Secondary Current Distribution (cd)

Current distribution in an electrolyte with electrode kinetics (primary: equilibrium electrodes; secondary: Butler-Volmer, Tafel, linearized or user-defined kinetics).

Interface settings

Setting Symbol Unit Default Choices / type

Setting	Default	Choices / type
Discretization
Element order	Quadratic	Linear, Quadratic
Physics Model
Current distribution type	Secondary	Primary, Secondary
Elmer Solver
Elmer solver Automatic: StatCurrentSolve for primary electrodes, HeatSolve when kinetics need a Robin condition, ModelPDE when a Heat Transfer interface already uses HeatSolve	Automatic	Automatic, StatCurrentSolve (no kinetics), HeatSolve (Robin form), ModelPDE (Robin form)
Maximum Newton iterations (kinetics)	`200`	Integer
Newton damping factor Fixed damping of every Newton step; with the line search on, 1 (full steps) is best	`1`	Text
Line search (backtrack steps that increase the residual)	On	Check box
Additional Elmer solver keywords Lines appended verbatim to this interface’s Solver section in the .sif file		Text (multi-line)

Discretization

Element order

Quadratic

Linear, Quadratic

Physics Model

Current distribution type

Secondary

Primary, Secondary

Elmer Solver

Elmer solver
Automatic: StatCurrentSolve for primary electrodes, HeatSolve when kinetics need a Robin condition, ModelPDE when a Heat Transfer interface already uses HeatSolve

Automatic

Automatic, StatCurrentSolve (no kinetics), HeatSolve (Robin form), ModelPDE (Robin form)

Maximum Newton iterations (kinetics)

200

Integer

Newton damping factor
Fixed damping of every Newton step; with the line search on, 1 (full steps) is best

1

Text

Line search (backtrack steps that increase the residual)

Check box

Additional Elmer solver keywords
Lines appended verbatim to this interface’s Solver section in the .sif file

Text (multi-line)

Domain features

Electrolyte

Selection: Domains (all by default) · Physics ribbon › Domains, or right-click the interface in the Model Builder

\(\nabla\cdot\mathbf{i}_l = Q_l,\quad \mathbf{i}_l = -\sigma_l\nabla\phi_l\)

Setting Symbol Unit Default Choices / type

Setting	Symbol	Unit	Default	Choices / type
Electrolyte Charge Conservation
Electrical conductivity	σ		From material	From material, User defined
sigma	σ	`S/m`	`0`	Expression

Electrolyte Charge Conservation

Electrical conductivity

From material

From material, User defined

sigma

S/m

0

Expression

Current Source

Selection: Domains · Physics ribbon › Domains, or right-click the interface in the Model Builder

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

Setting Symbol Unit Default Choices / type

Setting	Symbol	Unit	Default	Choices / type
Current Source
Current source	Q_l	`A/m^3`	`0`	Expression

Current Source

Current source

Q_l

A/m^3

0

Expression

Initial Values

Selection: Domains (all by default) · Physics ribbon › Domains, or right-click the interface in the Model Builder

Setting Symbol Unit Default Choices / type

Setting	Symbol	Unit	Default	Choices / type
Initial Values
Electrolyte potential	φ_l	`V`	`0`	Expression

Initial Values

Electrolyte potential

φ_l

V

0

Expression

Boundary features

Electrode Surface

Electrode reaction at the electrolyte boundary: overpotential eta = phis_ext - Eeq - phil, local current density i_loc(eta) entering the electrolyte (anodic positive).

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{i}_l = i_{loc}(\eta),\quad \eta = \phi_{s,ext}-E_{eq}-\phi_l\)

Setting Symbol Unit Default Choices / type

Setting	Symbol	Unit	Default	Choices / type
Electrode Phase Potential Condition
Electric potential (of the electrode) Sweep it with a parameter for a polarization curve	φ_s,ext	`V`	`0[V]`	Expression
Electrode Kinetics
Kinetics expression type			Butler-Volmer	Primary condition (thermodynamic equilibrium), Butler-Volmer, Linearized Butler-Volmer, Anodic Tafel equation, Cathodic Tafel equation, Butler-Volmer with limiting current, User defined i(eta)
Equilibrium Potential
Equilibrium potential	E_eq	`V`	`0[V]`	Expression
Electrode Kinetics
Exchange current density	i₀	`A/m^2`	`1[A/m^2]`	Expression
Anodic transfer coefficient	α_a	`1`	`0.5`	Expression
Cathodic transfer coefficient	α_c	`1`	`0.5`	Expression
Anodic Tafel slope	A_a	`V`	`0.118[V]`	Expression
Cathodic Tafel slope	A_c	`V`	`0.118[V]`	Expression
Limiting current density	i_lim	`A/m^2`	`100[A/m^2]`	Expression
Local current density i(eta) Any expression of the overpotential eta (V), parameters and functions, e.g. an Interpolation function of tabulated polarization data: pol(eta)			`1[A/m^2]*sinh(eta/0.05[V])`	Text
Temperature	T	`K`	`298.15[K]`	Expression
Dissolution
Molar mass (dissolving metal) Used by the dissolution-rate result variable cd.es<n>.vdis = M i / (n F rho)	M	`kg/mol`	`55.845[g/mol]`	Expression
Number of participating electrons	n	`1`	`2`	Expression
Metal density	ρ_m	`kg/m^3`	`7870[kg/m^3]`	Expression
Deforming electrode surface (the boundary dissolves or grows) In a Time Dependent study with a Deforming Domain, the electrode recedes along its normal at the dissolution velocity M i_loc/(n F rho) (anodic current dissolves, cathodic deposits)			Off	Check box
Current efficiency of the dissolution Fraction of the current that removes metal; may depend on the local current density iloc (A/m^2), e.g. iloc > 10[A/cm^2] for a passivation threshold	ε		`1`	Text

Electrode Phase Potential Condition

Electric potential (of the electrode)
Sweep it with a parameter for a polarization curve

φ_s,ext

V

0[V]

Expression

Electrode Kinetics

Kinetics expression type

Butler-Volmer

Primary condition (thermodynamic equilibrium), Butler-Volmer, Linearized Butler-Volmer, Anodic Tafel equation, Cathodic Tafel equation, Butler-Volmer with limiting current, User defined i(eta)

Equilibrium Potential

Equilibrium potential

E_eq

V

0[V]

Expression

Electrode Kinetics

Exchange current density

i₀

A/m^2

1[A/m^2]

Expression

Anodic transfer coefficient

α_a

1

0.5

Expression

Cathodic transfer coefficient

α_c

1

0.5

Expression

Anodic Tafel slope

A_a

V

0.118[V]

Expression

Cathodic Tafel slope

A_c

V

0.118[V]

Expression

Limiting current density

i_lim

A/m^2

100[A/m^2]

Expression

Local current density i(eta)
Any expression of the overpotential eta (V), parameters and functions, e.g. an Interpolation function of tabulated polarization data: pol(eta)

1[A/m^2]*sinh(eta/0.05[V])

Text

Temperature

K

298.15[K]

Expression

Dissolution

Molar mass (dissolving metal)
Used by the dissolution-rate result variable cd.es<n>.vdis = M i / (n F rho)

kg/mol

55.845[g/mol]

Expression

Number of participating electrons

1

2

Expression

Metal density

ρ_m

kg/m^3

7870[kg/m^3]

Expression

Deforming electrode surface (the boundary dissolves or grows)
In a Time Dependent study with a Deforming Domain, the electrode recedes along its normal at the dissolution velocity M i_loc/(n F rho) (anodic current dissolves, cathodic deposits)

Off

Check box

Current efficiency of the dissolution
Fraction of the current that removes metal; may depend on the local current density iloc (A/m^2), e.g. iloc > 10[A/cm^2] for a passivation threshold

1

Text

Electrolyte Potential

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(\phi_l = \phi_{l,bnd}\)

Setting Symbol Unit Default Choices / type

Setting	Symbol	Unit	Default	Choices / type
Electrolyte Potential
Electrolyte potential	φ_l,bnd	`V`	`0[V]`	Expression

Electrolyte Potential

Electrolyte potential

φ_l,bnd

V

0[V]

Expression

Electrolyte Current Density

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{i}_l = i_n\)

Setting Symbol Unit Default Choices / type

Setting	Symbol	Unit	Default	Choices / type
Electrolyte Current Density
Inward electrolyte current density	i_n	`A/m^2`	`0[A/m^2]`	Expression

Electrolyte Current Density

Inward electrolyte current density

i_n

A/m^2

0[A/m^2]

Expression

Insulation

Selection: Boundaries (all by default) · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{i}_l = 0\)

No settings.

Symmetry

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{i}_l = 0\)

No settings.

Result variables

Name Description Unit

Name	Description	Unit
`cd.phil`	Electrolyte potential	`V`
`cd.Il`	Electrolyte current density vector	`A/m^2`
`cd.IlMag`	Electrolyte current density magnitude	`A/m^2`
`cd.Ilx`	Electrolyte current density, x component	`A/m^2`
`cd.Ily`	Electrolyte current density, y component	`A/m^2`
`cd.Ilz`	Electrolyte current density, z component	`A/m^2`
`cd.iloc`	Local current density into the electrolyte (from the reaction)	`A/m^2`
`cd.nIl`	Normal electrolyte current density (outward)	`A/m^2`
`cd.disp`	Mesh displacement (deforming electrodes)	`m`
`cd.dispMag`	Mesh displacement magnitude	`m`
`cd.dx`	Mesh displacement, x component	`m`
`cd.dy`	Mesh displacement, y component	`m`
`cd.dz`	Mesh displacement, z component	`m`

cd.phil

Electrolyte potential

V

cd.Il

Electrolyte current density vector

A/m^2

cd.IlMag

Electrolyte current density magnitude

A/m^2

cd.Ilx

Electrolyte current density, x component

A/m^2

cd.Ily

Electrolyte current density, y component

A/m^2

cd.Ilz

Electrolyte current density, z component

A/m^2

cd.iloc

Local current density into the electrolyte (from the reaction)

A/m^2

cd.nIl

Normal electrolyte current density (outward)

A/m^2

cd.disp

Mesh displacement (deforming electrodes)

m

cd.dispMag

Mesh displacement magnitude

m

cd.dx

Mesh displacement, x component

m

cd.dy

Mesh displacement, y component

m

cd.dz

Mesh displacement, z component

m