial value.

1 On the Physics toolbar, click Global and choose Global Constraint.

2 In the Global Constraint settings window, locate the Global Constraint section.

3 In the Constraint expression edit field, type intop2(cm)-(cref*L).

G L O B A L O D E S A N D D A E S

Global Equations 1Finally, add the equation for the cell potential.

1 In the Model Builder window, under Component 1>Global ODEs and DAEs click Global

Equations 1.

2 In the Global Equations settings window, locate the Global Equations section.


M E S H 1

Edit the default meshing sequence, make the mesh parameter dependent to make sure the mesh is always a well resolved at the boundaries. (The parametric sweep will change the size of the geometry during the solver process.)

Name f(u,ut,utt,t) (1) Initial value (u_0) (1)

Initial value (u_t0) (1/s)

Description

V intop1(iloc)-icell

0 0 Cell voltage



Edge 1In the Model Builder window, under Component 1 right-click Mesh 1 and choose Edit

Physics-Induced Sequence.

Size 11 In the Model Builder window, under Component 1>Mesh 1 right-click Edge 1 and

choose Size.

2 In the Size settings window, locate the Element Size section.

3 Click the Custom button.

4 Locate the Element Size Parameters section. Select the Maximum element size check box.

5 In the associated edit field, type L/20.

Size 21 Right-click Edge 1 and choose Size.




5 Locate the Element Size section. Click the Custom button.


7 In the associated edit field, type lambdaD/10.

Edge 1Right-click Edge 1 and choose Build Selected.

S T U D Y 1

Solve the problem using a parametric sweep.

Parametric Sweep1 In the Model Builder window, right-click Study 1 and choose Parametric Sweep.


3 Click Add.



epsilon 0.001 0.01 0.1



14 | M O

5 In the Model Builder window, click Study 1.

6 In the Study settings window, locate the Study Settings section.

7 Clear the Generate default plots check box.


R E S U L T S

Reproduce the figures from the Results and Discussion section in the following way:


2 On the 1D plot group toolbar, click Line Graph.


4 In the Line Graph settings window, locate the y-Axis Data section.

5 In the Expression edit field, type (cp+cm)/(2*cref).

6 Locate the x-Axis Data section. From the Parameter list, choose Expression.

7 In the Expression edit field, type x/L.

8 Click to expand the Legends section. Select the Show legends check box.

9 In the Model Builder window, click 1D Plot Group 1.

10 In the 1D Plot Group settings window, locate the Data section.

11 From the Data set list, choose Solution 2.

12 Click to expand the Title section. From the Title type list, choose Manual.

13 In the Title text area, type Dimensionless concentration.

14 Click to expand the Legend section. From the Position list, choose Lower right.


16 Right-click 1D Plot Group 1 and choose Rename.

17 Go to the Rename 1D Plot Group dialog box and type Dimensionless concentration in the New name edit field.

18 Click OK.

Dimensionless concentration 11 Right-click 1D Plot Group 1 and choose Duplicate.

2 In the 1D Plot Group settings window, locate the Title section.

3 In the Title text area, type Dimensionless charge density.

4 Locate the Legend section. From the Position list, choose Upper right.



5 In the Model Builder window, expand the Dimensionless concentration 1 node, then click Line Graph 1.


7 In the Expression edit field, type (cp-cm)/(2*cref).


9 In the Model Builder window, right-click Dimensionless concentration 1 and choose Rename.

10 Go to the Rename 1D Plot Group dialog box and type Dimensionless charge density in the New name edit field.

11 Click OK.

Dimensionless charge density 11 Right-click Dimensionless concentration 1 and choose Duplicate.


3 In the Title text area, type Dimensionless potential.

4 Locate the Legend section. From the Position list, choose Lower right.

5 In the Model Builder window, expand the Dimensionless charge density 1 node, then click Line Graph 1.


7 In the Expression edit field, type phi*Z*F_const/(R_const*T).


9 In the Model Builder window, right-click Dimensionless charge density 1 and choose Rename.

10 Go to the Rename 1D Plot Group dialog box and type Dimensionless potential in the New name edit field.

11 Click OK.



16 | M O


De s a l i n a t i o n i n an E l e c t r o d i a l y s i s C e l l

Introduction

Electrodialysis is a separation process for electrolytes based on the use of electric fields and ion selective membranes. Some common applications of the electrodialysis process are:

• Desalination of process streams, effluents, and drinking water

• pH regulation in order to remove acids from, for example, fruit juices and wines

• Metal electrowinning of precious metals

This tutorial demonstrates the basics of electrodialysis in a desalination cell.

1 | D E S A L I N A T I O N I N A N E L E C T R O D I A L Y S I S C E L L


2 | D E S

Model Definition

Figure 1shows the basic principle of the desalination stack. The stack has one repetitive cell unit, shown in Figure 2.

Figure 1: Schematic picture of a desalination stack with 3 desalination units (in reality 10 - 20 unit cells are used)

Figure 2: The repetitive unit cell with one desalination unit.

Cathode:NegativeElectrode

Anode:PositiveElectrode

Diluate

Diluate

Concentrate

Concentrate

ElectrodeStream

ElectrodeStream

ElectrodeStream

ElectrodeStream

OH ‐

SO42‐

Na+

Cl ‐Cl ‐

Na+ Na+Na+

Cl ‐

Na+Na+ Na+

Cl ‐Cl ‐ Cl ‐SO42‐ SO4

2‐

Na+

H+

Na +

Cl-

Na + Na +

Cl -Cl -

A L I N A T I O N I N A N E L E C T R O D I A L Y S I S C E L L


The model geometry for this example is based on the repetitive unit, excluding the inlet and outlet flow regions. The model geometry is shown in Figure 3.

Figure 3: Model geometry.

The cell contains two ion selective membrane domains, the left is assumed to be permeable to cations only, and the right to anions only. The middle domain is a free flowing electrolyte domain, where salt is to be removed. This domain is named the dilutate domain. The rightmost and leftmost domains are free flowing electrolyte domains, where the ion concentration will increase during cell operation. These domains are called the concentrate domains. In the free electrolyte domains, the flow enters at the bottom inlets and exits at the top outlets at an average flow rate is 5 mm·s-1. An analytical expression (Poiseuille flow) for the fluid velocity is used in this model.

The cell is fed with a seawater electrolyte consisting of 0.5 M NaCl, and operated at a unit cell voltage of 1.5 V.

C H O I C E O F C U R R E N T D I S T R I B U T I O N I N T E R F A C E

In this model we may use of the Nernst-Planck equations for ion flux and charge transport by which the following equation describes the molar flux of species i (which is either Cl or Na in this model), Ni, due to diffusion, migration and convection:

Anion Selective Membrane

Dilutate Channel

Concentrate Channels

Outlets

Inlets

Cation Selective Membrane



4 | D E S

The first term is the diffusion flux, Di is the diffusion coefficient (m2/s). The migration term consists of the species charge number zi, the species mobility umob,i (s·mol/kg) and the electrolyte potential ( ). In the convection term, u denotes the fluid velocity vector (m/s).

The electrolyte current density is calculated using Faraday’s law by summing up the contributions from the molar fluxes, multiplied by the species charges, with the observation that the convective term vanishes due to the electroneutrality condition (see the Theory for the Tertiary Current Distribution, Nernst-Plank interface):

(1)

The conservation of current is then used to calculate the electrolyte potential.

where the Ri terms are the reaction sources due the porous electrode reactions.

This model uses Tertiary Current Distribution, Nernst-Planck interface when solving for the electrolyte potential in the free electrolyte domains.

In the dilutate and concentrate domains, where there is free electrolyte present, the gradients of Na+ and Cl- are not negligible. In the ion selective membranes, however, all the counter ions are fixed in a matrix, implying that the concentration of the permeable ion is constant. Therefore, due to electroneutrality, Equation 1 reduces to (here written for Na+ in the cation selective membrane)

(2)

in the membrane, where l is the electrolyte conductivity. Equation 2 is the equation used in the Secondary Current Distribution interface.

Note that Equation 2 only concerns the governing equation for the electrolyte potential in the membrane. Transport of additional trace species in the membrane, with minor impact on the total charge transport, can be added to the model using the Transport of Diluted Species interface.

Ni Dici– ziumob i Fcil– ciu+=

l

il F zi Dici– zium i Fcil–

i 1=

n

=

il F ziRi

i 1=

n

=

il um + F2c+l– ll–= =



M E M B R A N E — F R E E E L E C T R O L Y T E B O U N D A R Y C O N D I T I O N S

The boundary conditions at the boundaries between the Secondary Current Distribution interface for the membrane and the Tertiary Current Distribution, Nernst-Planck interface are set up in the following way:

The normal electrolyte current density is equal to the current density in the membrane:

(3)

The ion flux for the permeable ion is proportional to the current, according to Faraday’s law

(4)

For the non-permeable ion, the flux is zero.

Furthermore, we have the following relation between the potentials and the concentrations:

(5)

where ai,m is the permeable ion activity in the membrane, and ai,e the permeable ion activity in the free electrolyte. In this model the activities equal the respective concentrations, divided by 1000 mol/m3. The potential shift caused by Equation 5 is called Donnan potential, or dialysis potential.

Equation 3 -Equation 5 above represent a mix a Dirichlet and Neumann conditions for three different dependent variables (two potentials and one concentration). Use a point wise constraint with a user-defined reaction force term to define these conditions. The user-defined reaction force provides a proper scale between the fluxes (two current densities and one molecular flux) for the dependent variables. You can read more about pointwise constraints in the Comsol Multiphysics Reference Manual.

C O U P L I N G B E T W E E N T H E O U T E R C O N C E N T R A T E B O U N D A R I E S

Use a linear extrusion operator to set the concentration on the rightmost boundary to equal that of the leftmost boundary.

n il e n il m=

n Ni e nil mziF----------=

l m l eRTF

--------ai mai e----------- ln+=



6 | D E S


Figure 4 shows the ion concentration in the cell. The concentration increases in the concentrate domains, and decreases in the dilutate domain. A boundary layer with high concentration gradients forms close to the membrane surfaces. In a real desalination cell it is common to add spacers that, apart from adding mechanical stability to the cell, also induce convective mass transport in the direction perpendicular to the main flow, and hence may reduce the thickness of the boundary layer.

Figure 4: Ion concentration.

Figure 5 shows the electrolyte potential along a horizontal line placed at half the cell height. The main part of the potential losses occurs in the membranes. The Donnan potential discontinuity can be seen at the boundaries between the free electrolyte and membrane domains.



Figure 5: Electrolyte potential at half the cell height.

Figure 6 shows the concentration at half the cell height for three different cell potentials, 0.5, 1 and 1.5 V. The concentration gradients increase with higher potentials.



8 | D E S

Figure 6: Ion concentration at half the cell height for different unit cell voltages.

Figure 7 and Figure 8 show a comparison between the migrative and diffusive fluxes in the free electrolyte for Na+ and Cl-, respectively. The diffusive fluxes get prominent close to the membrane boundaries due to the high concentration gradients. The migrative fluxes govern in the middle of the channels and have different signs for Na+ and Cl- due to the different signs of the ion charges.



Figure 7: Comparison of fluxes, species Na+.

Figure 8: Comparison of fluxes, species Cl-.



10 | D E

Finally, Figure 9 shows the current density along the left free electrolyte – membrane boundary of the anion selective membrane for three different cell voltage levels. Depletion of chloride gets more prominent towards the top of the cell, resulting in a higher Donnan potential shift and a lower current density. Increasing the cell voltage further will increase the cell current somewhat more, but the cell will eventually reach a limiting current density.

Figure 9: Normal current density along the membrane – free electrolyte boundary for the anion selective membrane for three different cell voltages (0.5 V, 1.0 V and 1.5 V).

Model Library path: Electrochemistry_Module/Electrochemical_Engineering/electrodialysis



N E W


S A L I N A T I O N I N A N E L E C T R O D I A L Y S I S C E L L




2 In the Select physics tree, select Electrochemistry>Tertiary Current Distribution,

Nernst-Planck (tcdee).




(siec).


7 In the Electrolyte potential edit field, type phil_m.


9 In the tree, select Preset Studies for Selected Physics>Stationary.



Load the model parameters from a text file.




4 Browse to the model’s Model Library folder and double-click the file electrodialysis_parameters.txt.

G E O M E T R Y 1

Draw the geometry as a set of rectangles.

Rectangle 1 (r1)1 In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and

choose Rectangle.

2 In the Rectangle settings window, locate the Size section.

3 In the Width edit field, type 2*(W_ch+W_m).

4 In the Height edit field, type L.

cNa

cCl



12 | D E

5 Locate the Position section. In the x edit field, type -(W_ch+W_m).

6 Click the Build Selected button.

Rectangle 2 (r2)1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.


3 In the Width edit field, type (W_ch+2*W_m).


5 Locate the Position section. In the x edit field, type -(W_ch+2*W_m)/2.


Rectangle 3 (r3)1 Right-click Geometry 1 and choose Rectangle.


3 In the Width edit field, type W_ch.


5 Locate the Position section. In the x edit field, type -W_ch/2.



The finished geometry should look like Figure 3 in the Model Definition section.


Make some selections on the geometry to use later when you set up the physics.

Explicit 11 On the Definitions toolbar, click Explicit.

2 Select Domains 2 and 4 only.

3 Right-click Component 1 (comp1)>Definitions>Explicit 1 and choose Rename.

4 Go to the Rename Explicit dialog box and type Membranes in the New name edit field.

5 Click OK.


2 Select Domains 1, 3, and 5 only.


4 Go to the Rename Explicit dialog box and type Channels in the New name edit field.



5 Click OK.


2 In the Explicit settings window, locate the Input Entities section.


4 Select Boundaries 2, 8, and 14 only.


6 Go to the Rename Explicit dialog box and type Inlets in the New name edit field.

7 Click OK.




4 Select Boundaries 3, 9, and 15 only.


6 Go to the Rename Explicit dialog box and type Outlets in the New name edit field.

7 Click OK.

Analytic 1 (an1)Create an analytic function for the Nernst equation. The function will be used when setting up the membrane electrolyte potential boundary condtions.

1 On the Home toolbar, click Functions and choose Global>Analytic.

2 In the Analytic settings window, locate the Function Name section.

3 In the Function name edit field, type Nernst.

4 Locate the Definition section. In the Expression edit field, type E0+R_const*T/F_const*log(c_expr).

5 In the Arguments edit field, type E0, T, c_expr.

6 Locate the Units section. In the Arguments edit field, type V, K, 1.

7 In the Function edit field, type V.

8 Right-click Component 1 (comp1)>Definitions>Analytic 1 (an1) and choose Rename.

9 Go to the Rename Analytic dialog box and type Nernst Equation in the New name edit field.



14 | D E

10 Click OK.

Linear Extrusion 1 (linext1)Create a linear extrusion from the leftmost boundary to the rightmost boundary. It will be used later to couple the concentrations on each side of the cell together.

1 On the Definitions toolbar, click Component Couplings and choose Linear Extrusion.

2 In the Linear Extrusion settings window, locate the Source Selection section.



5 Locate the Source Vertices section. Select the Source vertex 1 toggle button.

6 Select Point 1 only.

7 Select the Source vertex 2 toggle button.


9 Locate the Destination Vertices section. Select the Destination vertex 1 toggle button.


11 Select the Destination vertex 2 toggle button.


Variables 1Now add some variables to the model.

1 In the Model Builder window, right-click Definitions and choose Variables.

2 In the Variables settings window, locate the Variables section.


The max() functions ensures a positive value for the aH and aCl variables. eps is the machine epsilon, i.e. a very small number.



3 From the Geometric entity level list, choose Domain.


aNa max(cNa,eps^2) mol/m³ Na activity

aCl max(cCl,eps^2) mol/m³ Cl activity















TE R T I A R Y C U R R E N T D I S T R I B U T I O N , N E R N S T - P L A N C K ( T C D E E )

Now start setting up the physics, start with the free electrolyte channels.

1 In the Model Builder window, under Component 1 (comp1) click Tertiary Current

Distribution, Nernst-Planck (tcdee).

2 In the Tertiary Current Distribution, Nernst-Planck settings window, locate the Domain

Selection section.

3 From the Selection list, choose Channels.


vel 3*v_avg*(1-((x)/(W_ch/2))^2)

m/s Channel velocity (middle channel)


vel 3*v_avg*(1-((x+W_ch+W_m)/(W_ch/2))^2)

m/s Channel velocity (left channel)


vel 3*v_avg*(1-((x-W_ch-W_m)/(W_ch/2))^2)

m/s Channel velocity (right channel)



16 | D E

Electrolyte 11 In the Model Builder window, under Component 1 (comp1)>Tertiary Current

Distribution, Nernst-Planck (tcdee) click Electrolyte 1.

2 In the Electrolyte settings window, locate the Model Inputs section.

3 Specify the u vector as


5 Locate the Diffusion section. In the DcNa edit field, type DNa.

6 In the DcCl edit field, type DCl.

7 Locate the Migration in Electric Field section. In the zcNa edit field, type 1.

8 In the zcCl edit field, type -1.

Electrolyte Potential 11 On the Physics toolbar, click Boundaries and choose Electrolyte Potential.


Electrolyte Potential 21 On the Physics toolbar, click Boundaries and choose Electrolyte Potential.



4 In the l,bnd edit field, type Vtot.

Concentration 11 On the Physics toolbar, click Boundaries and choose Concentration.



4 Select the Species cCl check box.

5 In the c0,cCl edit field, type comp1.linext1(cCl).

This condition sets the concentration on the rightmost boundary to be equal its corresponding value on the leftmost boundary.

Inflow 11 On the Physics toolbar, click Boundaries and choose Inflow.

2 In the Inflow settings window, locate the Boundary Selection section.

0 x

vel y



3 From the Selection list, choose Inlets.

4 Locate the Concentration section. In the c0,cCl edit field, type cCl_0.

Outflow 11 On the Physics toolbar, click Boundaries and choose Outflow.

2 In the Outflow settings window, locate the Boundary Selection section.

3 From the Selection list, choose Outlets.

Note that no conditions for the membrane boundaries have been set so far. These will be set later by the user-defined Pointwise Constraint in the Secondary Current Distribution interface.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Tertiary Current

Distribution, Nernst-Planck (tcdee) click Initial Values 1.


3 In the cCl edit field, type cCl_0.


Now set up the physics for the membranes.

1 In the Model Builder window, under Component 1 (comp1) click Secondary Current

Distribution (siec).

2 In the Secondary Current Distribution settings window, locate the Domain Selection section.

3 From the Selection list, choose Membranes.

Electrolyte 11 In the Model Builder window, under Component 1 (comp1)>Secondary Current

Distribution (siec) click Electrolyte 1.

2 In the Electrolyte settings window, locate the Electrolyte section.

3 From the l list, choose User defined. In the associated edit field, type sigma_m.

Set up the potential jump according to the Nernst equation by using a user-defined pointwise constraint. The user-defined force expressions will also result in fluxes for the appropriate potential and concentration variables in the tertiary current distribution interfaces.

4 In the Model Builder window’s toolbar, click the Show button and select Advanced

Physics Options in the menu.



18 | D E

Pointwise Constraint 11 On the Physics toolbar, click Boundaries and choose Pointwise Constraint.


3 In the Pointwise Constraint settings window, locate the Pointwise Constraint section.

4 From the Apply reaction terms on list, choose User defined.

5 In the Constraint expression edit field, type phil-Nernst(0,T,aNa/cNam)-phil_m.

6 In the Constraint force expression edit field, type test(phil-phil_m).

Pointwise Constraint 21 On the Physics toolbar, click Boundaries and choose Pointwise Constraint.


3 In the Pointwise Constraint settings window, locate the Pointwise Constraint section.

4 From the Apply reaction terms on list, choose User defined.

5 In the Constraint expression edit field, type phil-Nernst(0,T,aCl/cClm)-phil_m.

6 In the Constraint force expression edit field, type test(phil-cCl/F_const-phil_m).

Change to second order element to match the second order elements of the tertiary current distribution interface.

7 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu.

8 In the Model Builder window, click Secondary Current Distribution (siec).

9 In the Secondary Current Distribution settings window, click to expand the Discretization section.

10 Find the Value types when using splitting of complex variables subsection. From the Electrolyte potential list, choose Quadratic.

M E S H 1

The physics is now complete. A mapped mesh is suitable for this geometry. Use distributions to obtain thinner elements in the free electrolyte close to the membrane surfaces.

Mapped 1In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Mapped.



Distribution 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Mapped

1 and choose Distribution.


3 In the Distribution settings window, locate the Distribution section.

4 In the Number of elements edit field, type 100.

Distribution 21 Right-click Mapped 1 and choose Distribution.



4 From the Distribution properties list, choose Predefined distribution type.


6 In the Element ratio edit field, type 10.

7 Select the Symmetric distribution check box.







7 Select the Reverse direction check box.










20 | D E






7 Select the Reverse direction check box.













5 In the Model Builder window, right-click Mesh 1 and choose Build All.

S T U D Y 1

Step 1: StationaryThe problem is now ready for solving. Use an auxiliary sweep to solve for a range of potentials.

1 In the Model Builder window, under Study 1 click Step 1: Stationary.

2 In the Stationary settings window, click to expand the Study extensions section.

3 Locate the Study Extensions section. Select the Auxiliary sweep check box.

4 Click Add.


Solver 11 On the Study toolbar, click Show Default Solver.

2 In the Model Builder window, under Study 1>Solver Configurations>Solver 1 click Stationary Solver 1.

3 In the Stationary Solver settings window, locate the General section.

Auxiliary parameter Parameter value list

Vtot range(0.5,0.5,1.5)



22 | D E

4 In the Relative tolerance edit field, type 1e-5.


R E S U L T S

Concentration (tcdee)Reproduce the figures from the Results and Discussion section in the following way.


Data Sets1 On the Results toolbar, click Cut Line 2D.

2 In the Cut Line 2D settings window, locate the Line Data section.

3 In row Point 1, set x to -W_ch-W_m.

4 In row Point 1, set y to L/2.

5 In row Point 2, set x to W_ch+W_m.

6 In row Point 2, set y to L/2.

7 Click the Plot button.



3 From the Data set list, choose Cut Line 2D 1.

4 From the Parameter selection (Vtot) list, choose Last.




8 In the Expression edit field, type phil_m.


10 In the 1D Plot Group settings window, click to expand the Title section.

11 From the Title type list, choose None.









6 In the Expression edit field, type cNa.

7 Click to expand the Legends section. Select the Show legends check box.





4 From the Parameter selection (Vtot) list, choose Last.



7 In the Expression edit field, type tcdee.tfluxx_cNa.

8 Locate the Legends section. Select the Show legends check box.

9 From the Legends list, choose Manual.



12 Right-click Results>1D Plot Group 6>Line Graph 1 and choose Duplicate.


14 In the Expression edit field, type tcdee.mfluxx_cNa.

15 Locate the Legends section. In the table, enter the following settings:


17 Right-click Results>1D Plot Group 6>Line Graph 2 and choose Duplicate.


19 In the Expression edit field, type tcdee.dfluxx_cNa.

Legends

Total Flux

Legends

Migrative Flux


24 | D E





24 In the 1D Plot Group settings window, locate the Plot Settings section.

25 Select the x-axis label check box.

26 In the associated edit field, type Position.

27 Select the y-axis label check box.

28 In the associated edit field, type Flux.

29 Locate the Title section. From the Title type list, choose Manual.


31 In the Title text area, type Na+ flux, x component (mol/(m2*s)).

32 Click to expand the Legend section. From the Position list, choose Upper left.

1D Plot Group 71 Right-click 1D Plot Group 6 and choose Duplicate.

2 In the Model Builder window, expand the 1D Plot Group 7 node, then click Line Graph

1.


4 In the Expression edit field, type tcdee.tfluxx_cCl.

5 In the Model Builder window, under Results>1D Plot Group 7 click Line Graph 2.


7 In the Expression edit field, type tcdee.mfluxx_cCl.

8 In the Model Builder window, under Results>1D Plot Group 7 click Line Graph 3.


10 In the Expression edit field, type tcdee.dfluxx_cCl.



Legends

Diffusive Flux

13 In the Title text area, type Cl- flux, x component (mol/(m2*s)).





4 In the Line Graph settings window, locate the y-axis data section.

5 Click Replace Expression in the upper-right corner of the section and choose Normal

electrolyte current density (siec.nIl). Locate the x-Axis Data section. From the Parameter list, choose Expression.

6 In the Expression edit field, type y.

7 Locate the Legends section. Select the Show legends check box.


9 In the Model Builder window, right-click 1D Plot Group 8 and choose Rename.

10 Go to the Rename 1D Plot Group dialog box and type Electrolyte current density in the New name edit field.

11 Click OK.

1D Plot Group 71 In the Model Builder window, under Results right-click 1D Plot Group 7 and choose

Rename.

2 Go to the Rename 1D Plot Group dialog box and type Fluxes, Cl in the New name edit field.

3 Click OK.


Rename.

2 Go to the Rename 1D Plot Group dialog box and type Fluxes, Na in the New name edit field.

3 Click OK.


Rename.


26 | D E

2 Go to the Rename 1D Plot Group dialog box and type Concentration in the New

name edit field.

3 Click OK.


Rename.

2 Go to the Rename 1D Plot Group dialog box and type Potential in the New name edit field.

3 Click OK.



G l u c o s e S e n s o r

Introduction

Glucose sensing is one of the most widespread and commercially successful uses of electroanalysis. In an electrochemical glucose sensor, the concentration of glucose in a sample is measured using amperometry; that is, the measurement of an electric current. An applied voltage causes the oxidation of glucose, and the current due to this oxidation is measured at the electrode. In a well designed glucose sensor, there is a linear relationship between the glucose concentration and the current, enabling a calibrated measurement.

Typically, the oxidation of glucose does not occur directly at the working electrode where current is measured. Instead, the reaction is accomplished by a chemical oxidant and accelerated by a biological enzyme such as glucose oxidase (GOx), which makes the sensor specific to glucose and independent of the concentration of other oxidizable species that may be present in the analyte solution.

The reduced form of the oxidant, after its reaction with glucose, can be re-oxidized directly at the electrode. In nature, the oxidant is oxygen, but this suffers from slow kinetics and the rate of oxidation will be perturbed by the oxygen concentration dissolved from atmosphere into the analyte solution.

Instead an inorganic oxidant with fast electrode kinetics, such as the hexacyanoferrate(III) anion (commonly, “ferricyanide”), is suitable for use in a glucose sensor, since the measured current is made independent of oxygen concentration and is not limited by slow electrode kinetics (Ref. 1).

This example demonstrates a steady-state analysis of the current drawn in a unit cell of solution above an interdigitated electrode, where the counter electrode reacts ferricyanide to ferrocyanide. The linearity of the response of the sensor is demonstrated for a typical range of glucose concentrations.

Model Definition

The model contains a single 2D domain representing a 100 µm-wide unit cell of solution above an interdigitated electrode (Figure 1). The real geometry is a periodic repetition of this unit cell in the x-direction. The cell and electrode are assumed to extend sufficiently far out-of-plane of the model that the 2D approximation is suitable.

1 | G L U C O S E S E N S O R

2 | G L U

At the top of the unit cell is a bulk boundary where the concentrations are assumed to equal those in the bulk solution of the analyte. At the bottom of the unit cell, the y 0 axis is divided by four points into separate electrode and insulator boundaries. The anode (working electrode) is at the center of the cell in the range 37.5 µm < x < 62.5 µm. The unit cell contains half of each of the two neighboring cathodes (counter electrodes) in the ranges x < 12.5 µm and x > 87.5 µm. Between the anode and cathode surfaces, a solid insulating material is present.

Figure 1: Model Geometry.


A large quantity of supporting electrolyte is present. This is inert salt added in electroanalytical experiments to increase the conductivity of the electrolyte without otherwise interfering with the reaction chemistry. Under these conditions, the resistance of the solution is sufficiently low that the electric field is negligible, and we can assume a constant electrolyte potential l.

The Electroanalysis interface implements chemical species transport equations to describe the diffusion of the chemical species. The domain equation is the diffusion equation (also known as Fick’s 2nd law). At steady-state, this reduces to:

(1) Di ci 0=

C O S E S E N S O R


for each species i. In this model three species are modeled: the active redox couple—ferricyanide and ferrocyanide anions—as well as the concentration of the glucose analyte species. We ignore the products of the glucose oxidation since they do not affect the behavior of the sensor.

The enzyme-mediated reaction of the glucose with the ferricyanide anion occurs in the solution phase above the electrode:

(2)

The rate of this reaction (mol/m3) is given by a Michaelis–Menten rate law as (Ref. 2):

(3)

Here, the parameter Vmax is the maximum rate of the enzyme-catalyzed reaction, depending on the quantity of enzyme available, and the parameter Km is a characteristic Michaelis-Menten coefficient. At large glucose concentration, the rate becomes independent of the glucose concentration and solely depends on the enzyme kinetics.

B O U N D A RY E Q U AT I O N S

At the bulk boundary (y = 1 mm), we assume a uniform concentration of each chemical species equal to its bulk concentration. The glucose concentration here is equivalent to that in the analyte mixture being measured; the ferricyanide:ferrocyanide ratio here is 50 000:1, with the ferricyanide anion present in bulk in a concentration of 50 mM. Because the analytical process is oxidizing with respect to the glucose analyte, more oxidant must be supplied.

At the insulating (inert) surfaces, the normal fluxes of all species are equal to zero, since this surface is impermeable and no species reacts there.

At the electrode boundaries, current is drawn from the interconversion of ferrocyanide and ferricyanide. By convention, electrochemical reactions are written in the reductive direction:

(4)

The stoichiometric coefficient is –1 for ferricyanide, the “reactant” in the reductive direction, and +1 for ferrocyanide, the “product” in the reductive direction. This formulation is consistent at the anode also, although here the reaction proceeds

glucose ferri products ferro++

RcglucoseVmax

1 Kmcglucose+ -----------------------------------------=

ferri e– ferro+



4 | G L U

favorably in the opposite, oxidative direction. The number of electrons transferred, n, equals one.

The current density for this reaction is given by the electroanalytical Butler–Volmer equation for an oxidation:

(5)

in which k0 is the heterogeneous rate constant of the reaction, c is the cathodic transfer coefficient, and is the overpotential at the working electrode.


(6)


The total current recorded at the electrode can be extracted by integrating the local current density across the electrode surface. It is not sufficient to simply multiply by the area of the electrode, because the current density may be non-uniform. An Integration Component Coupling is used to define an electrode current variable according to:

(7)

where the integration is performed over the area of the working electrode.

The working electrode (anode) is held at +0.4 V vs. the ferro/ferricyanide redox couple. The counter electrode is constrained to deliver an opposite current to the anode.

S TAT I O N A RY S T U DY

This model calculates the steady-state current delivered under a constant applied potential. Therefore a Stationary study is chosen. A Parametric Sweep is used to compare the currents and concentration profiles for different external glucose concentrations in the analyte solution.

iloc nFk0 cferroexpn c– F

RT---------------------------- cferriexp

cF–

RT----------------- –

=

n– NiiilocnF

-------------=

Iel iloc AdS=

C O S E S E N S O R



Figure 2 shows a typical concentration profile for the ferrocyanide ion in the unit cell. Ferrocyanide is generated in the solution between the electrodes and bulk by the enzyme-catalyzed oxidation of glucose. It reacts at the anode in the center of the unit cell to provide the working electrode current used to measure the concentration of glucose. Ferrocyanide is regenerated at the cathode counter electrodes at the left and right of the cell.

The diffusion of ferrocyanide from the counter to the working electrode is an example of a “redox cycling” process where a single redox reaction is driven in opposite directions at two electrodes with a small geometric separation. This cycling effect amplifies the current and so ensures a linear response to a wide range of glucose concentrations, as illustrated in Figure 3.

Figure 2: Ferrocyanide concentration for an external glucose concentration of 1 mol/m3.



6 | G L U

Figure 3: Current density versus glucose concentration.

References

1. J. Wang, Chem. Rev. 108 (2008) pp 814

2. P. Atkins, J. de Paula, Physical Chemistry, 9th Edition

Model Library path: Electrochemistry_Module/Electrochemical_Engineering/glucose_sensor



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C O S E S E N S O R



Build the model in 2D with the Electroanalysis interface. Solve for three concentrations in a Stationary study.




4 In the Number of species edit field, type 3.





G E O M E T R Y 1

Set the length unit to micrometers and create the geometry using a rectangle and an array of points.

1 In the Model Builder window, under Component 1 (comp1) click Geometry 1.

2 In the Geometry settings window, locate the Units section.

3 From the Length unit list, choose µm.

Rectangle 1 (r1)1 Right-click Component 1 (comp1)>Geometry 1 and choose Rectangle.


3 In the Width edit field, type 100.

4 In the Height edit field, type 1000.

Point 1 (pt1)1 In the Model Builder window, right-click Geometry 1 and choose Point.

2 In the Point settings window, locate the Point section.

3 In the x edit field, type 12.5.

c_glucose

c_ferro

c_ferri



8 | G L U

Array 1 (arr1)1 On the Geometry toolbar, click Array.

2 Select the object pt1 only.

3 In the Array settings window, locate the Size section.

4 In the x size edit field, type 4.

5 Locate the Displacement section. In the x edit field, type 25.


Your finished geometry should now look like this:


Import the model parameters from a text file.




4 Browse to the model’s Model Library folder and double-click the file glucose_sensor_parameters.txt.

C O S E S E N S O R



Add an average operator that will be used to calculate the average of the current density over one of the electrode surfaces.

Average 1 (aveop1)1 On the Definitions toolbar, click Component Couplings and choose Average.

2 In the Average settings window, locate the Source Selection section.



Variables 11 In the Model Builder window, right-click Definitions and choose Variables.



The i_avg variable in marked in orange. This is because the itot variable has not yet been defined. It will be defined and added automatically to the model later on when you add the Electrode Surface feature.

E L E C T R O A N A L Y S I S ( E L A N )

Now start defining the physics.

Electrode Surface 11 On the Physics toolbar, click Boundaries and choose Electrode Surface.



4 In the s,ext edit field, type 0.4.

Electrode Reaction 11 In the Model Builder window, expand the Electrode Surface 1 node, then click

Electrode Reaction 1.

2 In the Electrode Reaction settings window, locate the Stoichiometric Coefficients section.

3 In the cferro edit field, type 1.


R_MM V_max*c_glucose/(Km+c_glucose)

mol/(m³·s) Reaction rate of glucose

i_avg aveop1(elan.itot) Average current density



10 | G L

4 In the cferri edit field, type -1.




4 From the Boundary condition list, choose Average current density.

5 In the il,average edit field, type -i_avg.

6 In the s,ext edit field, type -0.2.



2 In the Electrode Reaction settings window, locate the Stoichiometric Coefficients section.

3 In the cferro edit field, type 1.

4 In the cferri edit field, type -1.




4 Select the Species c_glucose check box.

5 In the c0,cglucose edit field, type c_glucose_ext.

6 Select the Species c_ferro check box.

7 In the c0,cferro edit field, type c_ferro_ext.

8 Select the Species c_ferri check box.

9 In the c0,cferri edit field, type c_ferri_ext.

Reactions 11 On the Physics toolbar, click Domains and choose Reactions.


3 In the Reactions settings window, locate the Reactions section.

4 In the Rcglucose edit field, type -R_MM.

5 In the Rcferro edit field, type R_MM.

U C O S E S E N S O R


6 In the Rcferri edit field, type -R_MM.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Electroanalysis (elan) click

Initial Values 1.


3 In the cglucose edit field, type c_glucose_ext.

4 In the cferro edit field, type c_ferro_ext.

5 In the cferri edit field, type c_ferri_ext.

M E S H 1

The physics settings are now complete. Now customize the mesh and solve the problem.

Size1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and

choose Edit Physics-Induced Sequence.

2 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size.


4 From the Predefined list, choose Extra fine.

Size 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free

Triangular 1 and choose Size.



4 Select Boundaries 2 and 4–7 only.

5 Locate the Element Size section. From the Predefined list, choose Extremely fine.

6 Click the Custom button.


8 In the associated edit field, type 1.



12 | G L

9 In the Model Builder window, right-click Mesh 1 and choose Build All.

S T U D Y 1

Use a parametric sweep to solve for a range of different external concentration values for c_ferro.



3 Click Add.



R E S U L T S

Concentration (elan)Modify the default concentration plot to show the ferro concentration.


c_glucose_ext range(50,50,1000)[umol/L]



1 In the Model Builder window, expand the Concentration (elan) node, then click Surface 1.

2 In the Surface settings window, locate the Expression section.

3 In the Expression edit field, type c_ferro.


1D Plot Group 2Create a plot of the average current density for different c_ferro_ext values as follows:

1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

2 On the 1D plot group toolbar, click Global.

3 In the Global settings window, locate the y-Axis Data section.


5 Click to expand the Legends section. Clear the Show legends check box.


Expression Unit Description

i_avg A/m^2 Average current density



14 | G L


E l e c t r o c h em i c a l Impedan c e S p e c t r o s c o p y

Introduction

Electrochemical impedance spectroscopy is a common technique in electroanalysis. It is used to study the harmonic response of an electrochemical system. A small, sinusoidal variation is applied to the potential at the working electrode, and the resulting current is analyzed in the frequency domain.

The real and imaginary components of the impedance give information about the kinetic and mass transport properties of the cell, as well as its capacitive properties. By measuring impedance at a range of frequencies, the relative influence of the various constituent physics of the system can be interpreted as a function of time scale.

As in electrical analysis, the real component of impedance corresponds to a resistance in-phase with the applied voltage. The imaginary component corresponds to a reactance 90° out-of-phase with the applied voltage; the reactance is caused by capacitive charging in the cell.

Capacitive charging occurs at the double layer adjacent to the working electrode surface. Here, the net charge on the electrode causes accumulation or depletion of charged ions in the neighboring solution; as the potential on the electrode changes, so does its charge, and so the double layer will charge and discharge with a characteristic capacitance.

In this model, the AC Impedance Stationary study type is used to perform a stationary analysis at the center potential of the applied potential waveform, followed by a linearized perturbation study in the frequency domain to resolve the magnitude and phase of the current response to the sinusoidal voltage. A Parametric Sweep is used to compare the response of the system with different electrode kinetics for the redox couple.

Model Definition

This model contains a single 1D domain of length L 1 mm, which is sufficiently large compared to the timescale of diffusion at the lowest frequency investigated.

1 | E L E C T R O C H E M I C A L I M P E D A N C E S P E C T R O S C O P Y


2 | E L E


We assume the presence of a large quantity of supporting electrolyte. This is inert salt that is added in electroanalytical experiments to increase the conductivity of the electrolyte without otherwise interfering with the reaction chemistry. Under these conditions, the resistance of the solution is sufficiently low that the electric field is negligible, and we can assume l = 0.

The Electroanalysis interface implements chemical transport equations for the reactant and product species of the redox couple subject to this assumption. The domain equation is the diffusion equation (also known as Fick’s 2nd law) to describe the chemical transport of the electroactive species A and B:

(1)


Both the oxidized and reduced species are equally concentrated at ci = 1 mM in bulk (x = L).

At the working electrode surface (x = 0), the reactant species A oxidizes (loses one electron) to form the product B. By convention, electrochemical reactions are written in the reductive direction:

(2)

The stoichiometric coefficient is –1 for B, the “reactant” in the reductive direction, and +1 for A, the “product” in the reductive direction. The number of electrons transferred, n, equals one.


(3)

in which k0 is the heterogeneous rate constant of the reaction, c is the cathodic transfer coefficient, and is the overpotential at the working electrode. This overpotential is the difference between the applied potential and the equilibrium potential (formal reduction potential) of the redox couple of species A and B, Eeq:

(4)

cit

------- Di ci =

B e– A+


RT---------------------------- cBexp

cF–

RT----------------- –

=

s ext Eeq–=

C T R O C H E M I C A L I M P E D A N C E S P E C T R O S C O P Y



(5)


An additional capacitance is applied at the working electrode. It is set equal to 20 µF/cm2 which is a typical value for a water-metal interface. The real value of this capacitance can be established by impedance spectroscopy of the blank solution containing only the supporting electrolyte, or by an alternative voltammetric method.

A C I M P E D A N C E S TAT I O N A RY S T U DY

The AC Impedance Stationary study is used to model a harmonic perturbation applied to a fixed center electrode potential, which in this case is fixed to the equilibrium potential of the redox couple:

(6)

where it is implied that only the real part of the complex term contributes.

The magnitude of this perturbation, , is small with respect to RT/F, so that the Butler-Volmer equation (Equation 3) can be linearized. Therefore, the system responds linearly to the perturbation, and the flux (Equation 5), the concentration profiles, and the current density are all also subject to a sinusoidal perturbation at the same frequency.

The AC Impedance Stationary study then makes the approximation that the dependent variables ci can be expressed as the sum of a stationary solution ci,0 due to the center voltage of the applied potential, and a sinusoidal perturbation ci,1 to the concentration resulting from the perturbation on the applied potential:

(7)

If ci,1 is complex, it implies that the response of the concentration profile is out-of-phase with the applied waveform.

The AC Impedance Stationary study contains two parts. First, a Stationary study is used to initialize the resting condition at the center voltage of the perturbation. The domain equation is:

n– NiiilocnF

-------------=

s ext Eeq ejt+=

ci x t ci 0 x ci 1 x ejt+=



4 | E L E

(8)

subject to ci,0 set to equal, fixed concentrations of 1 mM in bulk, and subject to the fixed center potential at the electrode surface.

(9)

Then, a Frequency Domain study solves for the perturbation, for a range of applied frequencies from 1 Hz to 1 kHz. In each case the domain equation is the frequency domain form of Fick’s 2nd Law:

(10)

solved subject to ci,1 = 0 in bulk and:

(11)

at the surface.

A Parametric Sweep is used to investigate different values of the heterogeneous rate constant of the electrode reaction, k0, from a value that is kinetically fast on the timescale of the study (0.1 cm/s) to one that it is slow on the same timescale (0.001 cm/s).


A Nyquist plot (Figure 1) is the most common means of plotting the results of an impedance experiment. It is an Argand diagram of the complex value of the impedance as a function of frequency; the real component of impedance (resistance) is plotted on the x-axis, and the imaginary component (reactance) is plotted on the y-axis.

For a fast electrochemical reaction with respect to the frequency of the electrochemical impedance study, the impedance always results from the limitation to the current due to the finite diffusivity of the redox species in the solution. It is known from theory that the real and imaginary impedances are linearly correlated in this “transport-controlled” regime (Figure 1).

For a slow electrochemical reaction with respect to the frequency, the mass transport is unimportant as the rate of electron transfer is limited by the rate of reaction at the surface: this is the “kinetically controlled” regime. This regime is characterized by a semi-circular Nyquist plot.

Di ci 0 0=

s ext Eeq=

jci 1 Di ci 1 =

s ext =



It is common to observe both regimes in a single plot, since the relevant timescale of the experiment changes with the frequency of the harmonic perturbation. At low frequency, mass transport dominates, but at high frequency (towards the bottom-left of the plot), there is a transition to kinetic control. This transition is most marked for the slowest electrochemical reaction studied, where k0 = 0.001 cm/s.

Figure 1: Nyquist plot showing the relation of real to imaginary impedance for a range of frequencies and a range of electrode kinetic heterogeneous rate constants.

In a Bode plot, either the magnitude (Figure 2) or the phase (Figure 3) of the complex impedance is plotted against frequency on the x-axis.



6 | E L E

Figure 2: Bode plot showing the magnitude of impedance as a function of frequency for a range of electrode kinetic heterogeneous rate constants.

Figure 3: Bode plot showing the phase of the impedance as a function of frequency for a range of electrode kinetic heterogeneous rate constants.



Depending on whether a reaction proceeds under kinetic or transport control at a certain frequency, the Bode plots have different characteristic appearances. Ref. 1. gives a comprehensive discussion and references to further literature concerning the analysis of electrochemical impedance spectra.

Reference


Model Library path: Electrochemistry_Module/Electroanalysis/impedance_spectroscopy



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Use an AC Impedance Stationary study type in this model. This study will set up a suitable solver sequence for your problem.

6 In the tree, select Preset Studies>AC Impedance Stationary.


cRed

cOx



8 | E L E






4 Browse to the model’s Model Library folder and double-click the file impedance_spectroscopy_parameters.txt.

G E O M E T R Y 1

Build the model geometry as a single interval of length L_el. (The L_el parameter was included in the list of parameters that was loaded from the text file.)

Interval 1 (i1)1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose

Interval.


3 In the Right endpoint edit field, type L_el.

4 Click the Build All Objects button.



Diffusion 1Now set up the physics. Start with the diffusion coefficients for cOx and cRed.

1 In the Model Builder window, under Component 1 >Electroanalysis (elan) click Diffusion 1.


3 In the DcRed edit field, type D.

4 In the DcOx edit field, type D.

Initial Values 11 In the Model Builder window, under Component 1 >Electroanalysis (elan) click Initial

Values 1.


3 In the cRed edit field, type c_bulk.



4 In the cOx edit field, type c_bulk.




4 Select the Species cRed check box.

5 In the c0,cRed edit field, type c_bulk.

6 Select the Species cOx check box.

7 In the c0,cOx edit field, type c_bulk.

Add an Electrode Surface boundary node and specify the voltage pertubation, the electrode reaction, and the double layer capacitance.



3 In the Electrode Surface settings window, click to expand the Harmonic perturbation section.

4 Locate the Harmonic Perturbation section. In the s,ext edit field, type V_app.




3 In the k0 edit field, type k0.

4 Locate the Stoichiometric Coefficients section. In the cRed edit field, type 1.

5 In the cOx edit field, type -1.

Double Layer Capacitance 11 In the Model Builder window, under Component 1 >Electroanalysis (elan) right-click

Electrode Surface 1 and choose Double Layer Capacitance.

2 In the Double Layer Capacitance settings window, locate the Double Layer Capacitance section.

3 In the Cdl edit field, type Cdl.

M E S H 1

1 In the Model Builder window, under Component 1 click Mesh 1.



10 | E L E

2 In the Mesh settings window, locate the Mesh Settings section.

3 From the Element size list, choose Extremely fine.

Modify the default mesh by specifying the maximum element size on the electrode surface as a multiple of the shortest diffusion length (highest frequency).

Size 11 Right-click Component 1 >Mesh 1 and choose Edit Physics-Induced Sequence.

2 In the Model Builder window, under Component 1 >Mesh 1 click Size 1.

3 In the Size settings window, click to expand the Element size parameters section.



6 In the associated edit field, type xdiff_min/25.

S T U D Y 1

The model is now ready for solving. Add a parametric sweep to study the effect when varying the k0 parameter value for a range of frequencies.



3 Click Add.


Step 2: Frequency-Domain, Perturbation1 In the Model Builder window, under Study 1 click Step 2: Frequency-Domain,

Perturbation.

2 In the Frequency-Domain, Perturbation settings window, locate the Study Settings section.

3 In the Frequencies edit field, type 10^range(0,0.1,3).



k0 10^range(-1,-1,-3)[cm/s]



R E S U L T S

Impedance vs Ground (elan)A Nyquist plot is created by default.

1 In the 1D Plot Group settings window, click to expand the Legend section.

2 From the Position list, choose Lower right.

Create Bode plots of the impedance and the phase angle as follows:


2 In the Model Builder window, under Results right-click 1D Plot Group 3 and choose Rename.

3 Go to the Rename 1D Plot Group dialog box and type Bode plot, absolute value of impedance in the New name edit field.

4 Click OK.



7 Locate the Legend section. From the Position list, choose Lower left.

Bode plot, absolute value of impedance1 On the 1D plot group toolbar, click Global.



4 Clear the Compute differential check box.

5 Locate the x-Axis Data section. From the Parameter list, choose Expression.

6 In the Expression edit field, type freq.

7 Select the Description check box.

8 In the associated edit field, type Frequency.


10 Click the x-Axis Log Scale button on the Graphics toolbar.

11 Click the y-Axis Log Scale button on the Graphics toolbar.


abs(elan.Zvsgrnd_els1) ohm*m^2 Impedance vs ground



12 | E L E

Bode plot, absolute value of impedance 11 In the Model Builder window, right-click Bode plot, absolute value of impedance and

choose Duplicate.

2 Right-click Bode plot, absolute value of impedance 1 and choose Rename.

3 Go to the Rename 1D Plot Group dialog box and type Bode plot, phase angle in the New name edit field.

4 Click OK.

Bode plot, phase angle1 In the Model Builder window, expand the Results>Bode plot, phase angle node, then

click Global 1.



4 Click the y-Axis Log Scale button on the Graphics toolbar.



arg(elan.Zvsgrnd_els1) rad Phase angle



Vo l t amme t r y a t a M i c r o d i s k E l e c t r o d e

Introduction

Cyclic voltammetry is a common electroanalytical technique. Since the 1980s, it has been common in voltammetry to use a microdisk electrode as the working electrode. This is a disk electrode with a radius of the order of microns, embedded in an insulator whose surface is flush with the electrode.

Figure 1: Schematic of the simulation geometry for a microdisk electrode.

These very small electrodes have advantageous mass transport properties that can maximize the measured current density, and so enable the study of electrochemical behavior that would not be observable by conventional voltammetry as performed on a large macroelectrode (See the model “Cyclic Voltammetry at a Macroelectrode in 1D”.)

This example demonstrates the use of a common approximation in which an electrode with microscale dimensions is assumed to have stationary (equilibrium) diffusion properties on the timescale of a voltammetry study. This simplifies the analysis because

Electrode

Electrolyte

Insulator

Symmetry Axis

1 | V O L T A M M E T R Y A T A M I C R O D I S K E L E C T R O D E


2 | V O L

a time-dependent model is not required. Instead, a Parametric Sweep is used to assemble a voltammogram under a quasi-static approximation.

Model Definition

The model contains a 2D axisymmetric domain surrounded by a concentric region in which Infinite Elements are used to extend the bulk solution in the model to ‘infinity’. The approximation that the bulk solution is infinitely distant is suitable if the electrochemical cell is several orders of magnitude larger than the electrode.

The z 0 axis is divided by a point at the electrode radius, re, which equals 10 m. At r re, this axis represents the working electrode (microdisk) where the electrochemical reaction takes place. At r re, this axis represents the surrounding insulator in-plane with the disk electrode.


We assume the presence of a large quantity of supporting electrolyte. This is inert salt that is added in electroanalytical experiments to increase the conductivity of the electrolyte without otherwise interfering with the reaction chemistry. Under these conditions, the resistance of the solution is sufficiently low that the electric field is negligible, and we can assume that the electrolyte potential l.

The Electroanalysis interface implements chemical species transport equations for the reactant and product species of the redox couple subject to this assumption. The domain equation is the diffusion equation (also known as Fick’s 2nd law), which describes the chemical transport of the electroactive species A and B. At steady-state, this reduces to:

(1)


At the bulk boundary ( ), we assume a uniform concentration equal to the bulk concentration for the reactant. The product has zero concentration here, as in bulk.

At the insulating (inert) surface, the normal flux of both species A and B equals zero, since this surface is impermeable and neither species reacts there.

At the electrode boundary, the reactant species A oxidizes (loses one electron) to form the product B. By convention, electrochemical reactions are written in the reductive direction:

Di ci 0=

r

T A M M E T R Y A T A M I C R O D I S K E L E C T R O D E


(2)

The stoichiometric coefficient is –1 for B, the “reactant” in the reductive direction, and +1 for A, the “product” in the reductive direction. This formulation is consistent even in examples such as this model where at certain applied potentials, the reaction proceeds favorably to convert A to B. The number of electrons transferred, n, equals one.


(3)

in which k0 is the heterogeneous rate constant of the reaction, c is the cathodic transfer coefficient, and is the overpotential at the working electrode.


(4)


The total current recorded at the disk electrode can be extracted by integrating the local current density across the electrode surface. It is not sufficient to simply multiply by the area of the electrode, because the current density may be non-uniform. An Integral Component Coupling is used to define an electrode current variable according to:

(5)

S TAT I O N A RY S T U DY

In contrast to macroelectrode voltammetry, a voltammogram recorded at a microdisk does not exhibit hysteresis. Diffusion is so fast on the timescale of the experiment that a stationary approximation is suitable. A quasi-static approximation applies when:

(6)

B e– A+


RT---------------------------- cBexp

cF–

RT----------------- –

=

n– NiiilocnF

-------------=

Iel iloc AdS=

re2

D------- RT

Fv--------«



4 | V O L

where v is the voltammetric scan rate (SI unit: V/s). The two terms in this inequality are respectively the diffusive and voltammetric timescales of the system.

Within the Stationary study, a parametric sweep is used to study the range of applied potentials achieved in the voltammogram.


The stationary concentration profile around a microdisk electrode (Figure 2) has a distinct shape. At large distances from the electrode, the concentration profile is roughly hemispherical, but close to the disk edge the flux is elevated. For fast kinetics the concentrations on the electrode surface are roughly equilibrated and so are uniform. This leads to unequal flux over the surface of the electrode—it is non-uniformly accessible.

Figure 2: Characteristic concentration profile for transport-controlled oxidation of species A at a microdisk electrode (2D cross-section).

The shape of the cyclic voltammogram (Figure 3) illustrates the relation between electrode kinetics and chemical species transport (diffusion).



Figure 3: Quasi-static (steady-state) cyclic voltammetry recorded at a microdisk electrode.

Initially, at reducing potentials, the oxidation reaction is not driven, and negligible current is drawn. As the potential moves towards the reduction potential of the redox couple (0 V), the oxidation reaction is accelerated and the current increases.

Once the oxidation reaction is fast enough that it consumes significant reactant at the electrode surface, the current becomes limited by the rate of transport of A towards the working electrode. Because the diffusion layer is equilibrated, this transport-limited current is constant in time, and independent of applied potential. The analytical Saito equation gives this limiting current as (Ref. 3):

(7)

where c is the bulk concentration of reactant.

Negative current is never observed for the “steady-state” voltammetry at a microdisk electrode, since the product species is effectively dispersed to bulk solution. Rapid diffusion on the voltammetric timescale ensures equilibration between the bulk and the electrode surface. Because of the absence of product in bulk, this equilibrium means that the reaction is always oxidative.

Ilim 4nFcDre=



6 | V O L


A refined mesh is required close to the electrode surface in order to accurately resolve the concentration profile, and hence the current. The mesh is refined further close to the singularity where the electrode and insulator boundaries meet. In the Infinite

Element domain, a Swept mesh is used.

Figure 4: Customized mesh used for the microdisk analysis.

References

1. R.G. Compton and C.E. Banks, Understanding Voltammetry, 2nd Edition, London, 2011.


3. Y. Saito, Review of Polarography (Japan), vol. 15, pp. 177-187, 1968.

Model Library path: Electrochemistry_Module/Electroanalysis/microdisk_voltammetry





N E W



1 In the Model Wizard window, click the 2D Axisymmetric button.












4 Browse to the model’s Model Library folder and double-click the file microdisk_voltammetry_parameters.txt.

G E O M E T R Y 1

Draw the model geometry as a quarter circle, and specify the electrode disk radius using a point.

Circle 11 In the Model Builder window, under Component 1 right-click Geometry 1 and choose

Circle.

2 In the Circle settings window, locate the Size and Shape section.

cRed

cOx



8 | V O L

3 In the Radius edit field, type r_max.

4 In the Sector angle edit field, type 90.

Point 11 In the Model Builder window, right-click Geometry 1 and choose Point.

2 In the Point settings window, locate the Point section.

3 In the r edit field, type re.

Circle 1Add a second circle that will be used to set up an Infinite Element domain.

Circle 21 In the Model Builder window, under Component 1>Geometry 1 right-click Circle 1 and

choose Duplicate.

2 In the Circle settings window, locate the Size and Shape section.

3 In the Radius edit field, type r_max*1.2.






Add a postprocessing variable for the total current over the electrode. (The expression will be colored orange until you have defined the integral operator in the subsequent step.)

Variables 11 In the Model Builder window, under Component 1 right-click Definitions and choose

Variables.



Integration 11 On the Definitions toolbar, click Component Couplings and choose Integration.

2 In the Integration settings window, locate the Source Selection section.



5 Locate the Advanced section. Select the Compute integral in revolved geometry check box.

Infinite Element Domain 1Add an Infinite Element domain, and assign it to the outer domain.

1 On the Definitions toolbar, click Infinite Element Domain.



Diffusion 1Now start setting up the physics. Start with the diffusion coefficients.

1 In the Model Builder window, under Component 1>Electroanalysis click Diffusion 1.


3 In the DcRed edit field, type D1.

4 In the DcOx edit field, type D2.


i_el intop1(elan.itot) Working electrode current



10 | V O

Electrode Surface 1Add an Electrode Surface condition. Set up the electrode kinetics in the Electrode Reaction subnode.

1 On the Physics toolbar, click Boundaries and choose Electrode Surface.


3 From the Boundary condition list, choose Electrode potential.

4 In the Evsref edit field, type E_appl.




2 In the Electrode Reaction settings window, locate the Equilibrium Potential section.

3 In the Eeq,ref edit field, type Ef.

4 Locate the Electrode Kinetics section. In the k0 edit field, type k0.

5 Locate the Stoichiometric Coefficients section. In the cRed edit field, type 1.

6 In the cOx edit field, type -1.

Concentration 1Specify the bulk composition at the outer boundary.




4 Select the Species cRed check box.

5 In the c0,cRed edit field, type c_bulk.

6 Select the Species cOx check box.

Initial Values 11 In the Model Builder window, under Component 1>Electroanalysis click Initial Values

1.


3 In the cRed edit field, type c_bulk.

M E S H 1

Edit the default mesh to ensure good numerical accuracy.

L T A M M E T R Y A T A M I C R O D I S K E L E C T R O D E


Size1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Edit

Physics-Induced Sequence.

2 In the Model Builder window, under Component 1>Mesh 1 click Size.


4 From the Predefined list, choose Finer.

5 Click to expand the Element size parameters section. Locate the Element Size

Parameters section. In the Maximum element growth rate edit field, type 1.1.

Free Triangular 11 In the Model Builder window, under Component 1>Mesh 1 click Free Triangular 1.

2 In the Free Triangular settings window, locate the Domain Selection section.



Size 11 Right-click Component 1>Mesh 1>Free Triangular 1 and choose Size.


3 From the Geometric entity level list, choose Point.




7 In the associated edit field, type re/100.

Mapped 11 In the Model Builder window, right-click Mesh 1 and choose Mapped.

2 In the Mapped settings window, locate the Domain Selection section.



5 Right-click Mesh 1 and choose Build All.

S T U D Y 1

The problem is now ready for solving. Simulate a voltammogram by using a Parametric Sweep for a range of applied potentials.



12 | V O



3 Click Add.



R E S U L T S

Two default concentration plots are created by default: one in 2D and one revolved 3D plot.

1D Plot Group 3Create a plot of the voltammogram as follows.


2 In the 1D Plot Group settings window, locate the Plot Settings section.

3 Select the x-axis label check box.

4 In the associated edit field, type E_appl (V).






E_appl range(E_start,E_step,E_vertex)


i_el A Working electrode current

L T A M M E T R Y A T A M I C R O D I S K E L E C T R O D E


Orang e Ba t t e r y

Introduction

This tutorial example serves as an introduction to electrochemistry modeling in COMSOL, and models the currents and the concentration of dissolved metal ions in a battery (corrosion cell) made from an orange and two metal nails.

Figure 1: Modeled geometry. Orange and two metal nails.

This type of battery is commonly used in chemistry class tuition. Instead of an orange, also lemons or potatoes can be used.

Model Definition

The citric acid and various other ions in the orange serves as electrolyte, and using nails of different metals as electrodes creates a galvanic potential over the cell.

In this example a zinc nail is used as one of the electrodes, giving rise to the following electrode reaction:

(1)

Metal nails

Orange

Zn s Zn2+ 2e-+ E0 0.82–= V

1 | O R A N G E B A T T E R Y


2 | O R A

The other nail consists of copper, and here hydrogen evolution is assumed to take place:

(2)

The model for the currents in the orange and electrodes is set up using the Secondary Current Distribution interface. The electric currents in the nails, and the electrolyte current in the orange is thereby solved for by Ohms law. One nail is grounded and the other one is set to a cell potential of 0.5 V. Butler-Volmer type expressions are used for the electrode kinetics on the surface of the nails within the orange.

The initial values for the electric potential in the electrodes is set to the values at the end terminals; ground (0 V) for the zinc electrode, and the cell potential for the copper electrode. For the electrolyte potential the initial value is set to correspond to the potential of a cell at open circuit (i.e. no activation potential). Following the definition of the overpotential:

(3)

the initial value becomes:

(4)

In an extension of the model, the diffusion and migration of the dissolved zinc ions in the orange from the zinc electrode reaction is modeled by the Transport of Diluted Species interface in a time-dependent simulation. This assumes that the zinc ion transport can be described by the Nernst-Planck equations (without an electroneutrality condition due to a supporting electrolyte). In addition, the zinc electrode kinetics are modified to be dependent on the zinc concentration, which increases in the orange as more and more zinc is dissolved.

The zinc concentration is set to 0.01 mol/m3 at the start of the simulation. All boundaries except the zinc electrode are insulated.


Figure 2 shows the potential field in the orange. The potential decreases as the current flows from the zinc electrode (left) to the upper electrode (right). The main part of the cell voltage loss is due to Ohmic losses in the electrolyte.

2H+ 2e-+ H2 g E0 0 V=

s l– E0–=

l init s E0– – 0 E0 Zn– 0– E0 Zn–= = =

N G E B A T T E R Y


The performance of the battery could probably be increased by using an electrolyte of higher conductivity (for example, a lemon instead of an orange) or by decreasing the distance between the nails.

Figure 2: Potential field in the electrolyte.



4 | O R A

Figure 3 shows the electric currents in the nails. The current increases along the z-axis as more and more current is transferred from the electrolyte to the electrodes.

Figure 3: Electric currents in the nails.

Figure 4 shows an isosurface for the 0.2 mol/m3 concentration level of zinc ions after running the battery for 5 minutes. Figure 5 shows the how far the 0.2 mol/m3 isosurface level has reached after 1 hour.

N G E B A T T E R Y


Figure 4: 0.2 mol/m3 zinc concentration isosurface after 5 minutes.



6 | O R A

Figure 5: 0.2 mol/m3 zinc concentration isosurface after 1 hour.

Finally, Figure 6 shows how the cell current evolves with time. Due to the increase of zinc ions at the zinc nail electrode, the battery current decreases for a constant cell voltage.

N G E B A T T E R Y


Figure 6: Cell current vs. time.

Model Library path: Electrochemistry_Module/Tutorial_Models/orange_battery



N E W





(siec).




8 | O R A




G E O M E T R Y 1

Start by drawing the geometry; one sphere (the orange) and two cylinders (the metal nails).

Sphere 11 On the Geometry toolbar, click Sphere.

2 In the Sphere settings window, locate the Size section.

3 In the Radius edit field, type 5e-2.

Cylinder 11 On the Geometry toolbar, click Cylinder.

2 In the Cylinder settings window, locate the Size and Shape section.


4 In the Height edit field, type 5e-2.

5 Locate the Position section. In the x edit field, type -2e-2.

6 In the z edit field, type 2e-2.

Cylinder 21 On the Geometry toolbar, click Cylinder.

2 In the Cylinder settings window, locate the Size and Shape section.


4 In the Height edit field, type 5e-2.

5 Locate the Position section. In the x edit field, type 2e-2.

6 In the z edit field, type 2e-2.


Use selections to group different parts of the geometry together for easier selection when setting up the model.



3 Right-click Component 1>Definitions>Explicit 1 and choose Rename.

N G E B A T T E R Y


4 Go to the Rename Explicit dialog box and type Orange in the New name edit field.

5 Click OK.


Enabling transparency makes it easier to find and select objects within other objects.

2 Click the Transparency button on the Graphics toolbar.



5 Go to the Rename Explicit dialog box and type Zinc nail in the New name edit field.

6 Click OK.




4 Go to the Rename Explicit dialog box and type Copper nail in the New name edit field.

5 Click OK.




4 Select Boundaries 5–7, 13, and 15 only.


6 Go to the Rename Explicit dialog box and type Zinc nail active electrode surface in the New name edit field.

7 Click OK.




4 Select Boundaries 21–23, 29, and 31 only.



10 | O R


6 Go to the Rename Explicit dialog box and type Copper nail active electrode surface in the New name edit field.

7 Click OK.





S E C O N D A R Y C U R R E N T D I S T R I B U T I O N

Now, start setting up the Secondary Current Distribution model. Start with the current conduction in the metal nails.

Electrode 11 On the Physics toolbar, click Domains and choose Electrode.

2 In the Electrode settings window, locate the Domain Selection section.

3 From the Selection list, choose Zinc nail.

4 Locate the Electrode section. From the s list, choose User defined. In the associated edit field, type 1e7.

Electrode 21 On the Physics toolbar, click Domains and choose Electrode.

2 In the Electrode settings window, locate the Domain Selection section.

3 From the Selection list, choose Copper nail.

4 Locate the Electrode section. From the s list, choose User defined. In the associated edit field, type 1e8.

Electrolyte 1The Electrolyte is the default domain type, and there is no need to select the domain in this case.

Name Expression Value Description

E_eq_Zn -0.82[V] -0.8200 V Zinc electrode reference potential

E_cell 0.5[V] 0.5000 V Cell voltage

A N G E B A T T E R Y


1 In the Model Builder window, under Component 1>Secondary Current Distribution click Electrolyte 1.

2 In the Electrolyte settings window, locate the Electrolyte section.

3 From the l list, choose User defined. In the associated edit field, type 0.01.

The following steps set up the Zn electrode and its corresponding electrode reaction:

Electrolyte-Electrode Domain Interface 11 On the Physics toolbar, click Boundaries and choose Electrolyte-Electrode Domain

Interface.

2 In the Electrolyte-Electrode Domain Interface settings window, locate the Boundary

Selection section.

3 From the Selection list, choose Zinc nail active electrode surface.

Electrode Reaction 11 In the Model Builder window, expand the Electrolyte-Electrode Domain Interface 1


2 In the Electrode Reaction settings window, locate the Equilibrium Potential section.

3 In the Eeq,ref edit field, type E_eq_Zn.

4 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Butler-Volmer.

5 In the i0 edit field, type 1e-1.

Electrolyte-Electrode Domain Interface 2The following steps set up the hydrogen evolution reaction on the copper electrode:

1 On the Physics toolbar, click Boundaries and choose Electrolyte-Electrode Domain

Interface.

2 In the Electrolyte-Electrode Domain Interface settings window, locate the Boundary

Selection section.

3 From the Selection list, choose Copper nail active electrode surface.

Electrode Reaction 11 In the Model Builder window, expand the Electrolyte-Electrode Domain Interface 2



3 In the i0 edit field, type 10.



12 | O R

Electric Ground 1Ground the Zn electrode, set the other electrode to the cell voltage.

1 On the Physics toolbar, click Boundaries and choose Electric Ground.


Electric Potential 11 On the Physics toolbar, click Boundaries and choose Electric Potential.


3 In the Electric Potential settings window, locate the Electric Potential section.

4 In the s,bnd edit field, type E_cell.

Initial Values 2Provide as good initial values as possible to shorten the computing time.

1 On the Physics toolbar, click Domains and choose Initial Values.

2 In the Initial Values settings window, locate the Domain Selection section.

3 From the Selection list, choose Copper nail.

4 Locate the Initial Values section. In the phis edit field, type E_cell.

Initial Values 11 In the Model Builder window, under Component 1>Secondary Current Distribution

click Initial Values 1.


3 In the phil edit field, type -E_eq_Zn.

S T U D Y 1

The model is now ready for solving.

On the Home toolbar, click Compute.

R E S U L T S

Electrolyte Potential (siec)Use an isosurface for visualizing the potential field in the electrolyte.






3 Go to the Rename 3D Plot Group dialog box and type Potential Isosurface in the New name edit field.

4 Click OK.

Potential Isosurface1 Right-click Results>3D Plot Group 3 and choose Isosurface.

2 In the Isosurface settings window, locate the Levels section.

3 In the Total levels edit field, type 25.



3D Plot Group 4The following steps create a plot of the norm of the current density that is suitable here to visualize the currents within the metal nails.



3 Go to the Rename 3D Plot Group dialog box and type Electrode Current in the New name edit field.

4 Click OK.

Electrode Current1 Right-click Results>3D Plot Group 4 and choose Surface.

2 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Secondary Current

Distribution>Electrode current density, norm (siec.NormIs).



C O M P O N E N T 1

Now, extend the model to investigate the concentration of the dissolved zinc ions in the orange over time. Start by adding an interface to handle the mass transport.

1 On the Home toolbar, click Add Physics.



14 | O R

A D D P H Y S I C S

1 Go to the Add Physics window.

2 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species

(chds).

3 In the Add physics window, click Add to Component.

4 On the Home toolbar, click Add Physics.

TR A N S P O R T O F D I L U T E D S P E C I E S

1 In the Model Builder window, under Component 1 click Transport of Diluted Species.

2 In the Transport of Diluted Species settings window, locate the Domain Selection section.

3 From the Selection list, choose Orange.

4 Locate the Transport Mechanisms section. Clear the Convection check box.


Diffusion and Migration1 In the Model Builder window, expand the Transport of Diluted Species node, then click

Diffusion and Migration.

2 In the Diffusion and Migration settings window, locate the Model Inputs section.

3 In the V edit field, type phil.

4 Locate the Migration in Electric Field section. In the zc edit field, type 2.

Electrode-Electrolyte Interface Coupling 1The following steps couples the electrochemical reaction currents to the ion flux at the electrode surface.

1 On the Physics toolbar, click Boundaries and choose Electrode-Electrolyte Interface

Coupling.


3 In the Electrode-Electrolyte Interface Coupling settings window, locate the Boundary

Selection section.

4 From the Selection list, choose Zinc nail active electrode surface.

Reaction Coefficients 11 In the Model Builder window, under Component 1>Transport of Diluted

Species>Electrode-Electrolyte Interface Coupling 1 click Reaction Coefficients 1.

2 In the Reaction Coefficients settings window, locate the Model Inputs section.



3 From the iloc list, choose Local current density (siec/bei1/er1).

4 Locate the Stoichiometric Coefficients section. In the nm edit field, type 2.

5 In the c edit field, type -1.

Initial Values 11 In the Model Builder window, under Component 1>Transport of Diluted Species click

Initial Values 1.


3 In the c edit field, type c_ref.


Note that c_ref is colored orange, this is because the parameter is not yet defined. Define it now.

Parameters1 In the Model Builder window, under Global Definitions click Parameters.



S E C O N D A R Y C U R R E N T D I S T R I B U T I O N

Also, modify the electrode kinetics to be dependent on the local concentration of zinc ions.

Electrode Reaction 11 In the Model Builder window, under Component 1>Secondary Current

Distribution>Electrolyte-Electrode Domain Interface 1 click Electrode Reaction 1.


3 From the Kinetics expression type list, choose Concentration dependent kinetics.

4 In the CO edit field, type c/c_ref.

R O O T

Create a new time-dependent study for the concentration simulation.

1 On the Home toolbar, click Add Study.


c_ref 0.01[mol/m^3] 0.01000 mol/m³ Reference concentration



16 | O R

A D D S T U D Y

1 Go to the Add Study window.

2 Find the Studies subsection. In the tree, select Preset Studies>Time Dependent.

3 In the Add study window, click Add Study.

4 On the Home toolbar, click Add Study.

S T U D Y 2

Step 1: Time Dependent1 In the Time Dependent settings window, locate the Study Settings section.

2 In the Times edit field, type range(0,60,3600).


R E S U L T S



3 Go to the Rename 3D Plot Group dialog box and type Concentration Isosurface in the New name edit field.

4 Click OK.



7 From the Time (s) list, choose 300.

Concentration Isosurface1 Right-click Results>3D Plot Group 9 and choose Isosurface.

2 In the Isosurface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Transport of Diluted

Species>Species c>Concentration (c).

3 Locate the Levels section. From the Entry method list, choose Levels.

4 In the Levels edit field, type 0.2.






The following steps create an animation of the zinc ion isosurface during the simulated time:

Export1 On the Results toolbar, click Player.

2 In the Player settings window, locate the Scene section.

3 From the Subject list, choose Concentration Isosurface.

At t=0 there is no concentration gradient in the orange, de-select the first time-step.

4 Locate the Animation Editing section. From the Time selection list, choose From list.

5 In the Times (s) list, select all times except 0.

6 Right-click Results>Export>Player 1 and choose Play.


Derived ValuesFinally, create a plot for how the cell current changes with time. Do this by first integrating the current over the terminal boundary, and then plot that data.

1 On the Results toolbar, click More Derived Values and choose Integration>Surface

Integration.

2 In the Surface Integration settings window, locate the Data section.



5 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Secondary Current Distribution>Electrode current density

vector>Electrode current density vector, z component (siec.Isz).

6 Right-click Results>Derived Values>Surface Integration 1 and choose Evaluate>New

Table.



18 | O R


Th i n L a y e r Ch r onoampe r ome t r y

Introduction

Chronoamperometry is a technique in electroanalysis in which current drawn at an electrode is measured after a rapid step in the applied voltage.

In a thin-layer cell, the anode and cathode are separated by a microscale distance. This means that chemical species transport across the cell is fast, so an analyte in the cell can be consumed exhaustively after only a few seconds. By integrating the current transient (chronoamperogram), the initial concentration of analyte can be determined.

If the kinetics of the electrochemical reaction are always fast, there is no need to resolve the current density as a function of applied potential. Instead, the concentration of the analyte can be assumed to be driven to zero at the working electrode surface. Under this approximation, only the chemical species transport needs to be resolved.

Model Definition

This model contains a single 1D domain of length L 60 m, which is the thickness of the thin layer. Transport in plane with the anode and cathode is ignored; only normal transport is considered, which is assumed to be uniform across the cell.


The transport of the analyte obeys the diffusion equation (Fick’s 2nd Law):

(1)

The solution is assumed to be static (“quiescent”) so there is no mass transport by convection. A supporting electrolyte is present in high concentration, so the electric field is also taken to be zero. We do not model the product species as its concentration does not influence the current density.

B O U N D A RY E Q U AT I O N S

A high overpotential is applied so that the analyte undergoes a very fast electrochemical reaction at the working electrode surface (x 0). To model this, the analyte concentration here is rapidly stepped to zero. The facing surface (x L), is impermeable to the analyte—no flux is passed. We assume the counter reaction of the

ct

----- D c =

1 | T H I N L A Y E R C H R O N O A M P E R O M E T R Y


2 | T H I

electrochemical cell to either take place at a physically separate counter electrode, or to involve a distinct chemical species, present in excess, which we ignore in this model.

T I M E - D E P E N D E N T S T U DY

The Einstein equation gives the time for the mean position of a diffusion layer to cross a distance L, as a function of the diffusion coefficient D:

(2)

In the thin layer, the Einstein time is 0.9 s. After a few Einstein times, the analyte will have reacted to near exhaustion, and so the duration of the simulation is set to 5 s.


The concentration profiles through time demonstrate the growth of the diffusion layer across the cell (Figure 1).

Figure 1: Concentration profiles of the analyte across the thickness of the cell, as the experiment proceeds (top-left to bottom-right)

Once the diffusion layer encounters the outer boundary of the cell, the concentration here begins to diminish as the continuing electrochemical reaction exhausts the available analyte.

t L2

4D--------=

N L A Y E R C H R O N O A M P E R O M E T R Y


As the diffusion layer expands, the flux at the working electrode becomes smaller. Correspondingly the current also decreases (Figure 2).

Figure 2: Measured chronoamperogram for the thin-layer cell

From transport theory, the chronoamperometric current for an infinite expanse of bulk solution falls off inversely proportionally to the square root of time, as given by the Cottrell equation, where i is the current density, n is the number of electrons transferred per molecule of analyte, c is the bulk concentration of analyte and D is its diffusion coefficient:

(3)i nFc Dt-----=



4 | T H I

Figure 3: Simulated chronoamperogram compared on a logarithmic scale to the Cottrell equation for chronoamperometry with unlimited available analyte. The deviation at long times is caused by the finite quantity of analyte the cell.

By comparing the simulated results with the Cottrell equation, plotted on a logarithmic scale (Figure 3), good agreement is observed until roughly t 1 s. At this time—which is approximately the Einstein time noted above—the diffusion layer encounters the wall of the cell.

After this point, the current diminishes more quickly due to the exhaustion of available electroactive material for reaction. Under these conditions, the Cottrell equation no longer applies—the simulated current deviates negatively.

By integrating the concentration across the cell, we can calculate the proportion of the initial amount of analyte that has been consumed (Figure 4). After 5 seconds, 99% of the analyte has undergone an electrochemical reaction.



Figure 4: Proportion of the initial quantity of analyte that is consumed through the experiment

References

1. R.G. Compton and C.E. Banks, Understanding Voltammetry, 2nd ed., London, 2011.

2. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Second ed., Hoboken, 2001.

3. F.G. Cottrell, Zeitschrift für Physikalische Chemie, vol. 42, pp. 385–431, 1903.

Model Library path: Electrochemistry_Module/Electroanalysis/thin_layer_chronoamperometry





6 | T H I

N E W









7 In the tree, select Preset Studies>Time Dependent.



Add the model parameters from a text file.




4 Browse to the model’s Model Library folder and double-click the file thin_layer_chronoamperometry_parameters.txt.

G E O M E T R Y 1

Create the model geometry as a single interval.

Interval 1 (i1)1 In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and

choose Interval.




c




Add a smoothed step function that will be used to step the concentration at the electrode from initial conditions to zero as a continuous function of time.

Step 1 (step1)1 On the Home toolbar, click Functions and choose Global>Step.

2 In the Step settings window, locate the Parameters section.

3 In the Location edit field, type 0.5.

4 Click to expand the Smoothing section. In the Size of transition zone edit field, type 1.

Variables 1Add some variables that will be used during postprocessing for comparing the simulated current with the Cottrell equation.

1 In the Model Builder window, right-click Definitions and choose Variables.



E L E C T R O A N A L Y S I S ( E L A N )

Diffusion 1Now start setting up the physics. Start with the domain settings for the diffusion coefficient and the initial concentration.

1 In the Model Builder window, under Component 1 (comp1)>Electroanalysis (elan) click Diffusion 1.


3 In the Dc edit field, type D.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Electroanalysis (elan) click

Initial Values 1.


3 In the c edit field, type c_bulk.


i_el -F_const*A_el*elan.tfluxx_c

A Total Current

i_Cottrell F_const*A_el*c_bulk*sqrt(D/(pi*t+eps))

A Cottrell equation current



8 | T H I

Concentration 1Set up the Concentration condition on the boundary using the step function defined above.




4 Select the Species c check box.

5 In the c0,c edit field, type c_bulk*(1-step1(t/t_rise)).

M E S H 1

Refine the default mesh.

Size1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and

choose Edit Physics-Induced Sequence.

2 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size.



Size 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Edge 1

and choose Size.






7 In the associated edit field, type x_step/5.

Edge 11 Right-click Edge 1 and choose Build Selected.



Your finished mesh should now look like this:

Figure 5: Mesh for the thin layer chronoamperometry study.

S T U D Y 1

Step 1: Time DependentYour model is now ready for solving. Solve for a time range of 5 s.

1 In the Model Builder window, under Study 1 click Step 1: Time Dependent.

2 In the Time Dependent settings window, locate the Study Settings section.

3 In the Times edit field, type range(0,0.1,5).


R E S U L T S

Concentration (elan)The first default plot group shows the concentration.

1D Plot Group 2Create the chronoamperogram as follows:


2 On the 1D plot group toolbar, click Point Graph.



10 | T H


4 In the Point Graph settings window, locate the y-Axis Data section.

5 In the Expression edit field, type i_el.


7 In the 1D Plot Group settings window, click to expand the Axis section.

8 Select the Manual axis limits check box.

9 In the x minimum edit field, type 0.1.

10 In the x maximum edit field, type 5.

11 In the y minimum edit field, type 0.

12 In the y maximum edit field, type 5e-6.


14 In the Model Builder window, collapse the 1D Plot Group 2 node.

15 Right-click 1D Plot Group 2 and choose Rename.

16 Go to the Rename 1D Plot Group dialog box and type Chronoamperogram in the New

name edit field.

17 Click OK.

Chronoamperogram 1Now, duplicate the chronoamperogram and compare, in log scale, the simulated curve to the Cottrell equation.

1 Right-click 1D Plot Group 2 and choose Duplicate.

2 In the Model Builder window, under Results right-click Chronoamperogram 1 and choose Rename.

3 Go to the Rename 1D Plot Group dialog box and type Comparison to Cottrell equation in the New name edit field.

4 Click OK.

Comparison to Cottrell equation1 On the 1D plot group toolbar, click Global.




i_Cottrell A Cottrell equation current

I N L A Y E R C H R O N O A M P E R O M E T R Y


4 In the Model Builder window, click Comparison to Cottrell equation.

5 In the 1D Plot Group settings window, locate the Axis section.

6 Clear the Manual axis limits check box.

7 Select the x-axis log scale check box.

8 Select the y-axis log scale check box.


10 Click to expand the Axis section. Select the Manual axis limits check box.

11 In the x minimum edit field, type 1e-5.


Derived ValuesFinally, plot the extent of reaction by calculating the average amount of reacted concentration in a table, and then plotting the table data.

1 On the Results toolbar, click More Derived Values and choose Average>Line Average.


3 In the Line Average settings window, locate the Expression section.

4 In the Expression edit field, type (c_bulk-c)/c_bulk.

5 Select the Description check box.

6 In the associated edit field, type Extent of reaction.

7 Right-click Results>Derived Values>Line Average 1 and choose Evaluate>New Table.

TA B L E

In the Table window, click Table Graph.

R E S U L T S

1D Plot Group 41 In the Model Builder window, right-click 1D Plot Group 4 and choose Rename.

2 Go to the Rename 1D Plot Group dialog box and type Extent of reaction in the New name edit field.

3 Click OK.



12 | T H
I N L A Y E R C H R O N O A M P E R O M E T R Y


E l e c t r o c h em i c a l T r e a tmen t o f T umo r s

Introduction

The electrochemical treatment of tumors implies that diseased tissue is treated with direct current through the use of metallic electrodes inserted in the tumor. When tissue is electrolyzed, two competing reactions take place at the anode: oxygen evolution and chlorine production. The oxygen-evolution reaction also produces H+ ions, which lower the pH close to the anode. It should be stressed that chlorine production also leads to lowered pH through the hydrolysis of chlorine. One effect of low pH is the permanent destruction of hemoglobin in the tissue, which results in destruction of tumor tissue.

(1)

One challenge in developing this method of cancer treatment is in predicting the doses required for tumor destruction. One possible tool for dose planning is by modeling the reactions that take place close to the electrodes.

This example presents a first simple model for the development of dose-planning methods. More advanced models for dose planning, including secondary effects of chlorine, are found in Ref. 1, which also presents and solves models for the cathode.

Model Definition

This model uses the Tertiary Current Distribution, Nernst-Planck interface to predict the transport and reaction in the electrolysis of tumor tissue in a liver. A needle electrode is placed in the tumor, and transport is assumed to take place radially to and from this electrode. Because you can assume rotational symmetry, the computational domain reduces to a line (ra, rr) where ra is 1 mm and rr is 6 cm (see Figure 1).

The species you consider in the model are the protons, chloride, and sodium. At the surface of the anode you account for the chlorine and oxygen evolution reactions; see

2Cl- Cl2 g 2e-+=

2H2O 4H+ O2 g 4e-+ +=

1 | E L E C T R O C H E M I C A L TR E A T M E N T O F TU M O R S


2 | E L E

Equation 1.

Figure 1: Diagram of the cylindrical modeling domain inside a tumor.

This simplified model considers only a 1D model of the transport between two points, that is, between the two electrodes. The material balance for the species i is given by

where ci is the concentration (mol/m3), Di give the diffusivities (m2/s), zi equals the charge, umi represents the mobility ((mol·m2)/(J· s)), and Ri is the production term for species i (mol/(m3·s)), F denotes Faraday’s constant (C/mol), and l is the electrolyte potential (V). The mobility, um i, can be expressed in terms of Di, R, and T as

The conservation of electric charge is obtained through the divergence of the current density:

Anode

Computational domain

cit------- Dici– ziumiFcil– + Ri=

umiDiRT--------=

F zi2 Dici– ziumiFcil–

i 1=

N

0=

C T R O C H E M I C A L TR E A T M E N T O F TU M O R S


At the electrode surface (r ra) you use the Electrolyte-Electrode Boundary Interface boundary node to specify the electrode reactions and the resulting fluxes for the ionic species that are included in the electrode reactions, H+ and Cl-. For the inert ionic species, Na+, the transport through the electrode surface equals zero. The expressions for molar fluxes at the boundary are based on the electrode reaction currents according to

where Ni is the flux, ij represents the stoichiometric coefficient for the ionic species i in reaction j, and nj is the number of electrons in reaction j.

You can express the current densities for the two reactions using the Electrode Reaction nodes with Concentration Dependent Kinetics expressions. Introducing dimensionless pressure, P = p/pb, and concentration, C = c/cb, (where b denotes the reference concentration), the current density for the oxygen evolution is

where jI0, is the exchange current density (A/m2) and I is the overpotential for the oxygen evolution reaction, defined as

(2)

where Eeq,I (V) is the equilibrium potential for the oxygen evolution reaction.

The chlorine evolution reaction is similarly given by the expression

Using the input values nInI1, H,I1, and Cl,II1, gives the fluxes at the electrode surface:

Ni nij jjnj F-----------=

jI jI 0, e

FI

2RT------------–

PO2 1 4 CH + e

FI

2RT------------

–

=

I s l– Eeq I–=

jII jII 0, CCl– e

FII

2RT------------–

PCl2 1 2 e

FII

2RT------------

–

=

NH njIF----=

NCl njIIF------–=



4 | E L E

At the exterior boundary, assume the concentration is constant, cici0, and set the potential to

where t denotes time.

The initial concentration is constant: ci = ci0. You obtain the initial potential profile from the solution of the domain equations and boundary conditions at t0, yielding

where Vra0 is the potential satisfying jIjIIj0 and 0 is the conductivity at t0.

The input data is listed in this table:

TABLE 1: INPUT DATA

PARAMETER VALUE DESCRIPTION

DNa 1.33·10-9 m2/s Diffusion coefficient, Na+

DH 9.31·10-9 m2/s Diffusion coefficient, H+

DCl 2.03·10-9 m2/s Diffusion coefficient, Cl-

cNa,0 0.16 mol/liter Inlet concentration, Na+

cH,0 10-7 mol/liter Inlet concentration, H+

cCl,0 0.16 mol/liter Inlet concentration, Cl-

j0 100 A/m2 Initial current density

jI,0 10-6 A/m2 Exchange current density, reaction I

jII,0 10 A/m2 Exchange current density, reaction II

Vra,0 -1.4787 V Initial anode potential

Eeq,I 1.23 V Equilibrium potential, reaction I

Eeq,II 1.36 V Equilibrium potential, reaction II

T 298 K Temperature

pO2 1 atm

pCl2 1 atm

V0 t 0.4977 0.2567 100 t1 s------+

ln+ V–=

V t 0= Vra0jra0ra0

----------------rar-----log+=




The plot in Figure 2 shows the pH for different time steps. You can see that values below pH 2 are reached somewhere between 1800 and 2400 s. A closer examination reveals that it occur after 2000 s. At this pH, tumor destruction starts to occur very rapidly according to the experimental and theoretical findings in Ref. 1.

Figure 2: pH-profiles at different time steps during the treatment.



6 | E L E

The corresponding H+ profile in Figure 3 shows that the concentration maximum is not at the anode surface.

Figure 3: Proton concentration in the domain at different time steps.

This result arises because the current density is not constant over time. At high current densities, large amounts of protons are produced and this front moves inwards in the domain as the current density is lowered.



The corresponding plot for chloride (Figure 4) shows a continuous decrease of chloride concentration close to the anode surface. This in turn decreases the production of chlorine and increases oxygen evolution.

Figure 4: Chloride concentration at different time steps.

The current-density plot in Figure 5 shows that the total current decreases rapidly as the concentration overvoltage for chlorine formation increases, due to lowered



8 | E L E

chloride concentration at the anode surface. The potential is then increased, which results in an increase in total current through increased oxygen evolution.

Figure 5: Total current density and current density for the competing reactions on the anode surface. Oxygen evolution is the lowest graph.

Reference

1. E. Nilsson, Modelling of the Electrochemical Treatment of Tumors, PhD. thesis, Dept. Chemical Engineering and Technology, Royal Inst. of Technology, Stockholm, Sweden, 2000.

Model Library path: Electrochemistry_Module/Electrochemical_Engineering/tumor



N E W





1 In the Model Wizard window, click the 1D Axisymmetric button.

2 In the Select physics tree, select Electrochemistry>Tertiary Current Distribution,

Nernst-Planck (tcdee).





7 In the tree, select Preset Studies>Time Dependent.



Start by loading the model parameters and variables from text files.




4 Browse to the model’s Model Library folder and double-click the file tumor_parameters.txt.


Variables 11 In the Model Builder window, under Component 1 (comp1) right-click Definitions and

choose Variables.



4 Browse to the model’s Model Library folder and double-click the file tumor_variables.txt.

Na

H

Cl



10 | E L E

G E O M E T R Y 1

Create the geometry as a single interval.

Interval 1 (i1)1 In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and

choose Interval.


3 In the Left endpoint edit field, type r_a.

4 In the Right endpoint edit field, type r_ext.



TE R T I A R Y C U R R E N T D I S T R I B U T I O N , N E R N S T - P L A N C K ( T C D E E )

Electrolyte 1Now start with the physics, begin with the electrolyte settings.

1 In the Model Builder window, under Component 1 (comp1)>Tertiary Current

Distribution, Nernst-Planck (tcdee) click Electrolyte 1.

2 In the Electrolyte settings window, locate the Model Inputs section.


4 Locate the Diffusion section. In the DNa edit field, type D_Na.

5 In the DH edit field, type D_H.

6 In the DCl edit field, type D_Cl.

7 Locate the Migration in Electric Field section. In the zNa edit field, type z_Na.

8 In the zH edit field, type z_H.

9 In the zCl edit field, type z_Cl.

Electrolyte Potential 1Now set the potential of the exterior end, and then the concentration.

1 On the Physics toolbar, click Boundaries and choose Electrolyte Potential.



4 In the l,bnd edit field, type V0.






4 Select the Species H check box.

5 In the c0,H edit field, type H0.

6 Select the Species Cl check box.

7 In the c0,Cl edit field, type Cl0.

Electrolyte-Electrode Boundary Interface 1Now set up the anode, and the anode reactions.

1 On the Physics toolbar, click Boundaries and choose Electrolyte-Electrode Boundary

Interface.






4 Locate the Equilibrium Potential section. In the Eeq,ref edit field, type E_eqI.

5 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Concentration dependent kinetics.

6 In the i0 edit field, type j_I0.

7 In the CO edit field, type H/H0.

8 Locate the Stoichiometric Coefficients section. In the H edit field, type -1.

Electrode Reaction 21 In the Model Builder window, under Component 1 (comp1)>Tertiary Current

Distribution, Nernst-Planck (tcdee) right-click Electrolyte-Electrode Boundary Interface

1 and choose Electrode Reaction.



4 Locate the Equilibrium Potential section. In the Eeq,ref edit field, type E_eqII.

5 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Concentration dependent kinetics.

6 In the i0 edit field, type j_II0.



12 | E L E

7 In the CR edit field, type Cl/Cl0.

8 Locate the Stoichiometric Coefficients section. In the Cl edit field, type 1.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Tertiary Current

Distribution, Nernst-Planck (tcdee) click Initial Values 1.


3 In the H edit field, type H0.

4 In the Cl edit field, type Cl0.

5 In the phil edit field, type V_init.

M E S H 1

Create a user defined mesh with a very fine resolution close to the anode.

1 In the Model Builder window, under Component 1 (comp1) click Mesh 1.

2 In the Mesh settings window, locate the Mesh Settings section.

3 From the Sequence type list, choose User-controlled mesh.

Size1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size.



Size 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Edge 1

and choose Size.






7 In the associated edit field, type 1e-8.

8 Select the Maximum element growth rate check box.

9 In the associated edit field, type 1.1.



S T U D Y 1

Step 1: Time DependentModify the default solver to store all steps taken by the solver, and solve the problem.

1 In the Model Builder window, under Study 1 click Step 1: Time Dependent.

2 In the Time Dependent settings window, locate the Study Settings section.

3 In the Times edit field, type 0 3600.

Solver 11 On the Study toolbar, click Show Default Solver.

2 In the Model Builder window, expand the Solver 1 node, then click Time-Dependent

Solver 1.

3 In the Time-Dependent Solver settings window, click to expand the Output section.

4 From the Times to store list, choose Steps taken by solver.


R E S U L T S

Concentration (tcdee)The following instructions show how to create the plots shown in Figure 2 to Figure 5.


2 From the Time selection list, choose Interpolated.

3 In the Times (s) edit field, type range(0,600,3600).

4 In the Model Builder window, expand the Concentration (tcdee) node, then click Line

Graph 1.

5 In the Line Graph settings window, click to expand the Legends section.

6 Select the Show legends check box.

7 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle.

8 Locate the y-Axis Data section. In the Expression edit field, type H.


Compare the plot with that shown in Figure 3.

10 In the Model Builder window, right-click Concentration (tcdee) and choose Rename.



14 | E L E

11 Go to the Rename 1D Plot Group dialog box and type Proton concentration in the New name edit field.

12 Click OK.

Proton concentration 11 Right-click Concentration (tcdee) and choose Duplicate.

2 In the Model Builder window, expand the Proton concentration 1 node, then click Line

Graph 1.


4 In the Expression edit field, type Cl.


Compare with the plot in Figure 4.

6 In the Model Builder window, right-click Proton concentration 1 and choose Rename.

7 Go to the Rename 1D Plot Group dialog box and type Chloride concentration in the New name edit field.

8 Click OK.

Chloride concentration 11 Right-click Proton concentration 1 and choose Duplicate.

2 In the Model Builder window, expand the Chloride concentration 1 node, then click Line Graph 1.


4 In the Expression edit field, type pH.


The plot should look like that in Figure 2.

6 In the Model Builder window, right-click Chloride concentration 1 and choose Rename.

7 Go to the Rename 1D Plot Group dialog box and type pH in the New name edit field.

8 Click OK.


Finally, reproduce the plot shown in Figure 5.

2 On the 1D plot group toolbar, click Point Graph.


4 In the Point Graph settings window, locate the y-axis data section.



5 Click Local current density (tcdee.iloc_er1) in the upper-right corner of the section. Click to expand the Legends section. Select the Show legends check box.

6 From the Legends list, choose Manual.


8 Right-click Results>1D Plot Group 6>Point Graph 1 and choose Duplicate.

9 In the Point Graph settings window, locate the y-Axis Data section.

10 In the Expression edit field, type tcdee.iloc_er2.

11 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Line list, choose Dash-dot.



14 Right-click Results>1D Plot Group 6>Point Graph 2 and choose Duplicate.

15 In the Point Graph settings window, locate the y-axis data section.

16 Click Total interface current density (tcdee.itot) in the upper-right corner of the section. Locate the Coloring and Style section. Find the Line markers subsection. From the Line list, choose Dashed.



19 In the Model Builder window, right-click 1D Plot Group 6 and choose Rename.

20 Go to the Rename 1D Plot Group dialog box and type Electrode reaction current densities in the New name edit field.

21 Click OK.

Legends

Oxygen evolution

Legends

Chlorine evolution

Legends

Total electrode current density



16 | E L E


Wi r e E l e c t r o d e

One of the most important aspects in the design of electrochemical cells is the current density distributions in the electrolyte and electrodes. Non-uniform current density distributions can be detrimental for the operation of electrochemical processes. In many cases the parts of an electrode that are subjected to high current density degrade at a faster rate. Knowledge of the current density distribution is also desired to optimize the utilization of the electrocatalysts, because these are often made of expensive noble metals. Non-uniform deposition and consumption, as well as unnecessarily high overvoltages, with resulting energy losses and possibly unwanted side-reactions, may be other effects that one would like to minimize.

This example models the primary, secondary, and tertiary current density distributions (Ref. 1) of an arbitrary electrochemical cell. It successively goes through the different classes of current density distributions so as to also show how complexity should be gradually introduced when modeling electrochemical cells.

Introduction

The same geometry is considered in all three cases: a wire electrode structure is placed between two flat electrode surfaces, and in the open volume between the wire and the flat surfaces electrolyte is allowed to flow; see Figure 1. The electrochemical cell can be seen as a unit cell of a larger wire-mesh electrode—an electrochemical cell setup common for many large-scale industrial processes.

1 | W I R E E L E C T R O D E


2 | W I R

.

Figure 1: Modeled electrochemical cell. Wire electrode (anode) between two flat electrodes (cathodes). Flow inlet to the left, outlet to the right. The top and bottom flat surfaces are inert.

Model Definition

P R I M A RY C U R R E N T D I S T R I B U T I O N

Figure 1 shows the investigated geometry. Firstly, this example considers primary current density distribution. This is the situation where the mixing of electrolyte is vigorous or where concentration gradients are small, so that ionic migration is the dominating transport mechanism. The general mass balance in the electrolyte, assuming steady-state conditions and that no homogeneous reactions occur, can be given by

(1)

where Ni is the flux of species i (mol·m2/s), which in turn is governed by:

(2)

where ci represents the concentration of the ion i (mol/m3), zi its valence, Di its diffusivity (m2/s), mi its mobility (mol·m2(s·V·A)), F denotes the Faraday constant

inlet

outletwire electrode

flat electrode

flat electrode

(anode)

(cathode)

(cathode)

Ni 0=

Dici– zimiFcil– ciu+ Ni=

E E L E C T R O D E


(As/mol), the ionic potential, and u the velocity vector (m/s). The components operated upon by the above transport equation are often described as the diffusion, migration, and convection transport mechanisms. The net current density can be described through:

(3)

where i is the current density vector (A/m2). Combining the three above equations, while assuming electroneutrality (which removes the convection term) and negligible concentration gradients (which removes diffusion) leaves:

. (4)

Current density is conserved throughout:

(5)

so that by combining the valence, ionic mobility, constant concentration and the Faraday constant to a representative conductivity, (1/(W·m2)), Equation 5 becomes:

. (6)

This final equation is basically Ohm’s Law.

The boundary conditions for the case of primary current density distribution assume that the kinetics on the electrode surfaces are fast, which allow the assumption of constant potential on these surfaces (all other boundaries are insulated). The solid phase (electronic conductor) potential on the cathode, (V), is a convenient choice of reference potential in the system:

. (7)

The electrode potential equals the difference between the potential of solid phase in the electrode, , and the potential in the adjacent electrolyte, :

. (8)

In the absence of kinetic losses, the cathode potential, Ec, equals the equilibrium potential, Eeq,c:

(9)

i

i F ziNi–=

i F zi2

– miFcil–=

i 0=

l– 0=

s c

s c 0=

e i

Eelectrode s l–=

Eeq c s c l c– l c–= =



4 | W I R

which sets the boundary condition for the cathode.

The potential difference over the whole cell, Ecell, is defined as the potential difference between the solid phases of the two electrodes

. (10)

In this way the boundary condition for the ionic potential at the anode can be set via:

. (11)

S E C O N D A RY C U R R E N T D I S T R I B U T I O N

Secondary current distribution takes into account the kinetics at the electrodes. Mixing is supposed to be good and the electroneutrality condition still relevant so that Ohm’s Law remains a good description for the equations in the domain. Yet the electrochemical reactions are no longer fast enough that a constant potential can be applied at the electrodes. The properties of the chemical species and their ability to react at the surface, i.e. the reaction driving forces (overvoltages), need to be considered.

In this model, the expressions for the local current density, i (A/m2), is based on the Butler-Volmer equation (Ref. 2) for a single electron reaction. For the secondary current distribution case (that is, without concentration dependence) it reads:

(12)

here T is the temperature and R is the gas constant (J/(K·mol). i0, the exchange current density, (A/m2), and the symmetry factor, are reaction and electrode dependent and are therefore different for each electrode. The overpotential, , is the difference between the electrode potential and the equilibrium potential for the electrode reaction, defined in the following way:

. (13)

This results in the following expressions for the overpotentials for the cathode and anode, respectively:

, (14)

. (15)

Ecell s a s c– s a= =

Eeq a s a l a– Ecell l a–= =

iloc i0 1 – F RT exp F RT exp– =

Eelectrode Eeq–=

c l c– Eeq c–=

a Ecell l a– Eeq a–=

E E L E C T R O D E


T E R T I A R Y C U R R E N T D I S T R I B U T I O N

In tertiary current density distribution, mass transport through diffusion, convection, and migration has to be considered (that is, all components of Equation 2).

For the net ionic charge transport the assumption for this model still is electroneutrality and a supporting electrolyte with negligible concentration gradients, which means that the potential distribution in the electrolyte can be described through Ohm’s Law (Equation 6)

To introduce a mass transport dependence in this model the species being oxidized at the anode now has mass transport limitations and its localized concentration, c (mol/m3), affects the electrode kinetics. The anodic branch of the Butler-Volmer expression at the anode therefore gets a concentration dependence, and the expression now reads:

. (16)

Here c0 (mol/m3) denotes a reference concentration (equal to the inlet concentration). Equation 16 is applied to the wire (anode) electrode, while the cathodes keep the expression for the local current density from the secondary current distribution model.

Also a momentum balance is introduced to describe the convection. In this case, the assumption is a stationary laminar incompressible flow, using the Navier-Stokes equation:

(17)

where is the dynamic viscosity (Ns/m2), density (kg/m3) and p pressure (Pa).

No-slip boundary conditions are applied to the electrode surfaces, and slip boundary conditions to the top and bottom to account for the periodically repeating unit cell in this spatial direction. At the inlet, a laminar inflow with a fixed mean velocity is specified, whereas a pressure condition specifying a zero reference pressure is used at the outlet.

Finally, Equation 2 accounts for the mass transport of the reacting species:

(18)

No-flux boundary conditions are applied for all boundaries except for the inlet, outlet and the anode. At the inlet, a fixed concentration is specified. Outflow conditions are

ia i0cc0----- 1 – F RT exp F RT exp– =

u u T+ u u p+ +– 0=

u 0=

Dc– zmFc– cu+ 0=



6 | W I R

applied for the outlet. Faraday’s law is used to specify the net molar flux at the anode where the species is consumed:

. (19)


Figure 2 shows the different polarization plots that results from using a parametric solver to solve for all the three cases of current distribution. The total current decreases as potential losses due to kinetics and mass transport are introduced in the model. The following sections cover each case more in detail.

Figure 2: Polarization plots comparing the three cases of current distribution.

P R I M A RY C U R R E N T D I S T R I B U T I O N

Figure 3 shows the potential distribution in the electrolyte and current density distribution at the anode at a cell voltage of 1.45 V. The current density distribution is highest at the corners of the wires and close to zero at the central parts of the wire structure.

Naia

F----–=

E E L E C T R O D E


Figure 3: Primary current distribution, Ecell=1.45 V. Potential distribution in the electrolyte (top) and current density on the anode (bottom).

S E C O N DA RY C U R R E N T D I S T R I B U T I O N

Figure 4 shows the plots for the secondary current distribution. A higher cell voltage is chosen reach a total cell current comparable to Figure 3. Compared to the primary



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current distribution the secondary current distribution is smoother, this is due to the effect that a high local current density induces local over potential losses on the electrode surface.

Figure 4: Secondary current distribution, Ecell=1.65 V. Potential distribution in the electrolyte (left) and current density on the anode (right).

E E L E C T R O D E


T E R T I A R Y C U R R E N T D S I T R I B U T I O N

Figure 5 shows the flow velocity magnitude of the flow and the concentration of the reactant at 1.8 V. The convective flow is close to zero between the wires, and this results in a depletion zone with low concentration in these parts in the cell.

Figure 5: Flow field (top: slice plot, right: arrows) and concentration profile (bottom: slices and anode surface) at 1.8 V.



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Figure 6 shows the resulting potential and current density distribution. The low concentration between the wires now impacts severely on the smoothness of the current distribution.

Figure 6: Tertiary current distribution, Ecell=1.8 V. Potential distribution in the electrolyte (top) and current density on the anode (bottom).

R E E L E C T R O D E



You set up the model using the following physics interfaces:

• Primary Current Distribution for modeling the electrolyte potential, governed by Ohm’s Law (Equation 6). The secondary and tertiary current distributions are modeled by changing the current distribution type of the interface to Secondary.

• Transport of Diluted Species for the mass transport of the reacting species (Equation 18).

• Laminar Flow for the momentum balance to describe the convection (Equation 17).

References

1. J.S. Newman, Electrochemical Systems, 2nd ed., Prentice Hall, NJ, 1990.

2. J. O’M. Bockris and A.K.N. Reddy, Modern Electrochemistry, Plenum Press, NY, 1970.

Model Library path: Electrochemistry_Module/Electrochemical_Engineering/wire_electrode



N E W




2 In the Select physics tree, select Electrochemistry>Primary Current Distribution (siec).







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G E O M E T R Y 1

Import the model geometry from a file.

Import 1 (imp1)1 On the Home toolbar, click Import.

2 In the Import settings window, locate the Import section.

3 Click the Browse button.

4 Browse to the model’s Model Library folder and double-click the file wire_electrode.mphbin.



Create selections for the anode and the cathode in the geometry. They will be used later when setting up the physics.






6 Go to the Rename Explicit dialog box and type Cathodes in the New name edit field.

7 Click OK.




4 Select the All boundaries check box.

5 Select Boundaries 6–37 only.

This selection is easiest to achieve by selecting all boundaries (the 'All boundaries' check box), followed by deselecting all exterior surfaces.


7 Go to the Rename Explicit dialog box and type Anode in the New name edit field.

8 Click OK.



Integration 1 (intop1)Create some component couplings to be used when analyzing the results.

1 On the Definitions toolbar, click Component Couplings and choose Integration.

2 In the Integration settings window, locate the Operator Name section.

3 In the Operator name edit field, type anode_int.

4 Locate the Source Selection section. From the Geometric entity level list, choose Boundary.

5 From the Selection list, choose Anode.

Average 1 (aveop1)1 On the Definitions toolbar, click Component Couplings and choose Average.

2 In the Average settings window, locate the Operator Name section.

3 In the Operator name edit field, type anode_avg.

4 Locate the Source Selection section. From the Geometric entity level list, choose Boundary.



Now start defining the physics for the primary current distribution simulation. Begin with the model parameters.




M A T E R I A L S

Add water from the material library. Modify the material by adding the conductivity value.

1 On the Home toolbar, click Add Material.


Ecell 1.3[V] 1.300 V Cell voltage

Eeq_c 0[V] 0 V Cathode equilibrium potential

Eeq_a 1.2[V] 1.200 V Anode equilibrium potential



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A D D M A T E R I A L

1 Go to the Add Material window.

2 In the tree, select Built-In>Water, liquid.

3 In the Add material window, click Add to Component.

M A T E R I A L S

Water, liquid (mat1)1 In the Model Builder window, under Component 1 (comp1)>Materials click Water,

liquid (mat1).



4 Right-click Component 1 (comp1)>Materials>Water, liquid (mat1) and choose Rename.

5 Go to the Rename Material dialog box and type Electrolyte in the New name edit field.

6 Click OK.

P R I M A R Y C U R R E N T D I S T R I B U T I O N ( S I E C )

Electrolyte 1Now start setting up the physics. Only the electrolyte potential boundary values need to be set for the primary current distribution.


Interface.

2 In the Electrolyte-Electrode Boundary Interface settings window, locate the Boundary

Selection section.

3 From the Selection list, choose Cathodes.




Electrolyte conductivity sigmal 10[S/m]

S/m Electrolyte conductivity





4 Locate the Equilibrium Potential section. In the Eeq,ref edit field, type Eeq_c.


Interface.

2 In the Electrolyte-Electrode Boundary Interface settings window, locate the Boundary

Selection section.


4 Locate the Boundary Condition section. In the s,ext edit field, type Ecell.





4 Locate the Equilibrium Potential section. In the Eeq,ref edit field, type Eeq_a.

Initial Values 1Also, provide initial values for the electrolyte potential.

1 In the Model Builder window, under Component 1 (comp1)>Primary Current

Distribution (siec) click Initial Values 1.


3 In the phil edit field, type (Ecell-Eeq_a-Eeq_c)/2.

M E S H 1

The following steps creates a mesh with boundary layers adjacent to the anode and cathode surfaces. This is a convenient way of increasing the number of mesh elements close to a surface of special interest.

Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and

choose Boundary Layers.

2 In the Boundary Layer Properties settings window, locate the Boundary Selection section.




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4 Locate the Boundary Layer Properties section. In the Number of boundary layers edit field, type 6.

5 In the Boundary layer stretching factor edit field, type 1.3.

6 From the Thickness of first layer list, choose Manual.

7 In the Thickness edit field, type 2e-5.

Boundary Layer Properties 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click

Boundary Layers 1 and choose Boundary Layer Properties.

2 In the Boundary Layer Properties settings window, locate the Boundary Selection section.

3 From the Selection list, choose Cathodes.

4 Locate the Boundary Layer Properties section. In the Number of boundary layers edit field, type 2.

5 In the Boundary layer stretching factor edit field, type 1.3.

6 In the Thickness adjustment factor edit field, type 5.

Size1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size.


3 From the Calibrate for list, choose Fluid dynamics.

S T U D Y 1

The model is now ready for solving. Add a parametric sweep to solve for a range of cell potentials.

Step 1: Stationary1 In the Model Builder window, under Study 1 click Step 1: Stationary.

2 In the Stationary settings window, click to expand the Study extensions section.

3 Locate the Study Extensions section. Select the Auxiliary sweep check box.

4 Click Add.




Ecell range(1.25,0.05,1.8)






You have now solved the primary current distribution model.

R E S U L T S

Data SetsTo be able to study the anode surface in detail, create a selection of the solution.

1 On the Results toolbar, click More Data Sets and choose Solution.

2 In the Model Builder window, under Results>Data Sets right-click Solution 2 and choose Add Selection.

3 In the Selection settings window, locate the Geometric Entity Selection section.



S T U D Y 1

Solution 1 and 2 in the data sets will now be updated every time you update and solve the model. To store this particular solution, copy and store the primary current distribution solution in order to compare with these results later when you modify the model.

Copy 21 In the Model Builder window, under Study 1>Solver Configurations right-click Solver

1 and choose Solution>Copy.

2 Right-click Copy 2 and choose Rename.

3 Go to the Rename Solver dialog box and type Primary current distribution in the New name edit field.

4 Click OK.

R E S U L T S

Data Sets1 On the Results toolbar, click More Data Sets and choose Solution.

2 In the Solution settings window, locate the Solution section.

3 From the Solution list, choose Primary current distribution.

4 Right-click Results>Data Sets>Solution 4 and choose Add Selection.



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1D Plot Group 1Now create a polarization plot for the primary current distribution model.



3 From the Data set list, choose None.


5 In the Title text area, type Polarization plot.

6 Locate the Plot Settings section. Select the x-axis label check box.

7 In the associated edit field, type Cell voltage (V).

8 Select the y-axis label check box.

9 In the associated edit field, type Total current (A).

10 Click to expand the Legend section. From the Position list, choose Upper left.


12 In the Global settings window, locate the Data section.


14 Locate the y-Axis Data section. In the table, enter the following settings:

15 Click to expand the Legends section. From the Legends list, choose Manual.



3D Plot Group 2The following creates an isosurface of the potential in the electrolyte.




abs(anode_int(siec.nIl)) A

Legends

Primary current distribution




4 Right-click Results>3D Plot Group 2 and choose Isosurface.

5 In the Isosurface settings window, locate the Data section.


7 From the Parameter value (Ecell) list, choose 1.45.


3D Plot Group 3The following creates a normalized plot of the normal electrolyte current density on the anode surface.





5 In the Title text area, type Dimensionless current density distribution.

6 Locate the Plot Settings section. Clear the Plot data set edges check box.

7 Right-click Results>3D Plot Group 3 and choose Surface.

8 In the Surface settings window, locate the Data section.



Plot the normal current density divided by the average normal current density.

11 Locate the Expression section. In the Expression edit field, type (comp1.siec.nIl)/anode_avg(comp1.siec.nIl).


P R I M A R Y C U R R E N T D I S T R I B U T I O N ( S I E C )

Now modify the model to simulate the secondary current distribution.

1 In the Model Builder window, under Component 1 (comp1) click Primary Current

Distribution (siec).

2 In the Primary Current Distribution settings window, locate the Current Distribution

Type section.

3 From the Current distribution type list, choose Secondary.



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Add the needed parameters for the secondary model.





Electrode Reaction 1Now set up the new boundary conditions for the secondary current distribution by adding the needed kinetics parameters.

1 In the Model Builder window, under Component 1 (comp1)>Secondary Current

Distribution (siec)>Electrolyte-Electrode Boundary Interface 1 click Electrode Reaction

1.


3 From the Kinetics expression type list, choose Butler-Volmer.

4 In the i0 edit field, type i0_c.

5 In the a edit field, type be_c.

6 In the c edit field, type 1-be_c.

7 In the Model Builder window, click Secondary Current Distribution

(siec)>Electrolyte-Electrode Boundary Interface 2>Electrode Reaction 1.


9 From the Kinetics expression type list, choose Butler-Volmer.

10 In the i0 edit field, type i0_a.

11 In the a edit field, type be_a.


i0_c 100[A/m^2] 100.0 A/m² Cathode exchange current density

i0_a 100[A/m^2] 100.0 A/m² Anode exchange current density

be_c 0.5 0.5000 Cathode symmetry factor

be_a 0.5 0.5000 Anode symmetry factor

T 298[K] 298.0 K Temperature



12 In the c edit field, type 1-be_a.

S T U D Y 1

On the Home toolbar, click Compute.

R E S U L T S

3D Plot Group 2You have now solved the secondary current distribution problem.

S T U D Y 1

Solver 11 In the Model Builder window, under Study 1>Solver Configurations right-click Solver

1 and choose Rename.

2 Go to the Rename Solver dialog box and type Secondary current distribution in the New name edit field.

3 Click OK.

R E S U L T S

Proceed to look at the results by adding the secondary current distribution polarization plot.

1D Plot Group 11 In the Model Builder window, under Results>1D Plot Group 1 right-click Global 1 and

choose Duplicate.





Note that you have how updated the 3D plots to contain the results from the latest computation. (By choosing different data sets you may compare the primary and secondary current distribution results.)

3D Plot Group 21 In the Model Builder window, under Results>3D Plot Group 2 click Isosurface 1.

Legends

Secondary current distribution



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3D Plot Group 31 In the Model Builder window, under Results>3D Plot Group 3 click Surface 1.





Now modify the problem to model a tertiary current distribution problem by adding mass transport. Start by adding the parameters.






2 In the Model Builder window, under Component 1 (comp1)>Definitions right-click Explicit 3 and choose Rename.

3 Go to the Rename Explicit dialog box and type Inlet in the New name edit field.

4 Click OK.





D 1e-9[m^2/s] 1.000E-9 m²/s Diffusion coefficient

c_in 1e3[mol/m^3] 1000 mol/m³ Inlet concentration

u_in 5[mm/s] 0.005000 m/s Inlet flow velocity




2 In the Model Builder window, under Component 1 (comp1)>Definitions right-click Explicit 4 and choose Rename.

3 Go to the Rename Explicit dialog box and type Outlet in the New name edit field.

4 Click OK.




C O M P O N E N T 1 ( C O M P 1 )

On the Home toolbar, click Add Physics.

A D D P H Y S I C S


2 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species

(chds).


A D D P H Y S I C S


2 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).


TR A N S P O R T O F D I L U T E D S P E C I E S ( C H D S )

1 In the Model Builder window, under Component 1 (comp1) click Transport of Diluted

Species (chds).

2 In the Transport of Diluted Species settings window, locate the Transport Mechanisms section.


Convection, Diffusion, and Migration1 In the Model Builder window, expand the Transport of Diluted Species (chds) node,

then click Convection, Diffusion, and Migration.

2 In the Convection, Diffusion, and Migration settings window, locate the Model Inputs section.



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3 From the u list, choose Velocity field (spf/fp1).


5 In the V edit field, type phil.

6 Locate the Diffusion section. In the Dc edit field, type D.

7 Locate the Migration in Electric Field section. In the zc edit field, type -1.

Inflow 11 On the Physics toolbar, click Boundaries and choose Inflow.

2 In the Inflow settings window, locate the Boundary Selection section.

3 From the Selection list, choose Inlet.

4 Locate the Concentration section. In the c0,c edit field, type c_in.

Outflow 11 On the Physics toolbar, click Boundaries and choose Outflow.

2 In the Outflow settings window, locate the Boundary Selection section.

3 From the Selection list, choose Outlet.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Transport of Diluted

Species (chds) click Initial Values 1.


3 In the c edit field, type c_in.

Electrode-Electrolyte Interface Coupling 1Couple the flux on the anode surface to the electrode reaction currents by using a Electrode-Electrolyte Interface Coupling feature.

1 On the Physics toolbar, click Boundaries and choose Electrode-Electrolyte Interface

Coupling.

2 In the Electrode-Electrolyte Interface Coupling settings window, locate the Boundary

Selection section.


Reaction Coefficients 11 In the Model Builder window, expand the Electrode-Electrolyte Interface Coupling 1

node, then click Reaction Coefficients 1.

2 In the Reaction Coefficients settings window, locate the Model Inputs section.

3 From the iloc list, choose Local current density (siec/eebii2/er1).



4 Locate the Stoichiometric Coefficients section. In the c edit field, type 1.


Also modify the current density expression to be concentration dependent.

Electrode Reaction 11 In the Model Builder window, under Component 1 (comp1)>Secondary Current

Distribution (siec)>Electrolyte-Electrode Boundary Interface 2 click Electrode Reaction

1.


3 From the Kinetics expression type list, choose Concentration dependent kinetics.

4 In the CR edit field, type c/c_in.

L A M I N A R F L O W ( S P F )

Fluid Properties 11 In the Model Builder window, under Component 1 (comp1)>Laminar Flow (spf) click

Fluid Properties 1.

2 In the Fluid Properties settings window, locate the Model Inputs section.


Inlet 11 On the Physics toolbar, click Boundaries and choose Inlet.

2 In the Inlet settings window, locate the Boundary Selection section.

3 From the Selection list, choose Inlet.

4 Locate the Boundary Condition section. From the Boundary condition list, choose Laminar inflow.

5 Locate the Laminar Inflow section. In the Uav edit field, type u_in.

6 In the Lentr edit field, type 10e-3.

Outlet 11 On the Physics toolbar, click Boundaries and choose Outlet.

2 In the Outlet settings window, locate the Boundary Selection section.

3 From the Selection list, choose Outlet.

4 Locate the Pressure Conditions section. Select the Normal flow check box.

Wall 21 On the Physics toolbar, click Boundaries and choose Wall.



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3 In the Wall settings window, locate the Boundary Condition section.

4 From the Boundary condition list, choose Slip.

R O O T

Next set up the solver for the tertiary current distribution problem. Do this by adding a new study wherein you first solve for the flow problem, which does not depend on the other variables, and then the species transport and electric currents.

On the Home toolbar, click Add Study.

A D D S T U D Y

1 Go to the Add Study window.

2 Find the Studies subsection. In the tree, select Preset Studies>Stationary.

3 In the Add study window, click Add Study.

S T U D Y 2

Step 1: Stationary1 In the Model Builder window, under Study 2 click Step 1: Stationary.

2 In the Stationary settings window, locate the Physics and Variables Selection section.


Step 2: Stationary 21 On the Study toolbar, click Study Steps and choose Stationary>Stationary.

2 In the Stationary settings window, locate the Physics and Variables Selection section.


4 Click to expand the Study extensions section. Locate the Study Extensions section. Select the Auxiliary sweep check box.

5 Click Add.

Physics Solve for Discretization

Secondary Current Distribution × physics

Transport of Diluted Species × physics

Physics Solve for Discretization

Laminar Flow × physics








You have now solved the tertiary current distribution.

11 Right-click Study 2 and choose Rename.

12 Go to the Rename Study dialog box and type Tertiary current distribution in the New name edit field.

13 Click OK.

R E S U L T S

Data SetsAdd the tertiary current distribution to the polarization plot.

1D Plot Group 11 In the Model Builder window, under Results>1D Plot Group 1 right-click Global 2 and

choose Duplicate.





Data Sets1 On the Results toolbar, click More Data Sets and choose Solution.

2 In the Solution settings window, locate the Solution section.

3 From the Solution list, choose Solver 3.

4 Right-click Results>Data Sets>Solution 7 and choose Add Selection.



Ecell range(1.25,0.05,1.8)

Legends

Tertiary current distribution



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4 Right-click Results>3D Plot Group 4 and choose Slice.

5 In the Slice settings window, locate the Expression section.

6 Click Velocity magnitude (spf.U) in the upper-right corner of the section. Locate the Plane Data section. In the Planes edit field, type 7.


3D Plot Group 5Finish the post-processing by creating a arrow, slice and surface plot that visualizes the concentration and flow in the cell.




4 Locate the Plot Settings section. Clear the Plot data set edges check box.

5 Right-click Results>3D Plot Group 5 and choose Arrow Volume.

6 In the Arrow Volume settings window, locate the Expression section.

7 Click Velocity field (u,v,w) in the upper-right corner of the section. Locate the Arrow

Positioning section. In the Points edit field, type 15.



10 Locate the Coloring and Style section. From the Color list, choose Black.

11 Select the Scale factor check box.

12 Right-click Results>3D Plot Group 5>Arrow Volume 1 and choose Duplicate.

13 In the Arrow Volume settings window, locate the Arrow Positioning section.






17 On the 3D plot group toolbar, click More Plots and choose Multislice.

18 In the Multislice settings window, locate the Expression section.

19 Click Concentration (c) in the upper-right corner of the section. Locate the Multiplane Data section. In the Planes edit field, type 0.

20 In the Model Builder window, right-click 3D Plot Group 5 and choose Surface.



23 Locate the Expression section. Click Concentration (c) in the upper-right corner of the section. Locate the Inherit Style section. From the Plot list, choose Multislice 1.


3D Plot Group 21 In the Model Builder window, under Results>3D Plot Group 2 click Isosurface 1.




3D Plot Group 31 In the Model Builder window, under Results>3D Plot Group 3 click Surface 1.






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