******************
**Current Source**
******************

::

   Current Source = {model_name} <optional integer> <float_list> [E/M]

-----------------------
**Description / Usage**
-----------------------

This card is used to specify the model for the source term on the voltage potential
equation. Values for the permissible {model_names} and the associated <optional
integer> and <floatlist> parameters are given below.

+--------------------------+-------------------------------------------------------------------------------------+
|{model_name}              |Name of the model for the source term on the voltage equation having permissible     |
|                          |values                                                                               |
|                          |                                                                                     |
|                          | * **CONSTANT**                                                                      |
|                          | * **USER**                                                                          |
|                          | * **BUTLER_VOLMER**                                                                 |
|                          | * **ELECTRODE_KINETICS**                                                            |
|                          | * **FICKIAN_CHARGED**                                                               |
|                          | * **NET_CHARGE**                                                                    |
|                          | * **STEFAN_MAXWELL_CHARGED**                                                        |
+--------------------------+-------------------------------------------------------------------------------------+
|<optional integer>        |This is required for the **BUTLER_VOLMER** model only.                               |
+--------------------------+-------------------------------------------------------------------------------------+
|<float_list>              |One or more floating point numbers (<float1> through <floatn>) whose values are      |
|                          |determined by the selection for {model_name}. Note that not all models have a        |
|                          |<float_list>.                                                                        |
+--------------------------+-------------------------------------------------------------------------------------+

Source-term model choices and their parameters are discussed below. WARNING:
make sure the equation term multipliers for the source terms are set to unity (see the
Equation Cards segment in the previous chapter).

+--------------------------+-------------------------------------------------------------------------------------+
|**CONSTANT** <float1>     |For the **CONSTANT** current source term, there is a single input parameter          |
|                          |corresponding to the current density.                                                |
|                          |                                                                                     |
|                          | * <float1> - Current density [E/M]                                                  |
+--------------------------+-------------------------------------------------------------------------------------+
|**USER** <float_list>     |For a user-defined model, the set of parameters specified in the <floatlist> are     |
|                          |defined in file user_mp.c in the function usr_current_source.                        |
+--------------------------+-------------------------------------------------------------------------------------+
|**BUTLER_VOLMER**         |This is the homogeneous current source or sink term (in units of amphere per unit    |
|<integer> <float1>        |volume, e.g. A/cm3) as described by the Butler-Volmer kinetic model (see the Theory  |
|<float2> <float3> <float4>|section below).One integer and 8 flotas are required:                                |
|<float5> <float6> <float7>|                                                                                     |
|<float8>                  | * <integer> - Index of the species involved in the electrochemical reaction (here,  |
|                          |   we assume that only a single species is involved).                                |
|                          | * <float1> - Stoichiometric coefficient, s.                                         |
|                          | * <float2> - Product of interfacial area per unit volume by exchange current        |
|                          |   density, ai0, in units of A/cm3.                                                  |
|                          | * <float3> - Reaction order, β.                                                     |
|                          | * <float4> - Reference species concentration, cref, in units of moles/cm3.          |
|                          | * <float5> - Anodic transfer coefficient, αa.                                       |
|                          | * <float6> - Cathodic transfer coefficient, αc.                                     |
|                          | * <float7> - Temperature, T, in unit of K.                                          |
|                          | * <float8> - Open-circuit potential, U0, in unit of V.                              |
+--------------------------+-------------------------------------------------------------------------------------+
|**ELECTRODE_KINETICS**    |This is a toggle, turning the model on; no parameters are required.                  |
+--------------------------+-------------------------------------------------------------------------------------+
|**FICKIAN_CHARGED**       |This is a toggle, turning the model on; no parameters are required.                  |
+--------------------------+-------------------------------------------------------------------------------------+
|**NET_CHARGE**            |This is a toggle, turning the model on; no parameters are required.                  |
+--------------------------+-------------------------------------------------------------------------------------+
|**STEFAN_MAXWELL_CHARGED**|This is a toggle, turning the model on; no parameters are required.                  |
+--------------------------+-------------------------------------------------------------------------------------+

------------
**Examples**
------------

Sample card for the **CONSTANT** model:

::

   Current Source = CONSTANT 0.50

Sample card for the **BUTLER_VOLMER** model:

::

   Current Source = BUTLER_VOLMER 0 1. 1000. 0.5 4.e-5 1. 1. 353. 0.

-------------------------
**Technical Discussion**
-------------------------

The **CONSTANT** and **USER** models are those standardly available in *Goma*.

In the **BUTLER_VOLMER** model the current source or sink due to a homogeneuous
electrochemical reaction involving a single species (e.g., the hydrogen oxidation and
oxygen reduction reactions in a hydrogen-feuled polymer-electrolyte-membrane fuel
cell) is computed using the Butler-Volmer kinetic model as described below in the
Theory section.

In the **FICKIAN_CHARGED** model, current source or sink for electrochemical
processes involving dilute electrolyte solution and multiple species as in LIGA
electrodeposition is computed.

The **NET_CHARGE** model is used to compute the net charge or current source in a
region where the concentrations of positively and negatively charged species differ as
in the space layer of a atmospheric copper sulfidation process, in which the copper hole
and vacancy concentrations differ such that charge separation occur (see the reference
listed below in the Reference sub-section, Chen 2004, for further details).

In the **STEFAN_MAXWELL_CHARGED** and **ELECTRODE_KINETICS**
models, current sources or sinks for electrochemical processes involving concentrated
electrolyte solutions and multiple species as in thermal batteries are computed.

Further details of these models can be found in the SAND Reports and proceeding
paper referenced below in the Theory sub-section.

----------
**Theory**
----------

BUTLER_VOLMER model: for the Butler-Volmer kinetic model with the exchange
current density being dependent on a single species is given by (cf. Newman 1991,
Chen et al. 2000, Chen and Hickner 2006):

.. figure:: /figures/484_goma_physics.png
	:align: center
	:width: 90%

where j is the homogeneous current source or sink in units of A/cm3; s is the
stoichiometric coefficient with a sign convention such that j represents a source when s
> 0 and sink when s < 0; ai0 denotes the product of interfacial area per unit volume by
exchange current density, which has units of A/cm3; c and cref are, respectively, species
and reference molar concentrations in units of moles/cm3; β is reaction order; αa and αc
are, respetively, the anodic and cathodic transfer coefficients; F is the Faraday’s
constant ( ≡ 96487 C/mole) and R is the universal gasl constant ( ≡ 8.314 J/mole-
K); Φ1 and Φ2 are, respectively, the electrode and electrolyte potentials in unit of V;
U0 is the open-circuit potential in unit of V; and T is temperature in unit of K.

NET_CHARGE model: The net charge or current source in a region with charge
separation (e.g., in a space charge layer in which hole and vacancy concentrations
differ as in the atmospheric copper sulfidation corrosion process) is given by

.. figure:: /figures/485_goma_physics.png
	:align: center
	:width: 90%

where j is the net charge or current source in units of A/cm3; zi is the charge number and
ci is the molar concentration in units of moles/cm3, respectively, of species i; F is the
Faraday’s constant ( ≡ 96487 C/mole); and n is the number of charge species present.


--------------
**References**
--------------

J. Newman, Electrochemical Systems, 2nd Edition, Prentice-Hall, NJ (1991).

K. S. Chen, G. H. Evans, R. S. Larson, D. R. Noble, and W. G. Houf, “Final Report on
LDRD Project: A Phenomenological Model for Multicomponent Transport with
Simultaneous Electrochemical Reactions in Concentrated Solutions”, Sandia Report
SAND2000-0207 (2000).

K. S. Chen and G. H. Evans, “Multi-dimensional Multi-species Modeling of Transient
Electrodeposition in LIGA Microfabrication”, Sandia Report SAND2004-2864 (2004).

K. S. Chen, “Multi-dimensional Modeling of Atmospheric Copper-Sulfidation
Corrosion on non-Planar Substrates”, Sandia Report SAND2004-5878 (2004).

K. S. Chen and M. A. Hickner, “Modeling PEM fuel cell performance using the finiteelement
method and a fully-coupled implicit solution scheme via Newton’s technique”,
in ASME Proceedings of FUELCELL2006-97032 (2006).