# KKSPhaseChemicalPotential

KKS model kernel to enforce the pointwise equality of phase chemical potentials dFa/dca = dFb/dcb. The non-linear variable of this kernel is ca.

Enforces the point wise equality of the phase chemical potentials

\frac{dF_a}{dc_a}=\frac{dF_b}{dc_b}.

The non-linear variable of this Kernel is .

### Residual

R=\frac{dF_a}{dc_a} - \frac{dF_b}{dc_b}

### Jacobian

For the Jacobian we need to calculate

J=\frac \partial{\partial u_j}\left( \frac{dF_a}{dc_a} - \frac{dF_b}{dc_b} \right).

#### On-Diagonal

J = \phi_j \left( \frac{\partial2 F_a}{\partial c_a2} - \frac{\partial^2 F_b}{\partial c_a \partial c_b} \right)

#### Off-Diagonal

With the union of the argument vectors of and (represented in the code by _coupled_moose_vars[]) we get

\sum_i \left( \frac{\partial2 F_a}{\partial c_a \partial q_i}\frac{\partial q_i}{\partial u_j} - \frac{\partial2 F_b}{\partial c_b \partial q_i}\frac{\partial q_i}{\partial u_j} \right).

Again the is non-zero only if , which is the case if is the argument selected through jvar.

J = \frac{\partial2 F_a}{\partial c_a \partial q_\text{jvar}}\phi_j - \frac{\partial2 F_b}{\partial c_b \partial q_\text{jvar}}\phi_j.

Note that in the code jvar is not an index into _coupled_moose_vars[] but has to be resolved through the _jvar_map.

## Input Parameters

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

Description:The name of the variable that this Kernel operates on

• cbPhase b concentration

C++ Type:std::vector

Description:Phase b concentration

• fa_nameBase name of the free energy function Fa (f_name in the corresponding derivative function material)

C++ Type:MaterialPropertyName

Description:Base name of the free energy function Fa (f_name in the corresponding derivative function material)

• fb_nameBase name of the free energy function Fb (f_name in the corresponding derivative function material)

C++ Type:MaterialPropertyName

Description:Base name of the free energy function Fb (f_name in the corresponding derivative function material)

### Required Parameters

• args_aVector of further parameters to Fa (optional, to add in second cross derivatives of Fa)

C++ Type:std::vector

Description:Vector of further parameters to Fa (optional, to add in second cross derivatives of Fa)

• args_bVector of further parameters to Fb (optional, to add in second cross derivatives of Fb)

C++ Type:std::vector

Description:Vector of further parameters to Fb (optional, to add in second cross derivatives of Fb)

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

C++ Type:std::vector

Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Default:False

C++ Type:bool

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

C++ Type:std::vector

Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

• vector_tagsnontimeThe tag for the vectors this Kernel should fill

Default:nontime

C++ Type:MultiMooseEnum

Description:The tag for the vectors this Kernel should fill

• extra_vector_tagsThe extra tags for the vectors this Kernel should fill

C++ Type:std::vector

Description:The extra tags for the vectors this Kernel should fill

• matrix_tagssystemThe tag for the matrices this Kernel should fill

Default:system

C++ Type:MultiMooseEnum

Description:The tag for the matrices this Kernel should fill

• extra_matrix_tagsThe extra tags for the matrices this Kernel should fill

C++ Type:std::vector

Description:The extra tags for the matrices this Kernel should fill