Electrochemistry Module

Transcription

1 Electrochemistry Module Model Library Manual VERSION 4.4

2 Electrochemistry Module Model Library Manual COMSOL Protected by U.S. Patents 7,519,518; 7,596,474; 7,623,991; and 8,457,932. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement ( and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see Version: November 2013 COMSOL 4.4 Contact Information Visit the Contact COMSOL page at to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at Other useful links include: Support Center: Product Download: Product Updates: COMSOL Community: Events: COMSOL Video Center: Support Knowledge Base: Part number: CM024303

3 Current Distribution in a Chlor-Alkali Membrane Cell 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 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

4 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: 2 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

5 2Cl - Cl 2 (g) + 2e - 2H 2 O + 2e - H 2 (g) + 2OH - at the anode at the cathode (1) The domain in the model is half of the unit cell shown in Figure 1, as explained in Figure 2. Catholyte Cathode Anode Membrane Anolyte 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. 3 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

6 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: l a = s a E eq a (2) where E eq,a is the equilibrium potential of the anode electrode reaction, and s,a the electrode (metal phase) potential at the anode. The cell voltage, E cell, can be written as the difference in metal potential between the anode and cathode: E cell = s a s c (3) Since the cathode is grounded ( s,a =0) we get: E cell = s a (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 E cell = = s a E eq a E eq c + E pol (5) setting equilibrium potential of the cathode hydrogen reaction, E eq,c, to zero, and inserting Equation 5 into Equation 2 we get l a = E pol (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 4 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

7 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. 5 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

8 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 , J.S. Newman, Electrochemical Systems, 2nd ed., Prentice Hall, 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 , Model Library path: Electrochemistry_Module/ Electrochemical_Engineering/chlor_alkali Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 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. 6 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

9 GLOBAL DEFINITIONS Start by loading the model parameters from a text file. Parameters 1 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. GEOMETRY 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. MATERIALS 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 7 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

10 Material 2 (mat2) 1 In the Model Builder window, right-click Materials and choose New Material. 2 Select Domain 2 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Electrolyte conductivity sigmal K_m S/m Electrolyte conductivity Material 3 (mat3) 1 Right-click Materials and choose New Material. 2 Select Domain 1 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Electrolyte conductivity sigmal K_a S/m Electrolyte conductivity SECONDARY CURRENT DISTRIBUTION (SIEC) 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 1 1 On the Physics toolbar, click Boundaries and choose Electrolyte-Electrode Boundary Interface. 2 Select Boundaries and 18 only. Electrode Reaction 1 1 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 i 0 edit field, type i0_c. 8 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

11 Electrolyte Potential 1 Define 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 1 Provide 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. MESH 1 Create a triangular mesh with a higher resolution at the electrode surfaces and on the membrane boundaries. Free Triangular 1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Free Triangular. Size 1 1 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 only. 6 Locate the Element Size section. From the Predefined list, choose Extremely fine. 9 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

12 Free Triangular 1 1 In the Model Builder window, right-click Mesh 1 and choose Build All. The finalized mesh should now look like this: STUDY 1 The problem is now ready for solving. 1 On the Home toolbar, click Compute. RESULTS 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 In the Points edit field, type Locate the Coloring and Style section. From the Color list, choose White. 6 Click the Zoom Extents button on the Graphics toolbar. 10 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

13 Create a surface plot of the electrolyte current density norm to visualize the "hot spots" on the electrode surfaces. 2D Plot Group 2 1 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. 11 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

14 12 CURRENT DISTRIBUTION IN A CHLOR-ALKALI MEMBRANE CELL

15 Cyclic Voltammetry at a Macroelectrode in 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 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

16 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: L = 6 Dt max (1) Here, D is the diffusion coefficient of the reactant and t max is the duration of the cyclic voltammogram. DOMAIN EQUATIONS 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: c i = D t i c i (2) BOUNDARY EQUATIONS At the bulk boundary (x L), 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 (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 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

17 B+ e A (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: i loc nfk 0 c A exp n c F = RT c exp cf B RT (4) in which k 0 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: n N i = i i loc nf (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: I el = i loc A (6) CYCLIC VOLTAMMETRY STUDY 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. 3 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

18 Results and Discussion 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, I pf, 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 4 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

19 Ref. 1 and Ref. 2 for a detailed discussion and derivation): I pf = 0,446nFAc nf Dv RT (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, A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Second Edition, Hoboken, Model Library path: Electrochemistry_Module/Tutorial_Models/ cyclic_voltammetry_1d Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select Electrochemistry>Electroanalysis (elan). 3 Click the Add button. This model will solve for the two concentrations of a redox couple, change the default concentration variable names to ca and cb. 5 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

20 4 In the Concentrations table, enter the following settings: ca cb 5 Click the Study button. 6 In the tree, select Preset Studies>Cyclic Voltammetry. 7 Click the Done button. GLOBAL DEFINITIONS Add the model parameters from a text file. Parameters 1 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 cyclic_voltammetry_1d_parameters.txt. GEOMETRY 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 1 1 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. 6 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

21 4 In the Model Builder window, right-click Geometry 1 and choose Build All Objects. The completed geometry should now look like this: ELECTROANALYSIS Diffusion 1 Start 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 D ca edit field, type DA. 4 In the D cb edit field, type DB. Concentration 1 Set 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 c 0,cA edit field, type c_bulk. 7 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

22 6 Select the Species cb check box. Electrode Surface 1 Set up the electrode surface and the cyclic voltammetry settings on the left boundary. 1 On the Physics toolbar, click Boundaries and choose Electrode Surface. 2 Select Boundary 1 only. 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 1 Specify 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 k 0 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 1 Specify 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. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the ca edit field, type c_bulk. STUDY 1 Solve the problem for various sweep rates. 8 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

23 Parametric Sweep 1 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. 4 In the table, enter the following settings: Parameter names v Parameter value list (10^range(-3,1,0))[V/s] 5 On the Home toolbar, click Compute. RESULTS 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. 9 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

24 10 CYCLIC VOLTAMMETRY AT A MACROELECTRODE IN 1D

25 Modeling the Diffuse Double Layer 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. DOMAIN EQUATIONS The concentrations, c i (mol/m 3, i=+,-), of two ions of opposite charge (+1/-1) are solved for in the electrolyte phase. The fluxes (N i, mol/(m 2 s)) of these are described by the Nernst-Planck equation N i = D i c i u m i z i Fc i (1) with D i (m 2 /s) being the diffusion coefficient, u m,i (s mol/kg) the mobility, F (C/ mol) Faraday s constant, and (V) the potential. 1 MODELING THE DIFFUSE DOUBLE LAYER

26 Assuming no heterogenous reactions in the electrolyte, the governing equations for the two species become: N i = 0 (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: = Fc + c - (4) BOUNDARY CONDITIONS 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. S + + e - S(s) (5) The reaction rate r (mol/(m 2 s)) is a F r = K a c M exp K RT c c + exp c F RT (6) where K a and K c (m/s) are the anodic and cathodic rate constants, c M the metal species activity (mol/m 3, constant) and a and c the anodic and cathodic transfer coefficients. (V) is the difference in potential between the metal phase, M (V), and the reaction plane: = M (7) The electrode reaction renders an inward flux for the positive ion according to n N + = r (8) on both boundaries. For the negative ion, a zero flux condition is used. n N - = 0 (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 2 MODELING THE DIFFUSE DOUBLE LAYER

27 of a constant thickness, S (m), one can derive the following Robin type of boundary condition for the potential: + S n = M (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 n = S (11) CELL POTENTIAL EQUATION The problem formulated above can now be solved for given voltages of M in the metal electrode phase for each side. Typically one grounds one electrode and specifies the cell voltage as V so that M = 0 x = 0 M = V x = L (12) However, to solve for a given cell current density, i cell (A/m 2 ), with V not known a priori, an additional global equation, solving for V, is used, fulfilling the condition: i cell = Fr x = L (13) GLOBAL CONCENTRATION CONSTRAINT FOR THE NEGATIVE ION 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: L c 0 L = c - dx 0 (14) where c 0 is the initial ion concentration (mol/m 3 ), 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.) 3 MODELING THE DIFFUSE DOUBLE LAYER

28 DIMENSIONLESS NUMBERS AND PARAMETER VALUES 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 D = D L = RT 2F 2 c 0 D (15) where D is the Debye length. The current of the cell is defined via the dimensionless number j=0.9, j = i cell i D i D = 4FD + c 0 L (16) where i D is the Nernstian limiting current density. The cathodic reaction rate constant is defined using the dimensionless number k c 10, k c = K c L D + (17) The rate of the anodic reaction term is governed by the dimensionless number k r 10, k r = K r Lc M 4D + c 0 (18) and the Stern layer thickness is set using the dimensionless number =0.1, S = D (19) Results and Discussion The following dimensionless variables are used when presenting the results: 4 MODELING THE DIFFUSE DOUBLE LAYER

29 c x = Figure 1 shows the dimensionless concentration, c. The concentration gradients are steepest close to the electrodes. --- x L c + + c - = c 0 c + c - = c 0 F = RT (20) 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. 5 MODELING THE DIFFUSE DOUBLE LAYER

30 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. 6 MODELING THE DIFFUSE DOUBLE LAYER

31 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 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 Model Library path: Electrochemistry_Module/Tutorial_Models/ diffuse_double_layer 7 MODELING THE DIFFUSE DOUBLE LAYER

32 Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select AC/DC>Electrostatics (es). 3 Click the Add button. 4 In the Electric potential edit field, type phi. 5 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 6 Click the Add button. 7 Click Add Concentration. 8 In the Concentrations table, enter the following settings: cp cm 9 In the Select physics tree, select Mathematics>ODE and DAE Interfaces>Global ODEs and DAEs (ge). 10 Click the Add button. 11 Click the Study button. 12 In the tree, select Preset Studies for Selected Physics>Stationary. 13 Click the Done button. GLOBAL DEFINITIONS Start by loading some parameters from a text file. Parameters 1 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 diffuse_double_layer_parameters.txt. 8 MODELING THE DIFFUSE DOUBLE LAYER

33 GEOMETRY 1 Build the geometry as a single interval. Interval 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Interval. 2 In the Interval settings window, locate the Interval section. 3 In the Right endpoint edit field, type L. 4 Click the Build All Objects button. 5 Click the Zoom Extents button on the Graphics toolbar. DEFINITIONS 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 1 1 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. 3 From the Geometric entity level list, choose Boundary. 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 2 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 1 only. 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description phim 0[V] V Metal phase potential (ground) Variables 3 1 Right-click Definitions and choose Variables. 9 MODELING THE DIFFUSE DOUBLE LAYER

34 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 2 only. 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description phim V Metal phase potential (cell voltage) Integration 1 Add 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. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 2 only. Integration 2 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 Select Domain 1 only. ELECTROSTATICS Charge Conservation 1 Now 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 1 1 On the Physics toolbar, click Domains and choose Space Charge Density. 2 Select Domain 1 only. 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). 10 MODELING THE DIFFUSE DOUBLE LAYER

35 Surface Charge Density 1 1 On the Physics toolbar, click Boundaries and choose Surface Charge Density. 2 Select Boundaries 1 and 2 only. 3 In the Surface Charge Density settings window, locate the Surface Charge Density section. 4 In the s edit field, type rho_s. TRANSPORT OF DILUTED SPECIES 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 Migration 1 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. 4 In the T edit field, type T. 5 Locate the Diffusion section. In the D cp edit field, type Dp. 6 In the D cm edit field, type Dm. 7 Locate the Migration in Electric Field section. In the z cp edit field, type Z. 8 In the z cm edit field, type -Z. Flux 1 1 On the Physics toolbar, click Boundaries and choose Flux. 2 Select Boundaries 1 and 2 only. 3 In the Flux settings window, locate the Inward Flux section. 11 MODELING THE DIFFUSE DOUBLE LAYER

36 4 Select the Species cp check box. 5 In the N 0,cp edit field, type r. Initial Values 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 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 1 Add 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). GLOBAL ODES AND DAES Global Equations 1 Finally, 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. 3 In the table, enter the following settings: Name f(u,ut,utt,t) (1) Initial value (u_0) (1) V intop1(iloc)-ic ell Initial value (u_t0) (1/s) Description 0 0 Cell voltage MESH 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.) 12 MODELING THE DIFFUSE DOUBLE LAYER

37 Edge 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Edit Physics-Induced Sequence. Size 1 1 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 2 1 Right-click Edge 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. 4 Select Boundaries 1 and 2 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type lambdad/10. Edge 1 Right-click Edge 1 and choose Build Selected. STUDY 1 Solve the problem using a parametric sweep. Parametric Sweep 1 In the Model Builder window, right-click Study 1 and choose Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names Parameter value list epsilon MODELING THE DIFFUSE DOUBLE LAYER

38 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. 8 On the Home toolbar, click Compute. RESULTS Reproduce the figures from the Results and Discussion section in the following way: 1D Plot Group 1 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Line Graph. 3 Select Domain 1 only. 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 In the 1D Plot Group settings window, locate the Data section. 11 From the Data set list, choose Solution 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. 15 On the 1D plot group toolbar, click Plot. 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 1 1 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. 14 MODELING THE DIFFUSE DOUBLE LAYER

39 5 In the Model Builder window, expand the Dimensionless concentration 1 node, then click Line Graph 1. 6 In the Line Graph settings window, locate the y-axis Data section. 7 In the Expression edit field, type (cp-cm)/(2*cref). 8 On the 1D plot group toolbar, click Plot. 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 1 1 Right-click Dimensionless concentration 1 and choose Duplicate. 2 In the 1D Plot Group settings window, locate the Title section. 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. 6 In the Line Graph settings window, locate the y-axis Data section. 7 In the Expression edit field, type phi*z*f_const/(r_const*t). 8 On the 1D plot group toolbar, click Plot. 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. 15 MODELING THE DIFFUSE DOUBLE LAYER

40 16 MODELING THE DIFFUSE DOUBLE LAYER Solved with COMSOL Multiphysics 4.4

41 Desalination in an Electrodialysis Cell 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 DESALINATION IN AN ELECTRODIALYSIS CELL

42 Model Definition Figure 1shows the basic principle of the desalination stack. The stack has one repetitive cell unit, shown in Figure 2. Concentrate Diluate Electrode Stream Electrode Stream Cathode: Negative Electrode Electrode Stream Concentrate Na + Na + Na + Na + Na + Na + Na + Na + OH H + SO 4 2 SO 2 4 Cl Cl Cl Cl Cl Cl SO 2 4 Anode: Positive Electrode Electrode Stream Diluate Figure 1: Schematic picture of a desalination stack with 3 desalination units (in reality unit cells are used) Na + Na + Na + Cl - Cl - Cl - Figure 2: The repetitive unit cell with one desalination unit. 2 DESALINATION IN AN ELECTRODIALYSIS CELL

43 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. Outlets Dilutate Channel Cation Selective Membrane Anion Selective Membrane Concentrate Channels Inlets 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. CHOICE OF CURRENT DISTRIBUTION INTERFACE 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), N i, due to diffusion, migration and convection: 3 DESALINATION IN AN ELECTRODIALYSIS CELL

44 N i = D i c i z i u mob i Fc i l + c i u The first term is the diffusion flux, D i is the diffusion coefficient (m 2 /s). The migration term consists of the species charge number z i, the species mobility u mob,i (s mol/kg) and the electrolyte potential ( l ). 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): n i l = F z i D i c i z i u m i Fc i l i = 1 (1) The conservation of current is then used to calculate the electrolyte potential. i l = F z i R i i = 1 where the R i 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) n i l = u m + F 2 c + l = l l (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. 4 DESALINATION IN AN ELECTRODIALYSIS CELL

45 MEMBRANE FREE ELECTROLYTE BOUNDARY CONDITIONS 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: n i l e = n i l m (3) The ion flux for the permeable ion is proportional to the current, according to Faraday s law n N i e n i l m = z i F (4) For the non-permeable ion, the flux is zero. Furthermore, we have the following relation between the potentials and the concentrations: l m = l e + RT F a i m a i e ln (5) where a i,m is the permeable ion activity in the membrane, and a i,e the permeable ion activity in the free electrolyte. In this model the activities equal the respective concentrations, divided by 1000 mol/m 3. 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. COUPLING BETWEEN THE OUTER CONCENTRATE BOUNDARIES Use a linear extrusion operator to set the concentration on the rightmost boundary to equal that of the leftmost boundary. 5 DESALINATION IN AN ELECTRODIALYSIS CELL

46 Results and Discussion 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. 6 DESALINATION IN AN ELECTRODIALYSIS CELL

47 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. 7 DESALINATION IN AN ELECTRODIALYSIS CELL

48 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. 8 DESALINATION IN AN ELECTRODIALYSIS CELL

49 Figure 7: Comparison of fluxes, species Na +. Figure 8: Comparison of fluxes, species Cl -. 9 DESALINATION IN AN ELECTRODIALYSIS CELL

50 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 Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. 10 DESALINATION IN AN ELECTRODIALYSIS CELL

51 MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Electrochemistry>Tertiary Current Distribution, Nernst-Planck (tcdee). 3 Click the Add button. 4 In the Concentrations table, enter the following settings: cna ccl 5 In the Select physics tree, select Electrochemistry>Secondary Current Distribution (siec). 6 Click the Add button. 7 In the Electrolyte potential edit field, type phil_m. 8 Click the Study button. 9 In the tree, select Preset Studies for Selected Physics>Stationary. 10 Click the Done button. GLOBAL DEFINITIONS Load the model parameters from a text file. Parameters 1 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 electrodialysis_parameters.txt. GEOMETRY 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. 11 DESALINATION IN AN ELECTRODIALYSIS CELL

52 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. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type (W_ch+2*W_m). 4 In the Height edit field, type L. 5 Locate the Position section. In the x edit field, type -(W_ch+2*W_m)/2. 6 Click the Build Selected button. Rectangle 3 (r3) 1 Right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type W_ch. 4 In the Height edit field, type L. 5 Locate the Position section. In the x edit field, type -W_ch/2. 6 Click the Build Selected button. 7 Click the Zoom Extents button on the Graphics toolbar. The finished geometry should look like Figure 3 in the Model Definition section. DEFINITIONS Make some selections on the geometry to use later when you set up the physics. Explicit 1 1 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. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 Select Domains 1, 3, and 5 only. 3 Right-click Component 1 (comp1)>definitions>explicit 2 and choose Rename. 4 Go to the Rename Explicit dialog box and type Channels in the New name edit field. 12 DESALINATION IN AN ELECTRODIALYSIS CELL

53 5 Click OK. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 2, 8, and 14 only. 5 Right-click Component 1 (comp1)>definitions>explicit 3 and choose Rename. 6 Go to the Rename Explicit dialog box and type Inlets in the New name edit field. 7 Click OK. Explicit 4 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 3, 9, and 15 only. 5 Right-click Component 1 (comp1)>definitions>explicit 4 and choose Rename. 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. 13 DESALINATION IN AN ELECTRODIALYSIS CELL

54 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. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 1 only. 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. 8 Select Point 2 only. 9 Locate the Destination Vertices section. Select the Destination vertex 1 toggle button. 10 Select Point 11 only. 11 Select the Destination vertex 2 toggle button. 12 Select Point 12 only. Variables 1 Now 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. 3 In the table, enter the following settings: Name Expression Unit Description ana max(cna,eps^2) mol/m³ Na activity acl max(ccl,eps^2) mol/m³ Cl activity The max() functions ensures a positive value for the ah and acl variables. eps is the machine epsilon, i.e. a very small number. Variables 2 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 14 DESALINATION IN AN ELECTRODIALYSIS CELL

55 4 Select Domain 3 only. 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description vel 3*v_avg*(1-((x)/ (W_ch/2))^2) m/s Channel velocity (middle channel) Variables 3 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 1 only. 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description vel 3*v_avg*(1-((x+W_ch +W_m)/(W_ch/2))^2) m/s Channel velocity (left channel) Variables 4 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 5 only. 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description vel 3*v_avg*(1-((x-W_ch -W_m)/(W_ch/2))^2) m/s Channel velocity (right channel) TERTIARY CURRENT DISTRIBUTION, NERNST-PLANCK (TCDEE) 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. 15 DESALINATION IN AN ELECTRODIALYSIS CELL

56 Electrolyte 1 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. 3 Specify the u vector as 0 x vel y 4 In the T edit field, type T. 5 Locate the Diffusion section. In the D cna edit field, type DNa. 6 In the D ccl edit field, type DCl. 7 Locate the Migration in Electric Field section. In the z cna edit field, type 1. 8 In the z ccl edit field, type -1. Electrolyte Potential 1 1 On the Physics toolbar, click Boundaries and choose Electrolyte Potential. 2 Select Boundary 1 only. Electrolyte Potential 2 1 On the Physics toolbar, click Boundaries and choose Electrolyte Potential. 2 Select Boundary 16 only. 3 In the Electrolyte Potential settings window, locate the Electrolyte Potential section. 4 In the l,bnd edit field, type Vtot. Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 Select Boundary 16 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species ccl check box. 5 In the c 0,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 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 In the Inflow settings window, locate the Boundary Selection section. 16 DESALINATION IN AN ELECTRODIALYSIS CELL

57 3 From the Selection list, choose Inlets. 4 Locate the Concentration section. In the c 0,cCl edit field, type ccl_0. Outflow 1 1 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 1 1 In the Model Builder window, under Component 1 (comp1)>tertiary Current Distribution, Nernst-Planck (tcdee) click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the ccl edit field, type ccl_0. SECONDARY CURRENT DISTRIBUTION (SIEC) 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 1 1 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. 17 DESALINATION IN AN ELECTRODIALYSIS CELL

58 Pointwise Constraint 1 1 On the Physics toolbar, click Boundaries and choose Pointwise Constraint. 2 Select Boundaries 4 and 7 only. 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 2 1 On the Physics toolbar, click Boundaries and choose Pointwise Constraint. 2 Select Boundaries 10 and 13 only. 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. MESH 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 1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Mapped. 18 DESALINATION IN AN ELECTRODIALYSIS CELL

59 Distribution 1 1 In the Model Builder window, under Component 1 (comp1)>mesh 1 right-click Mapped 1 and choose Distribution. 2 Select Boundary 1 only. 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type 100. Distribution 2 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundaries 8 and 9 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Predefined distribution type. 5 In the Number of elements edit field, type In the Element ratio edit field, type Select the Symmetric distribution check box. Distribution 3 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundary 3 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Predefined distribution type. 5 In the Number of elements edit field, type In the Element ratio edit field, type Select the Reverse direction check box. Distribution 4 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundary 2 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Predefined distribution type. 5 In the Number of elements edit field, type In the Element ratio edit field, type 10. Distribution 5 1 Right-click Mapped 1 and choose Distribution. 19 DESALINATION IN AN ELECTRODIALYSIS CELL

60 2 Select Boundary 14 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Predefined distribution type. 5 In the Number of elements edit field, type In the Element ratio edit field, type Select the Reverse direction check box. Distribution 6 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundary 15 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Predefined distribution type. 5 In the Number of elements edit field, type In the Element ratio edit field, type 10. Distribution 7 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundaries 6 and 12 only. 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type DESALINATION IN AN ELECTRODIALYSIS CELL

61 5 In the Model Builder window, right-click Mesh 1 and choose Build All. STUDY 1 Step 1: Stationary The 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. 5 In the table, enter the following settings: Auxiliary parameter Vtot Parameter value list range(0.5,0.5,1.5) Solver 1 1 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. 21 DESALINATION IN AN ELECTRODIALYSIS CELL

62 4 In the Relative tolerance edit field, type 1e-5. 5 On the Home toolbar, click Compute. RESULTS Concentration (tcdee) Reproduce the figures from the Results and Discussion section in the following way. 1 On the 2D plot group toolbar, click Plot. Data Sets 1 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. 1D Plot Group 4 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 From the Parameter selection (Vtot) list, choose Last. 5 On the 1D plot group toolbar, click Line Graph. 6 On the 1D plot group toolbar, click Line Graph. 7 In the Line Graph settings window, locate the y-axis Data section. 8 In the Expression edit field, type phil_m. 9 In the Model Builder window, click 1D Plot Group In the 1D Plot Group settings window, click to expand the Title section. 11 From the Title type list, choose None. 12 On the 1D plot group toolbar, click Plot. 1D Plot Group 5 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 22 DESALINATION IN AN ELECTRODIALYSIS CELL

63 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D plot group toolbar, click Line Graph. 5 In the Line Graph settings window, locate the y-axis Data section. 6 In the Expression edit field, type cna. 7 Click to expand the Legends section. Select the Show legends check box. 8 On the 1D plot group toolbar, click Plot. 1D Plot Group 6 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 From the Parameter selection (Vtot) list, choose Last. 5 On the 1D plot group toolbar, click Line Graph. 6 In the Line Graph settings window, locate the y-axis Data section. 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. 10 In the table, enter the following settings: Legends Total Flux 11 On the 1D plot group toolbar, click Plot. 12 Right-click Results>1D Plot Group 6>Line Graph 1 and choose Duplicate. 13 In the Line Graph settings window, locate the y-axis Data section. 14 In the Expression edit field, type tcdee.mfluxx_cna. 15 Locate the Legends section. In the table, enter the following settings: Legends Migrative Flux 16 On the 1D plot group toolbar, click Plot. 17 Right-click Results>1D Plot Group 6>Line Graph 2 and choose Duplicate. 18 In the Line Graph settings window, locate the y-axis Data section. 19 In the Expression edit field, type tcdee.dfluxx_cna. 23 DESALINATION IN AN ELECTRODIALYSIS CELL

64 20 On the 1D plot group toolbar, click Plot. 21 Locate the Legends section. In the table, enter the following settings: Legends Diffusive Flux 22 On the 1D plot group toolbar, click Plot. 23 In the Model Builder window, click 1D Plot Group 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. 30 On the 1D plot group toolbar, click Plot. 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 7 1 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. 3 In the Line Graph settings window, locate the y-axis Data section. 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. 6 In the Line Graph settings window, locate the y-axis Data section. 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. 9 In the Line Graph settings window, locate the y-axis Data section. 10 In the Expression edit field, type tcdee.dfluxx_ccl. 11 In the Model Builder window, click 1D Plot Group In the 1D Plot Group settings window, locate the Title section. 24 DESALINATION IN AN ELECTRODIALYSIS CELL

65 13 In the Title text area, type Cl^- flux, x component (mol/ (m²*s)). 14 On the 1D plot group toolbar, click Plot. 1D Plot Group 8 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Line Graph. 3 Select Boundary 10 only. 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. 8 On the 1D plot group toolbar, click Plot. 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 7 1 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. 1D Plot Group 6 1 In the Model Builder window, under Results right-click 1D Plot Group 6 and choose Rename. 2 Go to the Rename 1D Plot Group dialog box and type Fluxes, Na in the New name edit field. 3 Click OK. 1D Plot Group 5 1 In the Model Builder window, under Results right-click 1D Plot Group 5 and choose Rename. 25 DESALINATION IN AN ELECTRODIALYSIS CELL

66 2 Go to the Rename 1D Plot Group dialog box and type Concentration in the New name edit field. 3 Click OK. 1D Plot Group 4 1 In the Model Builder window, under Results right-click 1D Plot Group 4 and choose Rename. 2 Go to the Rename 1D Plot Group dialog box and type Potential in the New name edit field. 3 Click OK. 26 DESALINATION IN AN ELECTRODIALYSIS CELL

67 Glucose Sensor Solved with COMSOL Multiphysics 4.4 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 GLUCOSE SENSOR

68 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. DOMAIN EQUATIONS 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: D i c i = 0 (1) 2 GLUCOSE SENSOR

69 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: glucose + ferri products + ferro (2) The rate of this reaction (mol/m 3 ) is given by a Michaelis Menten rate law as (Ref. 2): R = c glucose V max 1 + K m c glucose (3) Here, the parameter V max is the maximum rate of the enzyme-catalyzed reaction, depending on the quantity of enzyme available, and the parameter K m 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. BOUNDARY EQUATIONS 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 :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: ferri + e ferro (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 3 GLUCOSE SENSOR

70 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: i loc nfk 0 c ferro exp n c F = RT c exp cf ferri RT (5) in which k 0 is the heterogeneous rate constant of the reaction, c is the cathodic transfer coefficient, and is the overpotential at the working electrode. According to Faraday s laws of electrolysis, the flux of the reactant and product species are proportional to the current density drawn: n N i = i i loc nf (6) This is expressed in the Electrode Surface boundary condition. 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: I el = i loc da S (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. STATIONARY STUDY 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. 4 GLUCOSE SENSOR

71 Results and Discussion 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/m 3. 5 GLUCOSE SENSOR

72 Figure 3: Current density versus glucose concentration. References 1. J. Wang, Chem. Rev. 108 (2008) pp P. Atkins, J. de Paula, Physical Chemistry, 9th Edition Model Library path: Electrochemistry_Module/ Electrochemical_Engineering/glucose_sensor Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. 6 GLUCOSE SENSOR

73 MODEL WIZARD Build the model in 2D with the Electroanalysis interface. Solve for three concentrations in a Stationary study. 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Electrochemistry>Electroanalysis (elan). 3 Click the Add button. 4 In the Number of species edit field, type 3. 5 In the Concentrations table, enter the following settings: c_glucose c_ferro c_ferri 6 Click the Study button. 7 In the tree, select Preset Studies>Stationary. 8 Click the Done button. GEOMETRY 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. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type In the Height edit field, type 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 GLUCOSE SENSOR

74 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 Click the Build Selected button. Your finished geometry should now look like this: GLOBAL DEFINITIONS Import the model parameters from a text file. Parameters 1 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 glucose_sensor_parameters.txt. 8 GLUCOSE SENSOR

75 DEFINITIONS 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. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 5 only. Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description R_MM V_max*c_glucose/ (Km+c_glucose) mol/(m³ s) Reaction rate of glucose i_avg aveop1(elan.itot) Average current density 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. ELECTROANALYSIS (ELAN) Now start defining the physics. Electrode Surface 1 1 On the Physics toolbar, click Boundaries and choose Electrode Surface. 2 Select Boundary 5 only. 3 In the Electrode Surface settings window, locate the Boundary Condition section. 4 In the s,ext edit field, type 0.4. Electrode Reaction 1 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 Stoichiometric Coefficients section. 3 In the cferro edit field, type 1. 9 GLUCOSE SENSOR

76 4 In the cferri edit field, type -1. Electrode Surface 2 1 On the Physics toolbar, click Boundaries and choose Electrode Surface. 2 Select Boundaries 2 and 7 only. 3 In the Electrode Surface settings window, locate the Boundary Condition section. 4 From the Boundary condition list, choose Average current density. 5 In the i l,average edit field, type -i_avg. 6 In the s,ext edit field, type Electrode Reaction 1 1 In the Model Builder window, expand the Electrode Surface 2 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. 4 In the cferri edit field, type -1. Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 Select Boundary 3 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species c_glucose check box. 5 In the c 0,cglucose edit field, type c_glucose_ext. 6 Select the Species c_ferro check box. 7 In the c 0,cferro edit field, type c_ferro_ext. 8 Select the Species c_ferri check box. 9 In the c 0,cferri edit field, type c_ferri_ext. Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 Select Domain 1 only. 3 In the Reactions settings window, locate the Reactions section. 4 In the R cglucose edit field, type -R_MM. 5 In the R cferro edit field, type R_MM. 10 GLUCOSE SENSOR

77 6 In the R cferri edit field, type -R_MM. Initial Values 1 1 In the Model Builder window, under Component 1 (comp1)>electroanalysis (elan) click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the c glucose edit field, type c_glucose_ext. 4 In the c ferro edit field, type c_ferro_ext. 5 In the c ferri edit field, type c_ferri_ext. MESH 1 The physics settings are now complete. Now customize the mesh and solve the problem. Size 1 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. 3 In the Size settings window, locate the Element Size section. 4 From the Predefined list, choose Extra fine. Size 1 1 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. 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. 7 Locate the Element Size Parameters section. Select the Maximum element size check box. 8 In the associated edit field, type GLUCOSE SENSOR

78 9 In the Model Builder window, right-click Mesh 1 and choose Build All. STUDY 1 Use a parametric sweep to solve for a range of different external concentration values for c_ferro. Parametric Sweep 1 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. 4 In the table, enter the following settings: Parameter names c_glucose_ext Parameter value list range(50,50,1000)[umol/l] 5 On the Home toolbar, click Compute. RESULTS Concentration (elan) Modify the default concentration plot to show the ferro concentration. 12 GLUCOSE SENSOR

79 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. 4 On the 2D plot group toolbar, click Plot. 1D Plot Group 2 Create 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. 4 In the table, enter the following settings: Expression Unit Description i_avg A/m^2 Average current density 5 Click to expand the Legends section. Clear the Show legends check box. 6 On the 1D plot group toolbar, click Plot. 13 GLUCOSE SENSOR

80 14 GLUCOSE SENSOR Solved with COMSOL Multiphysics 4.4

81 Electrochemical Impedance Spectroscopy 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 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

82 DOMAIN EQUATIONS 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: c i = D t i c i (1) BOUNDARY EQUATIONS Both the oxidized and reduced species are equally concentrated at c i = 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: B + e A (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. The current density for this reaction is given by the electroanalytical Butler-Volmer equation for an oxidation: i loc nfk 0 c A exp n c F c F = c RT B exp RT (3) in which k 0 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, E eq : = s ext E eq (4) 2 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

83 According to Faraday s laws of electrolysis, the flux of the reactant and product species are proportional to the current density drawn: n N i = i i loc nf (5) This is expressed in the Electrode Surface boundary condition. An additional capacitance is applied at the working electrode. It is set equal to 20 µf/ cm 2 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. AC IMPEDANCE STATIONARY STUDY 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: s ext = E eq + e j t (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 c i can be expressed as the sum of a stationary solution c i,0 due to the center voltage of the applied potential, and a sinusoidal perturbation c i,1 to the concentration resulting from the perturbation on the applied potential: c i xt = c i 0 x + c i 1 x e j t (7) If c i,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: 3 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

84 D i c i 0 = 0 (8) subject to c i,0 set to equal, fixed concentrations of 1 mm in bulk, and subject to the fixed center potential at the electrode surface. s ext = E eq (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: j c i 1 = D i c i 1 (10) solved subject to c i,1 = 0 in bulk and: s ext = (11) at the surface. A Parametric Sweep is used to investigate different values of the heterogeneous rate constant of the electrode reaction, k 0, 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). Results and Discussion 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. 4 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

85 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 k 0 = 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. 5 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

86 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. 6 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

87 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 1. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Second Edition, Hoboken, Model Library path: Electrochemistry_Module/Electroanalysis/ impedance_spectroscopy Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select Electrochemistry>Electroanalysis (elan). 3 Click the Add button. 4 In the Concentrations table, enter the following settings: cred cox 5 Click the Study button. 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. 7 Click the Done button. 7 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

88 GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. Load the model parameters from a text file. 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 impedance_spectroscopy_parameters.txt. GEOMETRY 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. 2 In the Interval settings window, locate the Interval section. 3 In the Right endpoint edit field, type L_el. 4 Click the Build All Objects button. 5 Click the Zoom Extents button on the Graphics toolbar. ELECTROANALYSIS Diffusion 1 Now 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. 2 In the Diffusion settings window, locate the Diffusion section. 3 In the D cred edit field, type D. 4 In the D cox edit field, type D. Initial Values 1 1 In the Model Builder window, under Component 1 >Electroanalysis (elan) click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the cred edit field, type c_bulk. 8 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

89 4 In the cox edit field, type c_bulk. Concentration 1 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 cred check box. 5 In the c 0,cRed edit field, type c_bulk. 6 Select the Species cox check box. 7 In the c 0,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. Electrode Surface 1 1 On the Physics toolbar, click Boundaries and choose Electrode Surface. 2 Select Boundary 1 only. 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. Electrode Reaction 1 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 k 0 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 1 1 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 C dl edit field, type Cdl. MESH 1 1 In the Model Builder window, under Component 1 click Mesh 1. 9 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

90 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 1 1 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. 4 Locate the Element Size section. Click the Custom button. 5 Locate the Element Size Parameters section. Select the Maximum element size check box. 6 In the associated edit field, type xdiff_min/25. STUDY 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. Parametric Sweep 1 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. 4 In the table, enter the following settings: Parameter names k0 Parameter value list 10^range(-1,-1,-3)[cm/s] Step 2: Frequency-Domain, Perturbation 1 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). 4 On the Home toolbar, click Compute. 10 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

91 RESULTS 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: 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 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. 5 In the 1D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Solution 3. 7 Locate the Legend section. From the Position list, choose Lower left. Bode plot, absolute value of impedance 1 On the 1D plot group toolbar, click Global. 2 In the Global settings window, locate the y-axis Data section. 3 In the table, enter the following settings: Expression Unit Description abs(elan.zvsgrnd_els1) ohm*m^2 Impedance vs ground 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. 9 On the 1D plot group toolbar, click Plot. 10 Click the x-axis Log Scale button on the Graphics toolbar. 11 Click the y-axis Log Scale button on the Graphics toolbar. 11 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

92 Bode plot, absolute value of impedance 1 1 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 angle 1 In the Model Builder window, expand the Results>Bode plot, phase angle node, then click Global 1. 2 In the Global settings window, locate the y-axis Data section. 3 In the table, enter the following settings: Expression Unit Description arg(elan.zvsgrnd_els1) rad Phase angle 4 Click the y-axis Log Scale button on the Graphics toolbar. 5 On the 1D plot group toolbar, click Plot. 12 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

93 Voltammetry at a Microdisk Electrode 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. Symmetry Axis Electrolyte Electrode Insulator 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 1 VOLTAMMETRY AT A MICRODISK ELECTRODE

94 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, r e, which equals 10 m. At r r e, this axis represents the working electrode (microdisk) where the electrochemical reaction takes place. At r r e, this axis represents the surrounding insulator in-plane with the disk electrode. DOMAIN EQUATIONS 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: D i c i = 0 (1) BOUNDARY EQUATIONS At the bulk boundary ( r ), 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: 2 VOLTAMMETRY AT A MICRODISK ELECTRODE

95 B+ e A (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. The current density for this reaction is given by the electroanalytical Butler-Volmer equation for an oxidation: i loc nfk 0 c A exp n c F = RT c exp cf B RT (3) in which k 0 is the heterogeneous rate constant of the reaction, c is the cathodic transfer coefficient, and is the overpotential at the working electrode. According to Faraday s laws of electrolysis, the flux of the reactant and product species are proportional to the current density drawn: n N i = i i loc nf (4) This is expressed in the Electrode Surface boundary condition. 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: I el = i loc da S (5) STATIONARY STUDY 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: r e RT « D Fv (6) 3 VOLTAMMETRY AT A MICRODISK ELECTRODE

96 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. Results and Discussion 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). 4 VOLTAMMETRY AT A MICRODISK ELECTRODE

97 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): I lim = 4nFcDr e (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. 5 VOLTAMMETRY AT A MICRODISK ELECTRODE

98 Notes About the COMSOL Implementation 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, A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Second Edition, Hoboken, Y. Saito, Review of Polarography (Japan), vol. 15, pp , Model Library path: Electrochemistry_Module/Electroanalysis/ microdisk_voltammetry 6 VOLTAMMETRY AT A MICRODISK ELECTRODE

99 Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D Axisymmetric button. 2 In the Select physics tree, select Electrochemistry>Electroanalysis (elan). 3 Click the Add button. 4 In the Concentrations table, enter the following settings: cred cox 5 Click the Study button. 6 In the tree, select Preset Studies>Stationary. 7 Click the Done button. GLOBAL DEFINITIONS Load the model parameters from a text file. Parameters 1 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 microdisk_voltammetry_parameters.txt. GEOMETRY 1 Draw the model geometry as a quarter circle, and specify the electrode disk radius using a point. Circle 1 1 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. 7 VOLTAMMETRY AT A MICRODISK ELECTRODE

100 3 In the Radius edit field, type r_max. 4 In the Sector angle edit field, type 90. Point 1 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 r edit field, type re. Circle 1 Add a second circle that will be used to set up an Infinite Element domain. Circle 2 1 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* In the Model Builder window, right-click Geometry 1 and choose Build All Objects. 5 Click the Zoom Extents button on the Graphics toolbar. 8 VOLTAMMETRY AT A MICRODISK ELECTRODE

101 DEFINITIONS 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 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description i_el intop1(elan.itot) Working electrode current Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 2 only. 5 Locate the Advanced section. Select the Compute integral in revolved geometry check box. Infinite Element Domain 1 Add an Infinite Element domain, and assign it to the outer domain. 1 On the Definitions toolbar, click Infinite Element Domain. 2 Select Domain 2 only. ELECTROANALYSIS Diffusion 1 Now start setting up the physics. Start with the diffusion coefficients. 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 D cred edit field, type D1. 4 In the D cox edit field, type D2. 9 VOLTAMMETRY AT A MICRODISK ELECTRODE

102 Electrode Surface 1 Add 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. 2 In the Electrode Surface settings window, locate the Boundary Condition section. 3 From the Boundary condition list, choose Electrode potential. 4 In the E vsref edit field, type E_appl. 5 Select Boundary 2 only. Electrode Reaction 1 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 Equilibrium Potential section. 3 In the E eq,ref edit field, type Ef. 4 Locate the Electrode Kinetics section. In the k 0 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 1 Specify the bulk composition at the outer boundary. 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 Select Boundary 7 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species cred check box. 5 In the c 0,cRed edit field, type c_bulk. 6 Select the Species cox check box. Initial Values 1 1 In the Model Builder window, under Component 1>Electroanalysis click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the cred edit field, type c_bulk. MESH 1 Edit the default mesh to ensure good numerical accuracy. 10 VOLTAMMETRY AT A MICRODISK ELECTRODE

103 Size 1 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. 3 In the Size settings window, locate the Element Size section. 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 1 1 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. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 1 only. Size 1 1 Right-click Component 1>Mesh 1>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 Point. 4 Select Point 4 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type re/100. Mapped 1 1 In the Model Builder window, right-click Mesh 1 and choose Mapped. 2 In the Mapped settings window, locate the Domain Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 2 only. 5 Right-click Mesh 1 and choose Build All. STUDY 1 The problem is now ready for solving. Simulate a voltammogram by using a Parametric Sweep for a range of applied potentials. 11 VOLTAMMETRY AT A MICRODISK ELECTRODE

104 Parametric Sweep 1 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. 4 In the table, enter the following settings: Parameter names E_appl Parameter value list range(e_start,e_step,e_vertex) 5 On the Home toolbar, click Compute. RESULTS Two default concentration plots are created by default: one in 2D and one revolved 3D plot. 1D Plot Group 3 Create a plot of the voltammogram as follows. 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 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). 5 On the 1D plot group toolbar, click Global. 6 In the Global settings window, locate the y-axis Data section. 7 In the table, enter the following settings: Expression Unit Description i_el A Working electrode current 8 On the 1D plot group toolbar, click Plot. 12 VOLTAMMETRY AT A MICRODISK ELECTRODE

105 Orange Battery Solved with COMSOL Multiphysics 4.4 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. Metal nails Orange 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: Zn s Zn e - E 0 = 0.82 V (1) 1 ORANGE BATTERY

106 The other nail consists of copper, and here hydrogen evolution is assumed to take place: 2H + + 2e - H 2 g E 0 = 0 V (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: = s l E 0 (3) the initial value becomes: l init = s E 0 = 0 E 0 Zn 0 = E 0 Zn (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/m 3 at the start of the simulation. All boundaries except the zinc electrode are insulated. Results and Discussion 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. 2 ORANGE BATTERY

107 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. 3 ORANGE BATTERY

108 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/m 3 concentration level of zinc ions after running the battery for 5 minutes. Figure 5 shows the how far the 0.2 mol/m 3 isosurface level has reached after 1 hour. 4 ORANGE BATTERY

109 Figure 4: 0.2 mol/m 3 zinc concentration isosurface after 5 minutes. 5 ORANGE BATTERY

110 Figure 5: 0.2 mol/m 3 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. 6 ORANGE BATTERY

111 Figure 6: Cell current vs. time. Model Library path: Electrochemistry_Module/Tutorial_Models/ orange_battery Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Electrochemistry>Secondary Current Distribution (siec). 3 Click the Add button. 7 ORANGE BATTERY

112 4 Click the Study button. 5 In the tree, select Preset Studies>Stationary. 6 Click the Done button. GEOMETRY 1 Start by drawing the geometry; one sphere (the orange) and two cylinders (the metal nails). Sphere 1 1 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 1 1 On the Geometry toolbar, click Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 2e-3. 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 2 1 On the Geometry toolbar, click Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 2e-3. 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. DEFINITIONS Use selections to group different parts of the geometry together for easier selection when setting up the model. Explicit 1 1 On the Definitions toolbar, click Explicit. 2 Select Domain 1 only. 3 Right-click Component 1>Definitions>Explicit 1 and choose Rename. 8 ORANGE BATTERY

113 4 Go to the Rename Explicit dialog box and type Orange in the New name edit field. 5 Click OK. Explicit 2 1 On the Definitions toolbar, click Explicit. Enabling transparency makes it easier to find and select objects within other objects. 2 Click the Transparency button on the Graphics toolbar. 3 Select Domains 2 and 3 only. 4 Right-click Component 1>Definitions>Explicit 2 and choose Rename. 5 Go to the Rename Explicit dialog box and type Zinc nail in the New name edit field. 6 Click OK. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 Select Domains 4 and 5 only. 3 Right-click Component 1>Definitions>Explicit 3 and choose Rename. 4 Go to the Rename Explicit dialog box and type Copper nail in the New name edit field. 5 Click OK. Explicit 4 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 5 7, 13, and 15 only. 5 Right-click Component 1>Definitions>Explicit 4 and choose Rename. 6 Go to the Rename Explicit dialog box and type Zinc nail active electrode surface in the New name edit field. 7 Click OK. Explicit 5 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 21 23, 29, and 31 only. 9 ORANGE BATTERY

114 5 Right-click Component 1>Definitions>Explicit 5 and choose Rename. 6 Go to the Rename Explicit dialog box and type Copper nail active electrode surface in the New name edit field. 7 Click OK. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description E_eq_Zn -0.82[V] V Zinc electrode reference potential E_cell 0.5[V] V Cell voltage SECONDARY CURRENT DISTRIBUTION Now, start setting up the Secondary Current Distribution model. Start with the current conduction in the metal nails. Electrode 1 1 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 2 1 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 1 The Electrolyte is the default domain type, and there is no need to select the domain in this case. 10 ORANGE BATTERY

115 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 The following steps set up the Zn electrode and its corresponding electrode reaction: Electrolyte-Electrode Domain Interface 1 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 Zinc nail active electrode surface. Electrode Reaction 1 1 In the Model Builder window, expand the Electrolyte-Electrode Domain Interface 1 node, then click Electrode Reaction 1. 2 In the Electrode Reaction settings window, locate the Equilibrium Potential section. 3 In the E eq,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 i 0 edit field, type 1e-1. Electrolyte-Electrode Domain Interface 2 The 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 1 1 In the Model Builder window, expand the Electrolyte-Electrode Domain Interface 2 node, then click Electrode Reaction 1. 2 In the Electrode Reaction settings window, locate the Electrode Kinetics section. 3 In the i 0 edit field, type ORANGE BATTERY

116 Electric Ground 1 Ground the Zn electrode, set the other electrode to the cell voltage. 1 On the Physics toolbar, click Boundaries and choose Electric Ground. 2 Select Boundary 12 only. Electric Potential 1 1 On the Physics toolbar, click Boundaries and choose Electric Potential. 2 Select Boundary 28 only. 3 In the Electric Potential settings window, locate the Electric Potential section. 4 In the s,bnd edit field, type E_cell. Initial Values 2 Provide 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 1 1 In the Model Builder window, under Component 1>Secondary Current Distribution click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the phil edit field, type -E_eq_Zn. STUDY 1 The model is now ready for solving. On the Home toolbar, click Compute. RESULTS Electrolyte Potential (siec) Use an isosurface for visualizing the potential field in the electrolyte. 1 Click the Transparency button on the Graphics toolbar. 3D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 12 ORANGE BATTERY

117 2 In the Model Builder window, under Results right-click 3D Plot Group 3 and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Potential Isosurface in the New name edit field. 4 Click OK. Potential Isosurface 1 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 On the 3D plot group toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. 3D Plot Group 4 The following steps create a plot of the norm of the current density that is suitable here to visualize the currents within the metal nails. 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 4 and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Electrode Current in the New name edit field. 4 Click OK. Electrode Current 1 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). 3 On the 3D plot group toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. COMPONENT 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. 13 ORANGE BATTERY

118 ADD PHYSICS 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. TRANSPORT OF DILUTED SPECIES 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. 5 Select the Migration in electric field check box. Diffusion and Migration 1 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 z c edit field, type 2. Electrode-Electrolyte Interface Coupling 1 The 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. 2 Click the Transparency button on the Graphics toolbar. 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 1 1 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. 14 ORANGE BATTERY

119 3 From the i loc list, choose Local current density (siec/bei1/er1). 4 Locate the Stoichiometric Coefficients section. In the n m edit field, type 2. 5 In the c edit field, type -1. Initial Values 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the c edit field, type c_ref. GLOBAL DEFINITIONS Note that c_ref is colored orange, this is because the parameter is not yet defined. Define it now. Parameters 1 In the Model Builder window, under Global Definitions click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description c_ref 0.01[mol/m^3] mol/m³ Reference concentration SECONDARY CURRENT DISTRIBUTION Also, modify the electrode kinetics to be dependent on the local concentration of zinc ions. Electrode Reaction 1 1 In the Model Builder window, under Component 1>Secondary Current Distribution>Electrolyte-Electrode Domain Interface 1 click Electrode Reaction 1. 2 In the Electrode Reaction settings window, locate the Electrode Kinetics section. 3 From the Kinetics expression type list, choose Concentration dependent kinetics. 4 In the C O edit field, type c/c_ref. ROOT Create a new time-dependent study for the concentration simulation. 1 On the Home toolbar, click Add Study. 15 ORANGE BATTERY

120 ADD STUDY 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. STUDY 2 Step 1: Time Dependent 1 In the Time Dependent settings window, locate the Study Settings section. 2 In the Times edit field, type range(0,60,3600). 3 On the Home toolbar, click Compute. RESULTS 3D Plot Group 9 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 9 and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Concentration Isosurface in the New name edit field. 4 Click OK. 5 In the 3D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Solution 2. 7 From the Time (s) list, choose 300. Concentration Isosurface 1 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 On the 3D plot group toolbar, click Plot. 6 Click the Zoom Extents button on the Graphics toolbar. 16 ORANGE BATTERY

121 7 Click the Transparency button on the Graphics toolbar. The following steps create an animation of the zinc ion isosurface during the simulated time: Export 1 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. 7 Click the Zoom Extents button on the Graphics toolbar. Derived Values Finally, 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. 3 From the Data set list, choose Solution 2. 4 Select Boundary 28 only. 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. 17 ORANGE BATTERY

122 18 ORANGE BATTERY Solved with COMSOL Multiphysics 4.4

123 Thin Layer Chronoamperometry 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. DOMAIN EQUATIONS The transport of the analyte obeys the diffusion equation (Fick s 2nd Law): c ---- = D c t (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. BOUNDARY EQUATIONS 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 1 THIN LAYER CHRONOAMPEROMETRY

124 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. TIME-DEPENDENT STUDY 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: L 2 t = 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. Results and Discussion 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. 2 THIN LAYER CHRONOAMPEROMETRY

125 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: i = nfc ---- D t (3) 3 THIN LAYER CHRONOAMPEROMETRY

126 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. 4 THIN LAYER CHRONOAMPEROMETRY

127 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, A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Second ed., Hoboken, F.G. Cottrell, Zeitschrift für Physikalische Chemie, vol. 42, pp , Model Library path: Electrochemistry_Module/Electroanalysis/ thin_layer_chronoamperometry Modeling Instructions From the File menu, choose New. 5 THIN LAYER CHRONOAMPEROMETRY

128 NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select Electrochemistry>Electroanalysis (elan). 3 Click the Add button. 4 In the Number of species edit field, type 1. 5 In the Concentrations table, enter the following settings: c 6 Click the Study button. 7 In the tree, select Preset Studies>Time Dependent. 8 Click the Done button. GLOBAL DEFINITIONS Add the model parameters from a text file. Parameters 1 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 thin_layer_chronoamperometry_parameters.txt. GEOMETRY 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. 2 In the Interval settings window, locate the Interval section. 3 In the Right endpoint edit field, type L. 4 In the Model Builder window, right-click Geometry 1 and choose Build All Objects. 6 THIN LAYER CHRONOAMPEROMETRY

129 DEFINITIONS 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 Click to expand the Smoothing section. In the Size of transition zone edit field, type 1. Variables 1 Add 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. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description i_el -F_const*A_el*elan. A Total Current tfluxx_c i_cottrell F_const*A_el*c_bulk *sqrt(d/(pi*t+eps)) A Cottrell equation current ELECTROANALYSIS (ELAN) Diffusion 1 Now 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. 2 In the Diffusion settings window, locate the Diffusion section. 3 In the D c edit field, type D. Initial Values 1 1 In the Model Builder window, under Component 1 (comp1)>electroanalysis (elan) click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the c edit field, type c_bulk. 7 THIN LAYER CHRONOAMPEROMETRY

130 Concentration 1 Set up the Concentration condition on the boundary using the step function defined above. 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 Select Boundary 1 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species c check box. 5 In the c 0,c edit field, type c_bulk*(1-step1(t/t_rise)). MESH 1 Refine the default mesh. Size 1 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. 3 In the Size settings window, locate the Element Size section. 4 From the Predefined list, choose Extra fine. Size 1 1 In the Model Builder window, under Component 1 (comp1)>mesh 1 right-click Edge 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. 4 Select Boundary 1 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type x_step/5. Edge 1 1 Right-click Edge 1 and choose Build Selected. 8 THIN LAYER CHRONOAMPEROMETRY

131 Your finished mesh should now look like this: Figure 5: Mesh for the thin layer chronoamperometry study. STUDY 1 Step 1: Time Dependent Your 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). 4 On the Home toolbar, click Compute. RESULTS Concentration (elan) The first default plot group shows the concentration. 1D Plot Group 2 Create the chronoamperogram as follows: 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Point Graph. 9 THIN LAYER CHRONOAMPEROMETRY

132 3 Select Boundary 1 only. 4 In the Point Graph settings window, locate the y-axis Data section. 5 In the Expression edit field, type i_el. 6 In the Model Builder window, click 1D Plot Group 2. 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 In the x maximum edit field, type In the y minimum edit field, type In the y maximum edit field, type 5e On the 1D plot group toolbar, click Plot. 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 1 Now, 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 equation 1 On the 1D plot group toolbar, click Global. 2 In the Global settings window, locate the y-axis Data section. 3 In the table, enter the following settings: Expression Unit Description i_cottrell A Cottrell equation current 10 THIN LAYER CHRONOAMPEROMETRY

133 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. 9 On the 1D plot group toolbar, click Plot. 10 Click to expand the Axis section. Select the Manual axis limits check box. 11 In the x minimum edit field, type 1e On the 1D plot group toolbar, click Plot. Derived Values Finally, 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. 2 Select Domain 1 only. 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. TABLE In the Table window, click Table Graph. RESULTS 1D Plot Group 4 1 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. 11 THIN LAYER CHRONOAMPEROMETRY

134 12 THIN LAYER CHRONOAMPEROMETRY Solved with COMSOL Multiphysics 4.4

135 Electrochemical Treatment of Tumors 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. 2Cl - = Cl 2 g + 2e - 2H 2 O = 4H + + O 2 g + 4e - (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 (r a, r r ) where r a is 1 mm and r r 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 1 ELECTROCHEMICAL TREATMENT OF TUMORS

136 Equation 1. Anode Computational domain 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 c i + D t i c i z i u mi Fc i l = R i where c i is the concentration (mol/m 3 ), D i give the diffusivities (m 2 /s), z i equals the charge, u mi represents the mobility ((mol m 2 )/(J s)), and R i is the production term for species i (mol/(m 3 s)), F denotes Faraday s constant (C/mol), and l is the electrolyte potential (V). The mobility, u m i, can be expressed in terms of D i, R, and T as u mi = D i RT The conservation of electric charge is obtained through the divergence of the current density: N 2 F z i D i c i z i u mi Fc i l = 0 i = 1 2 ELECTROCHEMICAL TREATMENT OF TUMORS

137 At the electrode surface (r r a ) 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 N i n = ij j j n j F where N i is the flux, ij represents the stoichiometric coefficient for the ionic species i in reaction j, and n j 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/p b, and concentration, C = c/c b, (where b denotes the reference concentration), the current density for the oxygen evolution is j I = j I 0 F I 2RT, e 1 4 C + H e where j I0, is the exchange current density (A/m 2 ) and I is the overpotential for the oxygen evolution reaction, defined as P O2 F I 2RT I = s l E eq I (2) where E eq,i (V) is the equilibrium potential for the oxygen evolution reaction. The chlorine evolution reaction is similarly given by the expression j II = j II, 0 C Cl e Using the input values n I n I 1, H,I 1, and Cl,II 1, gives the fluxes at the electrode surface: F II RT 1 2 e P Cl2 F II RT N H n = j --- I F N Cl n j II = F 3 ELECTROCHEMICAL TREATMENT OF TUMORS

138 At the exterior boundary, assume the concentration is constant, c i c i0, and set the potential to where t denotes time. t V 0 t = ln s V The initial concentration is constant: c i = c i0. You obtain the initial potential profile from the solution of the domain equations and boundary conditions at t 0, yielding where V ra0 is the potential satisfying j I j II j 0 and 0 is the conductivity at t 0. The input data is listed in this table: TABLE 1: INPUT DATA PARAMETER VALUE DESCRIPTION D Na m 2 /s Diffusion coefficient, Na + D H m 2 /s Diffusion coefficient, H + D Cl m 2 /s Diffusion coefficient, Cl - c Na, mol/liter Inlet concentration, Na + c H, mol/liter Inlet concentration, H + c Cl, mol/liter Inlet concentration, Cl - j A/m 2 Initial current density j I, A/m 2 Exchange current density, reaction I j II,0 10 A/m 2 Exchange current density, reaction II V ra, V Initial anode potential E eq,i 1.23 V Equilibrium potential, reaction I E eq,ii 1.36 V Equilibrium potential, reaction II T 298 K Temperature p O2 p Cl2 1 atm 1 atm j ra0 r a r Vt = 0 V ra a = + log---- r 0 4 ELECTROCHEMICAL TREATMENT OF TUMORS

139 Results and Discussion 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. 5 ELECTROCHEMICAL TREATMENT OF TUMORS

140 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. 6 ELECTROCHEMICAL TREATMENT OF TUMORS

141 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 7 ELECTROCHEMICAL TREATMENT OF TUMORS

142 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, Model Library path: Electrochemistry_Module/ Electrochemical_Engineering/tumor Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. 8 ELECTROCHEMICAL TREATMENT OF TUMORS

143 MODEL WIZARD 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). 3 Click the Add button. 4 In the Number of species edit field, type 3. 5 In the Concentrations table, enter the following settings: Na H Cl 6 Click the Study button. 7 In the tree, select Preset Studies>Time Dependent. 8 Click the Done button. GLOBAL DEFINITIONS Start by loading the model parameters and variables from text files. Parameters 1 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 tumor_parameters.txt. DEFINITIONS Variables 1 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model s Model Library folder and double-click the file tumor_variables.txt. 9 ELECTROCHEMICAL TREATMENT OF TUMORS

144 GEOMETRY 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. 2 In the Interval settings window, locate the Interval section. 3 In the Left endpoint edit field, type r_a. 4 In the Right endpoint edit field, type r_ext. 5 Click the Build Selected button. 6 Click the Zoom Extents button on the Graphics toolbar. TERTIARY CURRENT DISTRIBUTION, NERNST-PLANCK (TCDEE) Electrolyte 1 Now 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. 3 In the T edit field, type T. 4 Locate the Diffusion section. In the D Na edit field, type D_Na. 5 In the D H edit field, type D_H. 6 In the D Cl edit field, type D_Cl. 7 Locate the Migration in Electric Field section. In the z Na edit field, type z_na. 8 In the z H edit field, type z_h. 9 In the z Cl edit field, type z_cl. Electrolyte Potential 1 Now set the potential of the exterior end, and then the concentration. 1 On the Physics toolbar, click Boundaries and choose Electrolyte Potential. 2 Select Boundary 2 only. 3 In the Electrolyte Potential settings window, locate the Electrolyte Potential section. 4 In the l,bnd edit field, type V0. Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 10 ELECTROCHEMICAL TREATMENT OF TUMORS

145 2 Select Boundary 2 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species H check box. 5 In the c 0,H edit field, type H0. 6 Select the Species Cl check box. 7 In the c 0,Cl edit field, type Cl0. Electrolyte-Electrode Boundary Interface 1 Now set up the anode, and the anode reactions. 1 On the Physics toolbar, click Boundaries and choose Electrolyte-Electrode Boundary Interface. 2 Select Boundary 1 only. Electrode Reaction 1 1 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 Equilibrium Potential section. In the E eq,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 i 0 edit field, type j_i0. 7 In the C O edit field, type H/H0. 8 Locate the Stoichiometric Coefficients section. In the H edit field, type -1. Electrode Reaction 2 1 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. 2 In the Electrode Reaction settings window, locate the Model Inputs section. 3 In the T edit field, type T. 4 Locate the Equilibrium Potential section. In the E eq,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 i 0 edit field, type j_ii0. 11 ELECTROCHEMICAL TREATMENT OF TUMORS

146 7 In the C R edit field, type Cl/Cl0. 8 Locate the Stoichiometric Coefficients section. In the Cl edit field, type 1. Initial Values 1 1 In the Model Builder window, under Component 1 (comp1)>tertiary Current Distribution, Nernst-Planck (tcdee) click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 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. MESH 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. Size 1 In the Model Builder window, under Component 1 (comp1)>mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 From the Predefined list, choose Extra fine. Size 1 1 In the Model Builder window, under Component 1 (comp1)>mesh 1 right-click Edge 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. 4 Select Boundary 1 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 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 ELECTROCHEMICAL TREATMENT OF TUMORS

147 STUDY 1 Step 1: Time Dependent Modify 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 Solver 1 1 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. 5 On the Home toolbar, click Compute. RESULTS Concentration (tcdee) The following instructions show how to create the plots shown in Figure 2 to Figure 5. 1 In the 1D Plot Group settings window, locate the Data section. 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. 9 On the 1D plot group toolbar, click Plot. Compare the plot with that shown in Figure In the Model Builder window, right-click Concentration (tcdee) and choose Rename. 13 ELECTROCHEMICAL TREATMENT OF TUMORS

148 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 1 1 Right-click Concentration (tcdee) and choose Duplicate. 2 In the Model Builder window, expand the Proton concentration 1 node, then click Line Graph 1. 3 In the Line Graph settings window, locate the y-axis Data section. 4 In the Expression edit field, type Cl. 5 On the 1D plot group toolbar, click Plot. 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 1 1 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. 3 In the Line Graph settings window, locate the y-axis Data section. 4 In the Expression edit field, type ph. 5 On the 1D plot group toolbar, click Plot. 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. 1D Plot Group 6 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. Finally, reproduce the plot shown in Figure 5. 2 On the 1D plot group toolbar, click Point Graph. 3 Select Boundary 1 only. 4 In the Point Graph settings window, locate the y-axis data section. 14 ELECTROCHEMICAL TREATMENT OF TUMORS

149 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. 7 In the table, enter the following settings: Legends Oxygen evolution 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. 12 Locate the Legends section. In the table, enter the following settings: Legends Chlorine evolution 13 On the 1D plot group toolbar, click Plot. 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. 17 Locate the Legends section. In the table, enter the following settings: Legends Total electrode current density 18 On the 1D plot group toolbar, click Plot. 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. 15 ELECTROCHEMICAL TREATMENT OF TUMORS

150 16 ELECTROCHEMICAL TREATMENT OF TUMORS Solved with COMSOL Multiphysics 4.4

151 Wire Electrode Solved with COMSOL Multiphysics 4.4 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 WIRE ELECTRODE

152 . flat electrode (cathode) wire electrode (anode) outlet flat electrode (cathode) inlet 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 PRIMARY CURRENT DISTRIBUTION 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 N i = 0 (1) where N i is the flux of species i (mol m2/s), which in turn is governed by: D i c i z i m i Fc i l + c i u = N i (2) where c i represents the concentration of the ion i (mol/m 3 ), z i its valence, D i its diffusivity (m 2 /s), m i its mobility (mol m 2 (s V A)), F denotes the Faraday constant 2 WIRE ELECTRODE

153 (As/mol), i 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: i = F z i N i (3) where i is the current density vector (A/m 2 ). Combining the three above equations, while assuming electroneutrality (which removes the convection term) and negligible concentration gradients (which removes diffusion) leaves: Current density is conserved throughout: 2 i = F z i. (4) m i Fc i l i = 0 (5) so that by combining the valence, ionic mobility, constant concentration and the Faraday constant to a representative conductivity, (1/(W m 2 )), Equation 5 becomes: This final equation is basically Ohm s Law. l = 0. (6) 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, of reference potential in the system: s c = 0 (V), is a convenient choice. (7) The electrode potential equals the difference between the potential of solid phase in the electrode,, and the potential in the adjacent electrolyte, : e E electrode = s l. (8) In the absence of kinetic losses, the cathode potential, E c, equals the equilibrium potential, E eq,c : s c i E eq c = = s c l c l c (9) 3 WIRE ELECTRODE

154 which sets the boundary condition for the cathode. The potential difference over the whole cell, E cell, is defined as the potential difference between the solid phases of the two electrodes E cell = s a s c = s a. (10) In this way the boundary condition for the ionic potential at the anode can be set via: E eq a = s a l a = E cell l a. (11) SECONDARY CURRENT DISTRIBUTION 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/m 2 ), 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: i loc = i 0 exp 1 F RT exp F RT (12) here T is the temperature and R is the gas constant (J/(K mol). i 0, the exchange current density, (A/m 2 ), 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: = E electrode E eq. (13) This results in the following expressions for the overpotentials for the cathode and anode, respectively: c = l c E eq c, (14) a = E cell l a E eq a. (15) 4 WIRE ELECTRODE

155 TERTIARY CURRENT DISTRIBUTION 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/ m 3 ), 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: c i a = i exp 1 F RT exp F RT. (16) c 0 Here c 0 (mol/m 3 ) 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: u + u T + u u + p = 0 u = 0 (17) where is the dynamic viscosity (Ns/m 2 ), density (kg/m 3 ) 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: D c zmfc + cu = 0 (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 5 WIRE ELECTRODE

156 applied for the outlet. Faraday s law is used to specify the net molar flux at the anode where the species is consumed: N = i ---- a a F. (19) Results and Discussion 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. PRIMARY CURRENT DISTRIBUTION 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. 6 WIRE ELECTRODE

157 Figure 3: Primary current distribution, E cell =1.45 V. Potential distribution in the electrolyte (top) and current density on the anode (bottom). SECONDARY CURRENT DISTRIBUTION 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 7 WIRE ELECTRODE

158 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, E cell =1.65 V. Potential distribution in the electrolyte (left) and current density on the anode (right). 8 WIRE ELECTRODE