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Pore nucleation, and the decrease of vacancy concentration due the formation of pores.

This commit is contained in:
Haiming Zhang 2014-11-11 16:03:06 +00:00
parent 3fcb2a6476
commit f0f04a25bf
1 changed files with 87 additions and 19 deletions
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106
code/vacancy_generation.f90
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@ -29,15 +29,23 @@ module vacancy_generation
vacancy_generation_freq, & vacancy_generation_freq, &
vacancy_generation_formationEnergy, & vacancy_generation_formationEnergy, &
vacancy_generation_diffusionEnergy, & vacancy_generation_diffusionEnergy, &
vacancy_generation_diffusionCoeff0, & !< the temperature-independent pre-exponential of diffusion coefficient D_0
vacancy_generation_stressCoeff, & vacancy_generation_stressCoeff, &
vacancy_generation_jogHeight, & !< the height of jogs in Burgers vectors vacancy_generation_jogHeight, & !< the height of jogs in Burgers vectors
vacancy_generation_jogSeparation, & !< the jog seperation vacancy_generation_jogSeparation, & !< the jog seperation
vacancy_generation_nLatticeSites, & !< the number of lattice sites per unit volume vacancy_generation_nLatticeSites, & !< the number of lattice sites per unit volume
vacancy_generation_burgersVec, & !< the Burgers vector vacancy_generation_burgersVec, & !< the Burgers vector
vacancy_generation_dislocationCoeff vacancy_generation_dislocationCoeff, &
vacancy_generation_equilibConcentration !< the equilibrium concentration of vacancy
real(pReal), dimension(:), allocatable, public :: &
pore_nucleation_surfaceEnergy, & !< surface energy of metal which controls the necleation of pores
pore_nucleation_atomVolume, & !< the volume of atom
pore_nucleation_shellThickness, & !< the thickness of spherical shell surrounding the pore
pore_nucleation_concentrationCoeff0 !< the pre-exponential of equilibrium concentration of critical pore
real(pReal), parameter, private :: & real(pReal), parameter, private :: &
kB = 1.38e-23_pReal !< Boltzmann constant in J/Kelvin kB = 1.38e-23_pReal !< Boltzmann constant in J/Kelvin
enum, bind(c) enum, bind(c)
enumerator :: undefined_ID, & enumerator :: undefined_ID, &
@ -138,9 +146,16 @@ subroutine vacancy_generation_init(fileUnit)
allocate(vacancy_generation_jogHeight(maxNinstance), source=0.0_pReal) allocate(vacancy_generation_jogHeight(maxNinstance), source=0.0_pReal)
allocate(vacancy_generation_jogSeparation(maxNinstance), source=0.0_pReal) allocate(vacancy_generation_jogSeparation(maxNinstance), source=0.0_pReal)
allocate(vacancy_generation_nLatticeSites(maxNinstance), source=0.0_pReal) allocate(vacancy_generation_nLatticeSites(maxNinstance), source=0.0_pReal)
allocate(vacancy_generation_burgersVec(maxNinstance), source=0.0_pReal) allocate(vacancy_generation_burgersVec(maxNinstance), source=0.0_pReal)
allocate(vacancy_generation_diffusionCoeff0(maxNinstance), source=0.0_pReal)
allocate(vacancy_generation_equilibConcentration(maxNinstance), source=0.0_pReal)
allocate(vacancy_generation_dislocationCoeff(maxNinstance), source=0.0_pReal) allocate(vacancy_generation_dislocationCoeff(maxNinstance), source=0.0_pReal)
allocate(pore_nucleation_surfaceEnergy(maxNinstance), source=0.0_pReal)
allocate(pore_nucleation_atomVolume(maxNinstance), source=0.0_pReal)
allocate(pore_nucleation_shellThickness(maxNinstance), source=0.0_pReal)
allocate(pore_nucleation_concentrationCoeff0(maxNinstance), source=0.0_pReal)
rewind(fileUnit) rewind(fileUnit)
phase = 0_pInt phase = 0_pInt
@ -175,33 +190,51 @@ subroutine vacancy_generation_init(fileUnit)
IO_lc(IO_stringValue(line,positions,2_pInt)) IO_lc(IO_stringValue(line,positions,2_pInt))
end select end select
case ('atol_vacancyGeneration') case ('atol_vacancygeneration')
vacancy_generation_aTol(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_aTol(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_frequency') case ('vacancy_frequency')
vacancy_generation_freq(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_freq(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_formationEnergy') case ('vacancy_formationenergy')
vacancy_generation_formationEnergy(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_formationEnergy(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_diffusionEnergy') case ('vacancy_equilibconcentration')
vacancy_generation_equilibConcentration(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_diffusionenergy')
vacancy_generation_diffusionEnergy(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_diffusionEnergy(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_stressCoeff') case ('vacancy_diffusioncoeff0')
vacancy_generation_diffusionCoeff0(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_stresscoeff')
vacancy_generation_stressCoeff(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_stressCoeff(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_jogHeight') case ('vacancy_jogheight')
vacancy_generation_jogHeight(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_jogHeight(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_jogSeparation') case ('vacancy_jogseparation')
vacancy_generation_jogSeparation(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_jogSeparation(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_nLatticeSites') case ('vacancy_nlatticesites')
vacancy_generation_nLatticeSites(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_nLatticeSites(instance) = IO_floatValue(line,positions,2_pInt)
case ('vacancy_burgersVec') case ('vacancy_burgersvec')
vacancy_generation_burgersVec(instance) = IO_floatValue(line,positions,2_pInt) vacancy_generation_burgersVec(instance) = IO_floatValue(line,positions,2_pInt)
case ('pore_surfacefnergy')
pore_nucleation_surfaceEnergy(instance) = IO_floatValue(line,positions,2_pInt)
case ('pore_atomvolume')
pore_nucleation_atomVolume(instance) = IO_floatValue(line,positions,2_pInt)
case ('pore_shellthickness')
pore_nucleation_shellThickness(instance) = IO_floatValue(line,positions,2_pInt)
case ('pore_concentrationcoeff0')
pore_nucleation_concentrationCoeff0(instance) = IO_floatValue(line,positions,2_pInt)
end select end select
endif; endif endif; endif
enddo parsingFile enddo parsingFile
@ -213,11 +246,11 @@ subroutine vacancy_generation_init(fileUnit)
!-------------------------------------------------------------------------------------------------- !--------------------------------------------------------------------------------------------------
! Calculate the coefficient for dislocation motion induced vacancy generation ! Calculate the coefficient for dislocation motion induced vacancy generation
vacancy_generation_dislocationCoeff(instance) = vacancy_generation_jogHeight(instance)/ & vacancy_generation_dislocationCoeff(instance) = vacancy_generation_jogHeight(instance)/ &
vacancy_generation_jogSeparation(instance)/ & vacancy_generation_jogSeparation(instance)/ &
vacancy_generation_nLatticeSites(instance)/ & vacancy_generation_nLatticeSites(instance)/ &
vacancy_generation_burgersVec(instance)/ & vacancy_generation_burgersVec(instance)/ &
vacancy_generation_burgersVec(instance) vacancy_generation_burgersVec(instance)
!-------------------------------------------------------------------------------------------------- !--------------------------------------------------------------------------------------------------
! Determine size of postResults array ! Determine size of postResults array
@ -313,7 +346,8 @@ subroutine vacancy_generation_dotState(nSlip, accumulatedSlip, Tstar_v, Temperat
vacancyState vacancyState
use math, only: & use math, only: &
math_Mandel6to33, & math_Mandel6to33, &
math_trace33 math_trace33, &
pi
implicit none implicit none
integer(pInt), intent(in) :: & integer(pInt), intent(in) :: &
@ -326,23 +360,57 @@ subroutine vacancy_generation_dotState(nSlip, accumulatedSlip, Tstar_v, Temperat
real(pReal), intent(in), dimension(6) :: & real(pReal), intent(in), dimension(6) :: &
Tstar_v !< 2nd Piola Kirchhoff stress tensor (Mandel) Tstar_v !< 2nd Piola Kirchhoff stress tensor (Mandel)
real(pReal), intent(in) :: & real(pReal), intent(in) :: &
Temperature !< 2nd Piola Kirchhoff stress tensor (Mandel) Temperature !< 2nd Piola Kirchhoff stress tensor (Mandel)
real(pReal) :: & real(pReal) :: &
pressure !< 2nd Piola Kirchhoff stress tensor (Mandel) pressure !< 2nd Piola Kirchhoff stress tensor (Mandel)
integer(pInt) :: & integer(pInt) :: &
instance, phase, constituent instance, phase, constituent
real(pReal) :: &
vacancyConcentration, & !< current vacancy concentration
vacancyDiffusion, & !< the diffusion coefficient D_v
poleZeldovichCoeff, & !< Zeldovich factor of pore nucleation
vacancyAbsorpRateCoeff, & !< vacancy absorption rate
chemicalPotential, & !< the chemical potential due to vacancy concentration
criticalRadius, & !< the critical pore radius
Gibbs4Pore, & !< the Gibbs free energy for generating a critical pore
equilibPoreConcentration, & !< the equilibrium pore concentration
nucleationRatePore !< the nucleation rate of pore
phase = mappingConstitutive(2,ipc,ip,el) phase = mappingConstitutive(2,ipc,ip,el)
constituent = mappingConstitutive(1,ipc,ip,el) constituent = mappingConstitutive(1,ipc,ip,el)
instance = phase_vacancyInstance(phase) instance = phase_vacancyInstance(phase)
pressure = math_trace33(math_Mandel6to33(Tstar_v)) pressure = math_trace33(math_Mandel6to33(Tstar_v))
!--------------------------------------------------------------------------------------------------
! Calculate nucleation rate of pore
vacancyDiffusion = vacancy_generation_diffusionCoeff0(instance)* &
exp( -vacancy_generation_diffusionEnergy(instance)/(kB*temperature) )
vacancyConcentration = vacancy_generation_getConcentration(ipc, ip, el)
chemicalPotential = kB*Temperature * log(vacancyConcentration/ &
vacancy_generation_equilibConcentration(instance))
criticalRadius = 2_pReal/chemicalPotential* &
pore_nucleation_surfaceEnergy(instance) * pore_nucleation_atomVolume(instance)
Gibbs4Pore = 4_pReal/3_pReal * pi * pore_nucleation_surfaceEnergy(instance)* &
criticalRadius * criticalRadius
equilibPoreConcentration = pore_nucleation_concentrationCoeff0(instance)* &
exp( -Gibbs4Pore/(kB*temperature) )
vacancyAbsorpRateCoeff = 2_pReal/pore_nucleation_shellThickness(instance) * &
vacancyDiffusion * vacancyConcentration
poleZeldovichCoeff = pore_nucleation_atomVolume(instance)* &
sqrt( pore_nucleation_surfaceEnergy(instance)/(kB*temperature) )
nucleationRatePore = poleZeldovichCoeff * vacancyAbsorpRateCoeff* equilibPoreConcentration
!--------------------------------------------------------------------------------------------------
! the net generating rate vacancy
vacancyState(phase)%dotState(1,constituent) = & vacancyState(phase)%dotState(1,constituent) = &
vacancy_generation_freq(instance)* & vacancy_generation_freq(instance)* &
exp(-(vacancy_generation_formationEnergy(instance) - vacancy_generation_stressCoeff(instance)*pressure)/ & exp(-(vacancy_generation_formationEnergy(instance) - vacancy_generation_stressCoeff(instance)*pressure)/ &
(kB*Temperature)) + & (kB*Temperature)) + &
sum(accumulatedSlip) * vacancy_generation_dislocationCoeff(instance) !< Induced by dislocation motion. sum(accumulatedSlip) * vacancy_generation_dislocationCoeff(instance)- & !< Induced by dislocation motion
nucleationRatePore * (4_pReal/3_pReal * pi * criticalRadius**3_pReal)/ & !< Reduced by the formation of pore
pore_nucleation_atomVolume(instance)
end subroutine vacancy_generation_dotState end subroutine vacancy_generation_dotState
!-------------------------------------------------------------------------------------------------- !--------------------------------------------------------------------------------------------------