The Nernst-Planck Equations Interface
Ionic solutions subjected to electric fields are affected by migration transport in addition to the existing mass transport by diffusion and convection. Migration implies that positive ions migrate from positive to negative potential, and vice versa for negatively charged ions. The flux of the ionic species i in the solution is given by the mass flux vector (SI unit: mol/(m2·s))
(3-95)
where ci denotes the concentration of species i (SI unit: mol/ m3), Di is the diffusion coefficient of species i (SI unit: m2/s), u is the solvent velocity (SI unit: m/s), F refers to Faraday’s constant (SI unit: s·A/mol), V denotes the electric potential (SI unit: V), zi is the charge number of the ionic species (dimensionless), and um,i its ionic mobility (SI unit: s·mol/kg).
There are three physics interfaces for this type of transport. In the Nernst-Planck Equations (npe) interface (), found under the Chemical Species Transport branch () when adding a physics interface, the transport of every charged species is accounted for, and is solved for in combination with the electroneutrality condition.
The Nernst-Planck Equations interface supports modeling of transport by convection, diffusion, and migration of dissolved ionic species in 1D, 2D, and 3D as well as for axisymmetric models in 1D and 2D. The dependent variables are the species concentrations, ci, and the electric potential, V. The physics interface is tailored for modeling of electrochemical cells and corrosion processes and offers a straightforward way to perform tertiary current distribution analysis.
The physics interface has the equations, boundary conditions, and rate expression terms for modeling mass transport of ionic species by convection, diffusion, and migration, solving for the species concentrations and the electric potential. The main feature is the Convection, Diffusion, and Migration node, which adds the equation for the species concentration and the potential and provides an interface for defining species properties and model inputs.
When this physics interface is added, these default physics nodes are also added to the Model Builder—Convection, Diffusion and Migration, No Flux (the default boundary condition for species concentrations), Electric Insulation (the default boundary condition for the electric field), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and rate expression terms. You can also right-click Nernst-Planck Equations to select physics features from the context menu.
Settings
The Label is the default physics interface name.
The Name is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers and underscores (_) are permitted in the Name field. The first character must be a letter.
The default Name (for the first physics interface in the model) is npe.
Species
Select the species that the physics interface computes from the electroneutrality condition in Equation 3-95. That is, its value comes from the fact that the net charge in every control volume is zero. Select the preferred species in the From electroneutrality list. To minimize the impact of any numerical errors, use the species with the highest concentration. By default, the software uses the first species.
Dependent Variables
Add or remove species in the model and also change the names of the dependent variables that represent the species concentrations.
Specify the Number of species. There must be at least two species. To add a single species, click the Add species button (). To remove a species, select it in the list and click the Remove species button ().Edit the names of the species directly in the table.
The species are dependent variables, and their names must be unique with respect to all other dependent variables in the model.
Advanced Settings
To display this section, click the Show button () and select Advanced Physics Options. Normally these settings do not need to be changed. Select a Convective term—Non-conservative form (the default) or Conservative form.
Convective Term Formulation
Discretization
To display this section, click the Show button () and select Discretization.
For more information about these settings, see Discretization in The Transport of Diluted Species Interface section.
Consistent and Inconsistent Stabilization
To display this section, click the Show button () and select Stabilization. Any settings unique to this physics interface are listed below.
The crosswind diffusion implementation used in the Nernst-Planck Equations interface is based on Codina (Ref. 1).
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When the Crosswind diffusion check box is selected, the Lower gradient limit glim (SI unit: mol/m4) field defaults to 0.1[mol/m^3)/npe.helem, where npe.helem is the local element size.
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For both consistent stabilization methods, select an Equation residual. Approximate residual is the default setting and it means that derivatives of the diffusion tensor components are neglected. This setting is usually accurate enough and is faster to compute. If required, select Full residual instead.
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Domain and Boundary Nodes for the Nernst-Planck Equations Interface
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Theory for the Nernst-Planck Equations Interface
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Convective Term Formulation
In the COMSOL Multiphysics Reference Manual see Table 2-3 for links to common sections and Table 2-4 to common feature nodes. You can also search for information: press F1 to open the Help window or Ctrl+F1 to open the Documentation window.