# GrandPotentialKernelAction

Automatically generate most or all of the kernels for the grand potential model

For more information on the grand potential model, see Plapp (2011) and Moelans et al. (2008). The following kernels are generated for each chemical potential variable:

The following kernels are generated for each order parameter:

Any additional kernels needed for your application will have to be entered manually into the input file. There is a large number of inputs for this action, but they can be categorized into four basic groups

### Global Inputs

These are applied throughout the kernels.

• switching_function_names: Vector of names of switching functions. These are used to distinguish between phases.

• use_displacesd_mesh: Standard option for kernels. This will be applied to all kernels generated.

• implicit: Standard option for kernels. This will be applied to all kernels generated.

### Chemical Potential Functions

These define the behavior of the chemical potential variables.

• chemical_potentials: Vector of names of chemical potential variables.

• susceptibilities: Vector of names of susceptibilities, chi. This vector should be the same length as "chemical_potentials" as each entry in this vector corresponds to the same entry in "chemical_potentials".

• mobilities: Vector of mobilities–either scalars or tensors–that correspond to the "chemical_potentials" variables. The entries should consist of diffusivities multiplied by susceptibilities.

• anisotropic: If the entries in "D" are tensors, set this to "true".

• free_energies_w: Vector of density functions that determine the densities corresponding with each "chemical_potentials". The total number of entries is the number of chemical potentials times the number of switching functions.

### Primary Set of Order Parameter Functions

Inputs affecting a group of auto-generated variables using the PolycrystalVarible action. These typically represent grains in a polycrystal system.

• op_num: Number of order parameters auto-generated in Variable block.

• var_name_base: Name of order parameters auto-generated in Variable block.

• free_energies_gr: Vector of chemical potential density functions used for this set of order parameters. Each entry corresponds to the phase in the same entry of "switching_function_names".

• mobility_name_gr: Name of scalar mobility used with this set of order parameters.

• energy_barrier_gr: Name of energy barrier coefficient (m in Moelans et al. (2008).) used with this set of order parameters.

• gamma_gr: Name of gamma coefficient used with this set of order parameters which controls interface energy between these order parameters.

• kappa_gr: Name of kappa coefficient to be used with this set of order parameters.

### Second Set of Order Parameter Functions

This optional set can include additional order parameters to distinguish phases or some other field not associated with the first set of order parameters.

• additional_ops: Vector of additional order parameters used in the model. Optional.

• free_energies_op: Vector of chemical potential density functions used for this set of order parameters. Each entry corresponds to the phase in the same entry of "switching_function_names".

• mobility_name_op: Name of scalar mobility used with this set of order parameters. If "additional_ops" is blank then this value and the others below will not be called and their values do not matter.

• energy_barrier_op: Name of energy barrier coefficient used with this set of order parameters.

• gamma_op: Name of gamma coefficient used with this set of order parameters which controls interface energy between these order parameters.

• gamma_grxop: Cross term gamma coefficient that controls the interface energy between the primary and second set of order parameters.

• kappa_op: Name of kappa coefficient to be used with this set of order parameters.

## Input Parameters

• var_name_basespecifies the base name of the grain variables

C++ Type:std::string

Options:

Description:specifies the base name of the grain variables

• susceptibilitiesList of susceptibilities that correspond to chemical_potentials

C++ Type:std::vector

Options:

Description:List of susceptibilities that correspond to chemical_potentials

C++ Type:std::vector

Options:

• free_energies_grList of free energies for each phase. Place in same order as switching_function_names.

C++ Type:std::vector

Options:

Description:List of free energies for each phase. Place in same order as switching_function_names.

• free_energies_wList of functions for each phase. Length should be length of chemical_potentials * length of switching_function_names.

C++ Type:std::vector

Options:

Description:List of functions for each phase. Length should be length of chemical_potentials * length of switching_function_names.

• mobilitiesVector of mobilities that must match chemical_potentials

C++ Type:std::vector

Options:

Description:Vector of mobilities that must match chemical_potentials

• chemical_potentialsList of chemical potential variables

C++ Type:std::vector

Options:

Description:List of chemical potential variables

• op_numspecifies the number of grains to create

C++ Type:unsigned int

Options:

Description:specifies the number of grains to create

### Required Parameters

• kappa_grkappaThe kappa used with the grains

Default:kappa

C++ Type:MaterialPropertyName

Options:

Description:The kappa used with the grains

• use_displaced_meshFalseWhether to use displaced mesh in the kernels

Default:False

C++ Type:bool

Options:

Description:Whether to use displaced mesh in the kernels

• mobility_name_grLName of mobility to be used with grains

Default:L

C++ Type:MaterialPropertyName

Options:

Description:Name of mobility to be used with grains

• gamma_opgammaName of the gamma used with additional order parameters

Default:gamma

C++ Type:MaterialPropertyName

Options:

Description:Name of the gamma used with additional order parameters

C++ Type:std::vector

Options:

Description:List of any additional order parameters which are not grains

• mobility_name_opLName of mobility to be used with additional_ops

Default:L

C++ Type:MaterialPropertyName

Options:

Description:Name of mobility to be used with additional_ops

• implicitTrueWhether kernels are implicit or not

Default:True

C++ Type:bool

Options:

Description:Whether kernels are implicit or not

• active__all__ If specified only the blocks named will be visited and made active

Default:__all__

C++ Type:std::vector

Options:

Description:If specified only the blocks named will be visited and made active

• inactiveIf specified blocks matching these identifiers will be skipped.

C++ Type:std::vector

Options:

Description:If specified blocks matching these identifiers will be skipped.

• free_energies_opList of free energies used by additional order parameters. Places in same order as switching_function_names.

C++ Type:std::vector

Options:

Description:List of free energies used by additional order parameters. Places in same order as switching_function_names.

• kappa_opkappaThe kappa used with additional_ops

Default:kappa

C++ Type:MaterialPropertyName

Options:

• anisotropicFalseSet to true if the diffusivity is a tensor

Default:False

C++ Type:bool

Options:

Description:Set to true if the diffusivity is a tensor

• gamma_grgammaName of the gamma used with grains

Default:gamma

C++ Type:MaterialPropertyName

Options:

Description:Name of the gamma used with grains

• gamma_grxopgammaName of the gamma used when grains interact with other order parameters

Default:gamma

C++ Type:MaterialPropertyName

Options:

Description:Name of the gamma used when grains interact with other order parameters

## Example Input File Syntax

[Modules]
[./PhaseField]
[./GrandPotential]
switching_function_names = 'hb hm'
anisotropic = false

chemical_potentials = 'w'
mobilities = 'chiD'
susceptibilities = 'chi'
free_energies_w = 'rhob rhom'

gamma_gr = gamma
mobility_name_gr = L
kappa_gr = kappa
free_energies_gr = 'omegab omegam'

gamma_grxop = gamma
mobility_name_op = L_phi
kappa_op = kappa
free_energies_op = 'omegab omegam'
[../]
[../]
[]

[Materials]
#REFERENCES
[./constants]
type = GenericConstantMaterial
prop_names =  'Va      cb_eq cm_eq kb   km  mu  gamma L      L_phi  kappa  kB'
prop_values = '0.04092 1.0   1e-5  1400 140 1.5 1.5   5.3e+3 2.3e+4 295.85 8.6173324e-5'
[../]
#SWITCHING FUNCTIONS
[./switchb]
type = SwitchingFunctionMultiPhaseMaterial
h_name = hb
all_etas = 'phi eta0'
phase_etas = 'phi'
[../]
[./switchm]
type = SwitchingFunctionMultiPhaseMaterial
h_name = hm
all_etas = 'phi eta0'
phase_etas = 'eta0'
[../]
[./omegab]
type = DerivativeParsedMaterial
f_name = omegab
args = 'w phi'
material_property_names = 'Va kb cb_eq'
function = '-0.5*w^2/Va^2/kb - w/Va*cb_eq'
derivative_order = 2
[../]
[./omegam]
type = DerivativeParsedMaterial
f_name = omegam
args = 'w eta0'
material_property_names = 'Va km cm_eq'
function = '-0.5*w^2/Va^2/km - w/Va*cm_eq'
derivative_order = 2
[../]
[./chi]
type = DerivativeParsedMaterial
f_name = chi
args = 'w'
material_property_names = 'Va hb hm kb km'
function = '(hm/km + hb/kb)/Va^2'
derivative_order = 2
[../]
#DENSITIES/CONCENTRATION
[./rhob]
type = DerivativeParsedMaterial
f_name = rhob
args = 'w'
material_property_names = 'Va kb cb_eq'
function = 'w/Va^2/kb + cb_eq/Va'
derivative_order = 1
[../]
[./rhom]
type = DerivativeParsedMaterial
f_name = rhom
args = 'w eta0'
material_property_names = 'Va km cm_eq(eta0)'
function = 'w/Va^2/km + cm_eq/Va'
derivative_order = 1
[../]
[./concentration]
type = ParsedMaterial
f_name = c
material_property_names = 'rhom hm rhob hb Va'
function = 'Va*(hm*rhom + hb*rhob)'
outputs = exodus
[../]
[./mobility]
type = DerivativeParsedMaterial
material_property_names = 'chi kB'
constant_names = 'T Em D0'
constant_expressions = '1400 2.4 1.25e2'
f_name = chiD
function = 'chi*D0*exp(-Em/kB/T)'
[../]
[]

(modules/phase_field/test/tests/actions/gpm_kernel.i)

## References

1. N. Moelans, B. Blanpain, and P. Wollants. Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems. Physical Review B, 78(2):024113, Jul 2008. URL: http://link.aps.org/doi/10.1103/PhysRevB.78.024113, doi:10.1103/PhysRevB.78.024113.[BibTeX]
2. Mathis Plapp. Unified derivation of phase-field models for alloy solidification from a grand-potential functional. Physical Review E, 84(3):031601, Sep 2011. URL: https://link.aps.org/doi/10.1103/PhysRevE.84.031601, doi:10.1103/PhysRevE.84.031601.[BibTeX]