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