When interpreting the behavior of an electrical circuit, we often speak of resistances and capacitances. Current passing through a resistor with a resistance *R* (Ω) scales with the applied voltage *V*, while current across a capacitor with a capacitance *C* (F) scales with the rate of change of voltage:

I_\mathrm{res} = \frac{V}{R}

I_\mathrm{cap} = C\,\frac{\partial V}{\partial t}

In an electrochemical cell, we can apply voltage and measure current, or *vice versa*. Therefore, the response of the cell is often considered in terms of resistances and capacitances. In AC impedance analysis, the resistive current is in phase with the applied voltage, while the capacitive current is out of phase. While we can recover resistances and capacitances from the measured data, a wide variety of physical effects contribute to the measured currents. Considering the solution to the equations describing the physical processes that cause resistance and capacitance can provide greater physical insight. In this way, we use simulation as a means of fitting measured data, interpreting it in terms of the underlying physical causes.

*A plot of electrolyte potential (color plot) illustrates the voltage associated with resistance to current flow (black arrows) in a plating cell for electrodeposition on a patterned wafer. The changing resistance of the deposited layer is also included in the model.*

Effects interpreted as resistance in an electrochemical cell most often arise due to the finite conductivity of the electrolyte. Passing current through the electrode, between the anode and cathode, requires a certain applied voltage, just like in a conventional resistor in a circuit. The *Primary Current Distribution* and *Secondary Current Distribution* physics interfaces in COMSOL Multiphysics solve Ohm’s law to predict the magnitude of this voltage, usually called the *ohmic drop*, as a function of electrolyte conductivity and electrode geometry.

When electrode reaction kinetics are important, an additional voltage, called *overpotential*, is required to overcome the activation energy for the electrochemical reaction occurring at an electrode surface. Common electrode reaction kinetic rate laws, such as the Tafel law or Butler-Volmer equations, define the current density in terms of an overpotential — this current-voltage relation can be interpreted as a resistance. However, unlike a simple resistor in an electrical circuit, the impedance due to electrode kinetics is usually nonlinear. This is shown in the Tafel law, which sets the overpotential proportional to the logarithm of current density. The relation is approximately linear only in the limit of small amplitudes of applied voltage, as typically used in electrochemical impedance spectroscopy.

In electroanalysis and impedance spectroscopy, mass transport by diffusion is also often discussed in terms of the measured resistance and capacitance. In traditional equivalent circuit analysis, the Warburg element is a lumped impedance that accounts for the observed effects due to diffusion. Similarly, the Randles circuit is typically used to describe the combination of all of the above effects: diffusion, electrode kinetics, and solution resistance.

However, the influence of kinetics and diffusion on an electrochemical cell’s behavior can be described directly by the corresponding physical equations. In the *Electroanalysis* interface, Fick’s laws of diffusion are solved together with electrode kinetic relations, such as the Tafel or Butler-Volmer equations, yielding impedance spectra directly in terms of the fundamental electrochemistry. In the tertiary current density interfaces, mass transport resistance may also include the contribution from migration and advection, in addition to diffusion.

To see an example of predicting an electrochemical impedance spectrum from physical equations, without using an equivalent circuit, check out our Electrochemical Impedance Spectroscopy tutorial and the lithium-ion battery examples.

Another important situation that introduces resistive and capacitive effects is the behavior of thin layers or surface films at the electrode-electrolyte interface. In these cases, it is often more straightforward to express the behavior directly in terms of a resistance or capacitance.

*Film resistance* occurs whenever a thin layer of material is created at an electrode surface, with significantly different electrical conductivity from the bulk electrode material. This additional resistance can lower the system’s efficiency. In the case of a battery, for example, less of the generated cell voltage from the electrochemical reactions is available to the external circuit as power.

When exposed to air, most metals quickly become oxidized at the surface and therefore are covered with a layer of metal oxide. The metal oxides tend to conduct electricity far more poorly than the original metal, thus introducing an additional film resistance into the system. This can also occur in subsea corrosion, as demonstrated in our anode film resistance example.

Another example of film resistance is in electrodeposition, where the bulk electrode material and depositing species are not the same. A layer of increasing thickness with altered conductivity is then deposited on the surface, with different properties than the bulk electrode.

Film resistance can be defined at an electrode surface in COMSOL Multiphysics. The most straightforward option is to set a given film resistance in Ω·m^{2}. The units of m^{2} are required to express the local resistance for a current density in A m^{-2} normal to the film, with a potential difference in V across the film that drives the current flow. Alternatively, we can provide a reference thickness, change in thickness, and thin film conductivity. This is useful when the film’s thickness changes with time.

*Film resistance settings. Left: A constant film resistance for an electrode with a surface shape and composition that doesn’t change. Right: A more general case where the film thickness can change during the simulation, under ongoing electrodeposition.*

At the interface between a charged electrode and the electrolyte, a region known as the *electrical double layer* (EDL) arises. Here, electrical attraction and repulsion cause a highly nonuniform distribution of ion concentrations. (In a future blog post, we will discuss the physics and modeling of the EDL in greater detail. In the meantime, check out this previous blog post describing the flow inside a battery.)

Changing the electrode potential through a time-varying applied voltage generates an accumulation or release of charge in the double layer. The amount of charge accumulated in the double layer is described by a *double-layer capacitance*. The scale of the double layer is given by the Debye length for the system, which is typically in the order of nanometers. As such, in comparison to the typical size of a diffusion layer or electrochemical cell on the micrometer-to-meter scale, this capacitance can be considered as a surface effect at the electrode-electrolyte interface.

In COMSOL Multiphysics, you will find several tools for modeling the EDL in detail (see our Diffuse Double Layer with Charge Transfer tutorial model). This is now easier than ever before with the new *Nernst-Planck-Poisson Equations* multiphysics interface available in COMSOL Multiphysics® version 5.2a. However, despite great theoretical effort, it is not possible to devise a model that accurately accounts for all of the experimentally observed phenomena in the EDL. For this reason, an empirical description of the double-layer capacitance may be preferred, where the capacitance is defined empirically by fitting data from experiments.

Double-layer capacitance is especially important in impedance spectroscopy, where the voltage is varied rapidly at high frequencies. Together with the finite electrode reaction rate and a finite rate of mass transport, it is a key factor in the characteristic “semicircle with a tail” shape of the Nyquist plot for many electrochemical systems.

*An experimentally measured Nyquist plot for a lithium-ion battery, derived from our Lithium-Ion Battery Impedance demo app.*

When defining an electrode surface, we can add a *Double Layer Capacitance* subnode. This electrode feature always requires a surface capacitance per area (in F m^{-2}), though certain circumstances may require additional inputs. For example, for a porous electrode, the *Porous Matrix Double Layer Capacitance* feature requires the input of a specific surface area to determine the true surface area of the electrode-electrolyte interface, where a double layer exists and at which capacitive charging occurs. You can also use the built-in tools to calculate the double layer area of a porous electrode based on the particle properties of spheres, cylinders, or flakes. In this case, there is no need to explicitly know the specific surface area of the electrode.

*An overview of the settings for the Double Layer Capacitance feature. The differences between nonporous and porous electrodes are shown, as well as the possibility to calculate the double layer surface area from some fundamental geometric properties instead of adding this directly as input.*

There may be occasional cases where it is desirable to connect an electrode to a more complex circuit in order to see how your electrochemical system couples with a complex driving circuit. Again, thanks to the flexibility of COMSOL Multiphysics, connecting any of the electrochemical physics interfaces to the *Electrical Circuit* interface is possible.

*A typical setup for coupling a current-driven electrochemical system to a circuit representing instrumentation that supplies the current. Note how the currents in the *Electrode Current* and *External Device* settings are set to a common variable I_couple. The value of this variable is determined by constraining the voltages associated with these two nodes to be equal.*

In today’s blog post, we have explained the approaches for describing electrochemical resistances and capacitances, while highlighting different cases in which each can be applied. We encourage you to use these tools and the knowledge that you have gained here to maximize the accuracy and usefulness of your electrochemical models.

- To learn more about including resistance and capacitance features in your electrochemical models, download the following tutorials:
- Watch this archived webinar for further details on simulating electrochemical systems in COMSOL Multiphysics

A tailored electroplating app can be built using the Application Builder and the Electrodeposition Module in COMSOL Multiphysics version 5.0. With this resource, PCB designers can use simulation to analyze a number of factors in the design and manufacturing process. They can evaluate whether a design will be adequate for achieving copper wiring specifications, assess the performance of such devices, and estimate the electroplating process’ manufacturing cost, without any prior knowledge of electroplating.

A common *printed circuit board* (PCB) uses a layer or multiple layers of copper wires to connect to the board’s active and passive devices. The more advanced PCBs, on the other hand, use *copper pattern plating* to generate the wires. Before the actual electroplating process can take place, the PCB needs to be prepared with a patterned insulating film. This particular process is done in a number of steps.

The first step is to cover the PCB with a thin conducting copper seed layer. Next, the PCB’s surface needs to be coated with a *photoresist* (a photosensitive polymer film), a process referred to as *photolithography*. This process exposes the resist to ultraviolet light through a patterned photomask and then dissolves the areas that were exposed. The result is a PCB with a patterned insulating film that exposes the seed layer on the bottom of the pattern.

*A seed layer is applied to the PCB (left). The PCB patterned with a photoresist using photolithography (right).*

During the electroplating process, the PCB is submerged into an electroplating bath, which is an electrolyte containing sulphuric acid and copper sulphate, together with copper anodes (e.g., solid copper bars). A voltage is applied between the anodes and the seed layer, the cathode, which results in the electrochemical reduction of the copper ions to copper metal that is plated (deposited) on the seed layer. The thickness of the deposited layer is directly proportional to the rate of the electrochemical reaction over time, which is given by the current density at different positions in the seed layer over time. As a result, the patterned photoresist’s cavities are filled with solid copper. The plating rate is maintained by controlling the average current density (i.e., the total current over the pattern area to be plated).

Finally, the remaining photoresist is stripped away and the thin seed layer is etched to isolate the plated copper wires from each other.

*Copper is electrodeposited on the conducting seed layer, thereby filling up the patterned photoresist’s cavities on the PCB (left). The photoresist is stripped away and the exposed seed layer is etched to isolate the copper wires from each other (right).*

A known issue with this process is that the plating rate is not always uniform across the entire PCB. The electric field in the electrolyte is concentrated to the conducting pattern that is surrounded by large insulating areas, as well as in the pattern close to the PCB’s edges. These nonuniformities in the electric field lead to locally higher current densities at the surface of the cathode in those areas — an effect commonly referred to as *current crowding*. The plated thickness is proportional to the current density over time, which results in an unwanted thickness variation of the copper wires across the PCB. This means that there will be a resistance variation between copper wires in different positions in the PCB. That variation can be the root cause of performance issues or, in the worst case scenario, device failure when the PCB is used in an electronic device.

*During the copper pattern plating step, the PCB is submerged into an electroplating bath (an electrolyte) together with copper anodes (left). When a voltage is applied between the anode and the PCB, copper is deposited to form the wire pattern. The electrical field from the anode to the conducting parts on the PCB is crowded in the pattern that is close to the large insulating areas and the edge of the PCB (illustrated by the colored electric field streamlines in the image to the left). This leads to locally higher copper thickness in these areas (seen in the red-colored parts of the wire pattern in the image to the right).*

To avoid reduced performance or device failure during the operation of electronic devices, the copper circuits must meet a set of thickness uniformity specifications. Normally, the printed circuit board designer would rely on simple design rules such as maximum and minimum lines, spaces, and pattern density. However, by using electroplating simulations, a much more accurate estimation of the expected copper thickness variation can be achieved. With this information, it is possible to modify the design at an early stage instead of waiting for prototype results.

In order to reduce current crowding, a “dummy” pattern can be included in the design where there would normally be large insulating areas. In that case, the dummy pattern would receive some of the current, which reduces the high current density in the actual wiring pattern. Parts of the dummy pattern may still receive a high current density, but since it is not part of the actual wiring, this is not an issue. With simulations, it is fast and easy to redesign and evaluate the resulting thickness uniformity for different pattern layouts.

*In order to reduce copper pattern thickness variation, a dummy pattern can be included where there would normally be large insulating areas. In the image to the left, the red-colored areas show the high thickness portions of the copper pattern that are close to insulating areas. The image to the right shows how a dummy pattern can be included to reduce the thickness variation in the copper wiring pattern.*

An additional step in reducing thickness variation is related to the set-up of the electroplating bath. To reduce the current crowding effect in the edges, a so-called *aperture* can be used.

The aperture is basically an insulating shield with an opening that is placed between the copper anodes and the PCB in the plating bath. The aperture opening must be smaller than the PCB dimensions in order to reduce the edge current crowding. Other than that, it is difficult to guess the optimal size and placement of the aperture.

Luckily, optimization through simulation is fairly quick and easy. In the illustrations below, an aperture with a rectangular opening is simulated. The length and width of the opening as well as the placement of the aperture in the bath are optimized to minimize the thickness variation across the PCB.

*To avoid the current crowding effect close to the edge of the PCB and aperture (as seen in the left image), an insulating shield with an opening can be placed between the anodes and the PCB in the electroplating bath. The right image shows the aperture where the size of its opening and placement in the bath has been optimized using simulation to give the minimum thickness variation.*

If PCB manufacturers want to be competitive, it is always essential to consider manufacturing costs. As mentioned above, there is usually a copper thickness uniformity specification that the final product needs to meet. The thickness uniformity is inherently dependent on the total plating rate used in the electroplating process; the higher the overall rate, the more thickness variation. Also, the total process time determines the throughput of the production line and, therefore, the manufacturing cost.

To minimize the manufacturing cost, the process is run at the highest possible rate while still achieving the thickness specifications. By using simulations to study the impact of the plating rate, it is possible to estimate which plating rate can be run for a given thickness uniformity specification. This enables the estimation of the manufacturing cost in the design phase.

By improving the design or using an aperture to improve the uniformity, it is possible to simulate how much of a higher plating rate this would enable as well as how much money would be saved during PCB production.

Electroplating simulation models are created by people with electrochemistry backgrounds and an understanding of simulation models and software. The PCB designer is normally skilled in electric design but has little or no knowledge about the electrochemical process that goes on during manufacturing.

Many of the benefits of electroplating simulations have been discussed, but how do you make the simulation models available for PCB designers?

One solution is to build an electroplating app with a tailored easy-to-use interface that allows PCB designers to study important parameters and run simulations with a few simple clicks.

With the Application Builder in COMSOL Multiphysics version 5.0, simulation experts can create these kinds of apps with very little effort and make simulations available to other stakeholders in an organization.

An electroplating app allows the PCB designer to import different designs (with or without dummy patterns), click Compute, and visualize the simulated thickness uniformity. It is also possible to change the dimensions of the plating bath and anodes as well as include an aperture. With a simple click, the app can also be run to optimize the dimensions and placement of the aperture. Finally, the app can be used to find out the maximum plating rate for a given thickness uniformity specification. With this information, the manufacturing cost can be estimated.

*The user interface from an electroplating app. It allows PCB designers to upload different designs, modify the dimensions of the electroplating bath, and (optionally) include an aperture with certain dimensions.*

*With an electroplating app, a user can run a simulation with a simple click. Users can study the copper wire thickness uniformity and how it is affected by different designs, plating rates, and plating bath set-ups. Moreover, the app can also be run to simulate the optimal aperture dimensions for reducing thickness variation. Finally, the app can be used to calculate the maximum plating rate for a given thickness uniformity target.*

We have discussed the importance of simulations for advanced PCBs that use copper pattern electroplating technology. By running electroplating simulations in the design stage, reduced performance or even device failure caused by unwanted thickness variation in the electroplating process can be mitigated.

Traditionally, these kinds of simulation models would not be carried out by PCB designers and are more likely to be ran by electroplating and simulation experts. However, by building an electroplating app with a tailored easy-to-use interface, we can bring electroplating simulations to PCB designers. The designers will be able to run and utilize all of the benefits of simulation in their daily work.

In the end, money can be saved by reducing the number of prototypes and optimizing the design and process to minimize manufacturing costs. Further, similar apps can be incorporated into the manufacturing process and run by the engineers and technicians responsible for this process. This allows them to make small tweaks and calibrations to the operation of the electroplating as well as assist in quality assurance.

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A U.S. penny is made of zinc with a 20 microns thick layer of copper on its surface. When we make the coin, its manufacturing process must be controlled. If pennies were produced in different weights, sizes, or colors, people would have no confidence in the currency. The greater the natural variation in a currency as minted, the easier it is to forge. Compared to people, the demand for quality control is even higher for vending machines and coin counters, as both rely on the weight and shape of coins to function properly.

*Electroplating* is a common manufacturing method that involves applying a thin layer of metal onto another. This takes advantage of the fact that we are plating a metal object that conducts electricity. The object being plated is referred to as the *substrate*.

*Schematic of an electroplating cell with a copper sulfate plating bath. The metal substrate is denoted “Me”. The transfer of electrons in the circuit is balanced by the transfer of ions in the solution.*

To understand how this works, let’s first think about a simple chemical reaction.

In school, you may have come across an experiment in which iron was placed into a copper salt solution. While in the solution, the metal iron spontaneously dissolved in the form of positively charged ferrous (iron) ions, leaving electrons behind. Meanwhile, the positively charged cupric (copper) ions acquired those electrons and deposited them as solid copper. On any metal surface, there is a constant chemical reaction of metal to ions in solution and ions to solid metal.

We can write this as a balanced chemical reaction:

Cu2+(aq) + Fe(s) → Cu(s) + Fe2+(aq)

where (aq) denotes a species dissolved in water and (s) represents a solid material. The superscript “2+” stands for the charge on a species.

When electroplating, a power source is used to apply a voltage to the substrate with respect to a positive electrode, called the *anode*. This is done to manipulate the equilibrium of this reaction. If the substrate becomes negative polarized, a *cathode*, the reaction of ions to metal will be favored. In other words, the voltage drives a reaction in which ions gain electrons to become solid metal. Because the electrons have to come from the conducting substrate and can’t just jump a long way into the solution, the copper is deposited on the cathode surface.

We can write this as:

Cu2+(aq) + 2e-(m) → Cu(s)

where the two electrons come from the metal.

The amount of metal deposited is proportional to the electric current passed. This is because for every atom of metal deposited, a precise number of electrons are drawn to the cathode to balance the ionic charge. In this case, we know that this number is two for cupric ions. Therefore, by measuring the passed current, we can stop the deposition when a certain amount of plating has taken place, i.e., when a certain mass of solid copper has been deposited.

We still don’t know exactly *where* that plating occurs, however. The electrons distribute themselves throughout the conductor, so it’s impossible to tell from the measured current whether the plating is uniform or if it only occurred in one part of the substrate.

In this case, we can turn to physical modeling to predict the *current distribution* of copper on the surface. My colleague, Melanie, recently blogged about the different theoretical types of current distribution.

For example, we find that because sharp edges and corners of solid objects are more exposed to the plating bath, the deposition rate will be greater here (you can see this effect in the Copper Deposition in a Trench model, which is available in our Model Gallery). Ions have to travel further or in competition with other ions between the anode and more “obscure” or distant sites on the substrate. This raises the resistance of the solution along these paths and lowers the driving force for deposition.

A number of options are available for controlling surface deposition. To see illustrations on how to model these, take a look at some models made with the Electrodeposition Module.

In superfilling, a catalyst is carefully localized to selectively accelerate the reaction rate in concave regions, balancing out the current distribution over the whole object.

An alternate method is to add chemical species at the cathode surface and through different mechanisms to inhibit deposition, i.e., contrary to the catalysts and force the current density to become more uniform.

Another route is to use forced flow in the plating bath in order to specifically direct the plating solution to certain regions. In COMSOL Multiphysics, it’s convenient to be able to solve for fluid flow and surface electrochemistry together. The Fountain Flow Effects on Electrodeposition on a Rotating Wafer model provides a good example of this.

Of course, the U.S. Mint isn’t going to tell us how they do it or we’d all be manufacturing money at home. The techniques involved are actually quite expensive. It costs more than a penny to make a penny, which is the price the U.S. government pays for a secure currency to be handled by its citizens. A lot of this cost is due to the rising price of raw metal; whereas the U.S. Mint uses zinc for the bulk of the penny, Canadian and British pennies are made of mild steel. Both methods keep their relative face values and production costs well above their value in raw metal. This proves that the more we know about electroplating, the wider our alternative options can be!

I set up a simple electrodeposition model in COMSOL Multiphysics to look at current flow to a disc-shaped substrate. I found that it’s not easy to get a uniform plating. My cell draws 1 A for five minutes, which is good enough to plate more than 20 microns thickness of copper over the surface of the substrate. However, without any particular attention, we get much more plating at the edges:

*Plot of deposition thickness vs. radial coordinate for electroplating of a disk.*

After five minutes, the plated penny would come out about 16 microns thicker at the edge than in the center. This is only 1% of the total thickness, which would be hard to tell for a non-specialist, but certainly noticeable to a machine or even a forger. One can only imagine how difficult it would be if the substrate to be plated has a complex pattern, such as tiny letters or Abraham Lincoln’s face, for that matter.

So, as it turns out, my first batch of pennies didn’t make the grade. Do you think you can you do better?

Governments around the world will always look for ways to save money. Many companies turn to modeling to better understand the required manufacturing process. One option they could choose is to use the Electrodeposition Module to predict the quality and expense of an operation for an electroplating cell.

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As you saw in the previous blog post, we can use an example model of a wire electrode to compare the three current distribution interfaces. Here is the geometry again:

*Wire electrode model solved using COMSOL Multiphysics. Electrolyte is allowed to flow in the open volume between the wire and flat surfaces.*

The same geometry is considered in all three cases presented here: a wire electrode structure is placed between two flat electrode surfaces. The electrochemical cell can be seen as a unit cell of a larger wire-mesh electrode, which is an electrochemical cell set-up common in many large-scale industrial processes.

Below are the essential equations we mentioned in detail previously:

Nernst-Planck equation:

(1)

\textbf{N}_i = -D_i\nabla c_i-z_i u_{m,i} F c_i\nabla \phi_l+c_i\textbf{u}

Current density expression with the Nernst-Planck equation:

(2)

\textbf{i}_l = -F \left(\nabla \sum_i z_iD_i c_i\right)-F^2\nabla \phi_l \sum_i z^2_i u_{m,i} c_i+\textbf{u}F\sum_i z_ic_i

General electrolyte current conservation:

\nabla\cdot\mathbf{i}_l=Q_l

The primary current distribution accounts only for losses due to solution resistance, neglecting electrode kinetic and concentration-dependent effects. The charge transfer in the electrolyte is assumed to obey Ohm’s law. We are making two assumptions here: first, that the electrolyte is electroneutral, which cancels out the convective contribution to the current density in equation (2), and second, that composition variations in the electrolyte are negligible (it is homogeneous), which cancels out the diffusive contribution to the current density in equation (2) and allows us to treat the ionic strength as a constant. Hence, the remaining term of equation (2) results in Ohm’s law for electrolyte current density.

At the electrode-electrolyte interface we assume that the electrolysis reaction is so fast that we can neglect the influence of electrode kinetics, and so the potential difference at the electrode-electrolyte boundary deviates negligibly from its equilibrium value. In other words, there is no activation overpotential and an arbitrary current density can occur through electrolysis. Therefore, the primary current distribution only depends on the geometry of the anode and cathode.

The *Primary Current Distribution* interface in COMSOL Multiphysics defines two dependent variables: one for the electric potential in the electrolyte () and another for the electric potential in the electrodes (). With the above described assumptions for a primary current distribution, you get the following equations to be solved:

Electrode: with

Electrolyte: with

Electrode-Electrolyte-Interface:

Here, denotes the conductivity of the electrolyte, which is constant by the above assumptions. The index represents the electrode and the electrolyte. denotes the equilibrium potential for reaction .

In the following picture, we show the primary current density distribution for our wire electrode example. As you can see, the current density distribution is highest at the corners of the wires directly facing the cathode plates, and close to zero at the central parts of the wire structure that are geometrically shielded from the cathode.

*Primary current distribution, Ecell = 1.65 V. Current density distribution on the anode (dimensionless).*

You can use this class of current distribution for modeling cells where you have a relatively high electrolyte concentration (in relation to current density) or vigorous mixing in the electrolyte, allowing the assumption of a uniform electrolyte concentration. In addition, the electrochemical reactions have to be fast enough for negligible resistance to be associated with the reaction, compared to the magnitude of the ohmic losses (solution resistance).

One application of these conditions is at the anodes in systems for imposed current cathodic protection, while a constant current corresponding to the mass transport-limited current for oxygen reduction can be set as a boundary condition at the cathode in the primary current density interface (here’s an example of this). This can also be a valuable approximation for electrochemical processes involving relatively fast reactions, such as the oxidation of chloride ions in the chlor-alkali process.

Since the *Primary Current Distribution* interface is easy to solve and involves no nonlinear kinetic expressions, it is often suitable to use in order to calculate a baseline approximation before approaching a more complex model.

The secondary current distribution accounts for the effect of the electrode kinetics in addition to solution resistance. The assumptions about the electrolyte composition and behavior are the same as for the primary current distribution, resulting in Ohm’s law for electrolyte current. The difference between the primary and secondary current distributions lies in the description of the electrochemical reaction at the interface between an electrolyte and an electrode.

Here, the influence of electrode kinetics is included; the potential difference may differ from its equilibrium value due to additional impedance associated with the finite rate of the electrolysis reaction. The difference between the actual potential difference and the equilibrium potential difference is the activation overpotential (). Thus, you get the same domain equations as in the *Primary Current Distribution* interface, but the electrode-electrolyte interface equation differs according to the overpotential:

Electrode: with

Electrolyte: with

Electrode-Electrolyte-Interface:

In the *Secondary Current Distribution* interface, the current density due to the electrochemical reactions is described as a function of the overpotential. The physics interface can use any relation between current density and overpotential, with common examples such as the Butler-Volmer equation (3) and the Tafel equation included as built-in options.

(3)

i_{loc,m} = i_{0,m}\left(e^\frac{\alpha_{a,m} F \eta_m}{RT}-e^\frac{-\alpha_{c,m} F \eta_m }{RT}\right)

In the Butler-Volmer equation above, for reaction : denotes the local charge transfer current density, the exchange current density, and the anodic and the cathodic charge transfer coefficient. is the universal gas constant. This equation describes the case when the charge transfer of one electron is the rate determining step in the net charge transfer reaction. The expression can be derived by analogy to the Arrhenius equation for a homogeneous chemical reaction, by assuming the free energy of the charged species to be influenced by the potential. Hence, the activation energy changes with the potential difference at the electrode-electrolyte interface.

The sum of all electrode reaction currents is implemented as a current density condition on the boundary between an electrode and an electrolyte domain according to:

-\textbf{i}_s \cdot \textbf{n} = \textbf{i}_l \cdot \textbf{n} = i_\mathrm{DL} + \sum_m{i_{loc,m}}

The additional capacitive current arises from charge and discharge of the electrical double layer.

In general, accounting for the effect of electrode kinetics by means of an activation overpotential will tend to make the current distribution more uniform. You can see this in the wire electrode example in the figure below.

Compared to the primary current distribution, the secondary current distribution is smoother, with a smaller difference between the minimum and maximum values. When the activation overpotential is included, a high local current density would introduce a high local activation overpotential at the electrode surface, which causes the current to naturally take a different path. To look at this another way, you can understand the electrochemical reaction as proceeding at a finite rate. In some regions, the reaction is kinetically limited, and so the distribution of current densities over the surface is less extreme than in the case when the reaction can proceed arbitrarily quickly.

*Secondary current distribution, Ecell = 1.65 V. Current density distribution on the anode (dimensionless).*

*Secondary Current Distribution* is the workhorse interface for modeling industrial applications in electrochemistry. You can use this class of current distribution for modeling cells where you can neglect concentration overpotential, due to good mixing or relatively high electrolyte concentration, but when the electrode kinetics cause losses that are not negligible compared to the ohmic losses. In industrial applications it is usually not a problem to provide an electrolyte of high concentration with vigorous mixing. You can also use the *Secondary Current Distribution* interface as a first step in your simulation of electrochemical cells to estimate the activation losses, before you eventually introduce concentration-dependent reaction kinetics.

The tertiary current distribution accounts for the effect of variations in electrolyte composition and ionic strength on the electrochemical process, as well as solution resistance and electrode kinetics. To do this, it solves the Nernst-Planck equation (1) explicitly for each chemical species to describe its mass transport through diffusion, migration, and convection. Additionally, the species concentrations are subject to the electroneutrality approximation. The kinetic expressions for the electrochemical reactions account for both activation and concentration overpotential, meaning that the rate of an electrolysis reaction can be transport-limited by exhaustion of the reactant at the electrode-electrolyte interface. This implies that all ions and all electroactive species in the electrolyte must be included in the model.

Unlike the primary and secondary current distributions, the electrolyte current density is no longer assumed to follow Ohm’s law in the tertiary current distribution. The imposition of electroneutrality still means that convective flux does not contribute to the current density, due to equation (2), but now the influence of the concentration variations in the electrolyte cannot be neglected. Therefore, the diffusion term in equation (2) may be non-zero.

At the electrode-electrolyte interface, the current density of charge transfer reactions is expressed as a function not only of the overpotential, but also of the concentration of the electroactive species at the interface. For a reaction rate determined by a one-electron charge transfer step, the reaction kinetics is expressed using a Butler-Volmer expression for the charge transfer current density (compare with equation (3)), which in this case can contain concentration dependencies.

The *Tertiary Current Distribution* interface in the COMSOL software solves for the electrolyte potential (), the electrode potential (), and the set of species concentrations . With the assumptions described above you get the following equations:

Electrode: with

Electrolyte: with

Electrolyte electroneutrality:

Electrode-Electrolyte-Interface:

Typical current density expression:

It is essential that the reference concentration is the same for all species involved in a reaction. This ensures that at zero current density (equilibrium) the overpotential obeys the thermodynamic Nernst equation.

In the image below, you can see the tertiary current distribution for the wire example. Due to the dependence of the concentration, the tertiary current distribution becomes influenced by the flow of the electrolyte and hence the availability of the reactant by mass transport. Where the flow velocity is small between the wires, electrolyte consumed to draw Faradaic current is not replenished, leading to a depletion zone of the reactant in these parts in the cell. This significantly lowers the local current density, which can be described as “mass-transport limited”, leading a greater amount of the current to be drawn from the outer edges of the wires. A corresponding increased voltage drop is observed due to the transport limitation of current: this is the “concentration overpotential”.

*Tertiary current distribution, Ecell = 1.65 V. Current density distribution on the anode (dimensionless).*

You can use this class of current distribution for modeling cells with poor mixing or relatively low electrolyte concentration (compared to net current density), such that the electrolyte composition varies significantly throughout the cell and the resistive losses cannot be described by Ohm’s law. Solving the Nernst-Planck equations for *all* species concentrations with concentration of current and electroneutrality makes the equation set nonlinear and very complicated for the tertiary current distribution, which results in more time and memory storage requirements for the simulation. It is good practice to predict and understand the likely behavior of an electrochemical cell with secondary current distribution before modeling the tertiary current distribution.

The primary, secondary, and tertiary current distributions distinguish successive levels of approximation in the analysis of the current-voltage relation of an electrochemical cell. There are other modeling approaches that may be suitable, however, to extract maximum information about a cell’s behavior while minimizing the complexity of the model as much as possible.

In cases where the current density may be limited by mass transport of the electroactive species, but the electrolyte composition remains near-constant, it may not be necessary to solve for the full tertiary current distribution. Instead, the constant ionic strength means that we can assume that the solution obeys Ohm’s law with a constant conductivity, and so *Secondary Current Distribution* is used to solve for the electrolyte potential. However, the kinetic rate law is made concentration-dependent by coupling to a chemical species transport model that solves for the diffusion of the chemical species (and, where necessary, their migration and convection).

In fact, this is the method used for the tertiary current distribution of the wire electrode example, since it is the depletion of the reactant rather than the bulk electrolyte species that has the dominating effect. You can see another example of this coupling in the orange battery model. Also, this partial coupling of charge transport with mass transport is a very common approach in the analysis of batteries and fuel cells.

A special case of the above occurs when the inert (supporting) electrolyte is in considerable excess compared to the quantity of reacting (electroactive) species. Hence, the ionic strength of the solution is large compared to the Faradaic current density. In this case, the electric field is small and so the electrolyte potential is almost constant — solution resistance does not contribute noticeably to the behavior of the electrochemical cell.

In cases where solution resistance is unimportant, but electrode kinetics (activation) and mass transport of the electroactive species are important, you can use the *Electroanalysis* interface. This is a chemical species transport interface solving the diffusion-convection equation for mass transport, which incorporates electrode kinetic boundary conditions to drive a flux of the chemical species at electrode-electrolyte interfaces as a function of the local overpotential.

The electroanalytical approximation of zero solution resistance applies to the standard experimental set-ups for electrochemical techniques such as cyclic voltammetry, chronoamperometry, and electrochemical impedance spectroscopy. You can see an example of a cyclic voltammetry model using this approximation in our Model Gallery.

This blog post has discussed the three current distribution interfaces available in the four electrochemical add-on modules for COMSOL Multiphysics, and when and why you should use each of them. The strength of the COMSOL Multiphysics software is that it offers you the ability to model all classes of current distributions (primary, secondary, and tertiary), and therefore provides you with the flexibility to gradually introduce and control the complexity of the theoretical model used to analyze an electrochemical cell.

If you are interested in using COMSOL Multiphysics for your electrochemical cell design, or have a question that isn’t addressed here, please contact us.

An electrochemical cell is characterized by the relation of the current it passes to the voltage across it. The current-voltage relation depends on diverse physical phenomena and is fundamental to performance. In a battery or fuel cell at zero current (*equilibrium*), a theoretical maximum voltage can be extracted, but we want to draw current in order to extract power.

When current is drawn, there are voltage losses; equally, the current density may not be uniformly distributed on the electrode surfaces. The performance and lifetime of electrochemical cells, such as electroplating cells or batteries, is often improved by a uniform current density distribution.

By contrast, bad design leads to poor performance, such as:

- Substantial losses and shortened lifetime of electrode material at practical operating currents in a battery or fuel cell
- Uneven plating thickness in electroplating
- Unprotected surfaces in a cathodic protection system

Simulating current distribution enables better understanding to avoid such problems.

The current distribution depends on several factors:

- Cell geometry
- Cell operating conditions
- Electrolyte conductivity
- Electrode kinetics (“activation overpotential”)
- Mass transport of the reactants (“concentration overpotential”)
- Mass transport of ions in the electrolyte

Because of this complexity, many applications benefit from suitable simplification when modeling. If one of these factors dominates the cell behavior, the others may not need to be taken into account. As a consequence, successive approximations are introduced by the classifications of primary, secondary, and tertiary current distribution.

Each of the three classes of current distribution is represented in COMSOL Multiphysics by its own interface: *Primary*, *Secondary*, and *Tertiary Current Distribution*. These interfaces are provided in all of the four different application-specific products available for modeling electrochemical cells: the Batteries & Fuel Cells Module, Electrodeposition Module, Corrosion Module, and Electrochemistry Module.

When modeling an electrochemical cell, you have to solve for the potential and current density in the electrodes and the electrolyte, respectively. You may also have to consider the contributing species concentrations and the involved electrolysis (Faradaic) reactions.

The electrodes in an electrochemical cell are normally metallic conductors and so their current-voltage relation obeys Ohm’s law:

with conservation of current

where denotes the current density vector (A/m^{2}) in the electrode, denotes the conductivity (S/m), the electric potential in the metallic conductor (V), and denotes a general current source term (A/m^{3}, normally zero).

In the electrolyte, which is an ionic conductor, the net current density can be described using the sum of fluxes of all ions:

\textbf{i}_l = F\sum_i{z_i\textbf{N}_i}

where denotes the current density vector (A/m^{2}) in the electrolyte, denotes the Faraday constant (C/mol), and the flux of species (mol/(m^{2}·s)) with charge number . The flux of an ion in an ideal electrolyte solution is described by the Nernst-Planck equation and accounts for the flux of solute species by diffusion, migration, and convection in the three respective additive terms:

(1)

\textbf{N}_i = -D_i\nabla c_i-z_i u_{m,i} F c_i\nabla \phi_l+c_i\textbf{u}

where represents the concentration of the ion (mol/m^{3}), the diffusion coefficient (m^{2}/s), its mobility (s·mol/kg), the electrolyte potential, and the velocity vector (m/s).

On substituting the Nernst-Planck equation into the expression for current density, we find:

(2)

\textbf{i}_l = -F \left(\nabla \sum_i z_iD_i c_i\right)-F^2\nabla \phi_l \sum_i z^2_i u_{m,i} c_i+\textbf{u}F\sum_i z_ic_i

with conservation of current including a general electrolyte current source term (A/m^{3}):

\nabla\cdot\mathbf{i}_l=Q_l

As well as conservation of current in the electrodes and electrolyte, you also have to consider the interface between the electrode and the electrolyte. Here, the current must also be conserved. Current is transferred between the electrode and electrolyte domains either by an electrochemical reaction, also called electrolysis or Faradaic current, or by dynamic charging or discharging of the charged double layer of ions adjacent to the electrode, also called capacitive or non-Faradaic current.

This general treatment of electrochemical theory is usually too complicated to be practical. By assuming that one or more of the terms in Equation (2) are small, the equations can be simplified and linearized. The three different current distribution classes applied in electrochemical analysis are based on a range of assumptions made to these general equations, depending on the relative influence of the different factors affecting the current distribution as listed above. In the next blog post in the series we’ll discuss the detailed content of these assumptions: going from primary to secondary to tertiary, fewer assumptions are made. Therefore, the complexity increases, but so does the level of detail available from the simulation.

Below you can see the geometry from a modeling example of a wire electrode. This example models the primary, secondary, and tertiary current distributions of an electrochemical cell. In the open volume between the wire and the flat surfaces, electrolyte is allowed to flow. You can think of the electrochemical cell as a unit cell of a larger wire-mesh electrode — a common electrochemical cell set-up in many large-scale industrial processes.

*Geometry of the 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.*

Now, you might be wondering which of the three current distribution interfaces you should use for your particular electrochemical cell simulations. In an upcoming blog post, we will use the wire electrode example shown here for a comparison of the three current distributions. Stay tuned!

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You may have wondered why car rims and motorcycle parts are so shiny, and the answer is: electroplating. Generally speaking, electroplating is used for many purposes, including augmenting undersized parts and altering an object’s surface properties. During the process, metal often goes to waste and companies need to figure out how to reduce the amount of metals that are consumed. Many times, this waste occurs as electrodeposited layers are applied unevenly, and an excess of material is required to ensure complete coverage. PEM is one such company, and by using COMSOL they figured out how to reduce the metal deposit during electrolysis by 10-30%. Several parameters are important when engaging in electroplating, and COMSOL’s Electrodeposition Module can be used to study these to ensure the desired outcome of the electrochemical process.

We have two tutorial models for the Electrodeposition Module that touch on electroplating. One of them is a decorative plating model, demonstrating how to use COMSOL to calculate the thickness of a deposited layer as well as the pattern created by the dissolution of the anode surface. The electroplating model is a secondary current density distribution model and uses full Butler-Volmer kinetics for the anode and the cathode. The planar anode dissolves, while the cathode is a furniture fitting of sorts that is to be decorated with a nice finishing metal. You can study the following important parameters using the Electrodeposition Module:

- Cell geometry
- Electrolyte composition and mixing
- Electrode kinetics
- Operating potential and average current density
- Temperature

*Electroplating: Thickness of the decorative layer in a furniture fitting.
Simulations based on secondary current distribution.*

The company based out of France, PEM, took this type of modeling to the next level, when they also considered the effects of fluid flow and convection. They are in the business of treating and finishing surfaces for the electronics industry and make this a viable process through reel-to-reel electroplating. The customer provides PEM with the reel containing tape holding a series of metal parts and PEM then unwinds the tape from the reel and sends it through an electroplating process. Here, the tape is turned into the cathode via contact with a voltage source, and the anodes are located in the plating cell. The electrical circuit is closed by an electrolyte. The big trick is to achieve a uniform plating layer on the tape as edge effects cause more metal to accumulate around the edges of the tape. PEM’s solution to this problem was to design a shield of sorts to even out the plating across the width of the tape.

The shape of this shield is dependent on the reel’s components, the plating material, the thickness of the plating material, the concentration of the electrolyte, and the traveling speed of the tape. Each new job or batch-run can first be simulated to design an appropriate shield, through COMSOL Multiphysics. They found that computer modeling eliminated the need for time-consuming experiments and physical prototypes — with a successful design now often reached on the first try! Next, PEM used COMSOL to design more efficient reactor cells. As Dr. Philippe Gendre of PEM states in COMSOL News: “If you can replenish the metal ions better, you can raise the current density and thus production.”

This was back in 2008, and these days we have the Electrodeposition Module to make these types of calculations even easier to perform.