Kim-Kim-Suzuki Example for three or more components
#
# KKS ternary (3 chemical component) system example in the split form
# We track c1 and c2 only, since c1 + c2 + c3 = 1
#
[Mesh]
type = GeneratedMesh
dim = 2
nx = 150
ny = 15
nz = 0
xmin = -25
xmax = 25
ymin = -2.5
ymax = 2.5
zmin = 0
zmax = 0
elem_type = QUAD4
[]
[AuxVariables]
[./Fglobal]
order = CONSTANT
family = MONOMIAL
[../]
[]
[Variables]
# order parameter
[./eta]
order = FIRST
family = LAGRANGE
[../]
# solute 1 concentration
[./c1]
order = FIRST
family = LAGRANGE
[../]
# solute 2 concentration
[./c2]
order = FIRST
family = LAGRANGE
[../]
# chemical potential solute 1
[./w1]
order = FIRST
family = LAGRANGE
[../]
# chemical potential solute 2
[./w2]
order = FIRST
family = LAGRANGE
[../]
# Liquid phase solute 1 concentration
[./c1l]
order = FIRST
family = LAGRANGE
initial_condition = 0.1
[../]
# Liquid phase solute 2 concentration
[./c2l]
order = FIRST
family = LAGRANGE
initial_condition = 0.05
[../]
# Solid phase solute 1 concentration
[./c1s]
order = FIRST
family = LAGRANGE
initial_condition = 0.8
[../]
# Solid phase solute 2 concentration
[./c2s]
order = FIRST
family = LAGRANGE
initial_condition = 0.1
[../]
[]
[Functions]
[./ic_func_eta]
type = ParsedFunction
expression = '0.5*(1.0-tanh((x)/sqrt(2.0)))'
[../]
[./ic_func_c1]
type = ParsedFunction
expression = '0.8*(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10)+0.1*(1-(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10))'
[../]
[./ic_func_c2]
type = ParsedFunction
expression = '0.1*(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10)+0.05*(1-(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10))'
[../]
[]
[ICs]
[./eta]
variable = eta
type = FunctionIC
function = ic_func_eta
[../]
[./c1]
variable = c1
type = FunctionIC
function = ic_func_c1
[../]
[./c2]
variable = c2
type = FunctionIC
function = ic_func_c2
[../]
[]
[Materials]
# Free energy of the liquid
[./fl]
type = DerivativeParsedMaterial
property_name = fl
coupled_variables = 'c1l c2l'
expression = '(0.1-c1l)^2+(0.05-c2l)^2'
[../]
# Free energy of the solid
[./fs]
type = DerivativeParsedMaterial
property_name = fs
coupled_variables = 'c1s c2s'
expression = '(0.8-c1s)^2+(0.1-c2s)^2'
[../]
# h(eta)
[./h_eta]
type = SwitchingFunctionMaterial
h_order = HIGH
eta = eta
[../]
# g(eta)
[./g_eta]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta
[../]
# constant properties
[./constants]
type = GenericConstantMaterial
prop_names = 'M L eps_sq'
prop_values = '0.7 0.7 1.0 '
[../]
[]
[Kernels]
# enforce c1 = (1-h(eta))*c1l + h(eta)*c1s
[./PhaseConc1]
type = KKSPhaseConcentration
ca = c1l
variable = c1s
c = c1
eta = eta
[../]
# enforce c2 = (1-h(eta))*c2l + h(eta)*c2s
[./PhaseConc2]
type = KKSPhaseConcentration
ca = c2l
variable = c2s
c = c2
eta = eta
[../]
# enforce pointwise equality of chemical potentials
[./ChemPotSolute1]
type = KKSPhaseChemicalPotential
variable = c1l
cb = c1s
fa_name = fl
fb_name = fs
args_a = 'c2l'
args_b = 'c2s'
[../]
[./ChemPotSolute2]
type = KKSPhaseChemicalPotential
variable = c2l
cb = c2s
fa_name = fl
fb_name = fs
args_a = 'c1l'
args_b = 'c1s'
[../]
#
# Cahn-Hilliard Equations
#
[./CHBulk1]
type = KKSSplitCHCRes
variable = c1
ca = c1l
fa_name = fl
w = w1
args_a = 'c2l'
[../]
[./CHBulk2]
type = KKSSplitCHCRes
variable = c2
ca = c2l
fa_name = fl
w = w2
args_a = 'c1l'
[../]
[./dc1dt]
type = CoupledTimeDerivative
variable = w1
v = c1
[../]
[./dc2dt]
type = CoupledTimeDerivative
variable = w2
v = c2
[../]
[./w1kernel]
type = SplitCHWRes
mob_name = M
variable = w1
[../]
[./w2kernel]
type = SplitCHWRes
mob_name = M
variable = w2
[../]
#
# Allen-Cahn Equation
#
[./ACBulkF]
type = KKSACBulkF
variable = eta
fa_name = fl
fb_name = fs
w = 1.0
coupled_variables = 'c1l c1s c2l c2s'
[../]
[./ACBulkC1]
type = KKSACBulkC
variable = eta
ca = c1l
cb = c1s
fa_name = fl
coupled_variables = 'c2l'
[../]
[./ACBulkC2]
type = KKSACBulkC
variable = eta
ca = c2l
cb = c2s
fa_name = fl
coupled_variables = 'c1l'
[../]
[./ACInterface]
type = ACInterface
variable = eta
kappa_name = eps_sq
[../]
[./detadt]
type = TimeDerivative
variable = eta
[../]
[]
[AuxKernels]
[./GlobalFreeEnergy]
variable = Fglobal
type = KKSGlobalFreeEnergy
fa_name = fl
fb_name = fs
w = 1.0
[../]
[]
[Executioner]
type = Transient
solve_type = 'PJFNK'
petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type'
petsc_options_value = 'asm ilu nonzero'
l_max_its = 100
nl_max_its = 100
num_steps = 50
dt = 0.1
[]
#
# Precondition using handcoded off-diagonal terms
#
[Preconditioning]
[./full]
type = SMP
full = true
[../]
[]
[Outputs]
exodus = true
[]
When additional chemical components are added to the KKS model, a Cahn-Hilliard equation must be added for each additional component. (For components, Cahn-Hilliard equations are required). Each additional Cahn-Hilliard equation requires the kernels:
To enforce the composition and chemical potential constraints, each additional component also requires the kernels
The Allen-Cahn equation is also modified when additional components are added. The residual becomes
where is the number of components. A single KKSACBulkF kernel is needed as in the 2-component case, and an additional KKSACBulkC kernel must added for each additional component.