From f0f04a25bf3756a102ab1977e94e4ef526c696f7 Mon Sep 17 00:00:00 2001 From: Haiming Zhang Date: Tue, 11 Nov 2014 16:03:06 +0000 Subject: [PATCH] Pore nucleation, and the decrease of vacancy concentration due the formation of pores. --- code/vacancy_generation.f90 | 106 +++++++++++++++++++++++++++++------- 1 file changed, 87 insertions(+), 19 deletions(-) diff --git a/code/vacancy_generation.f90 b/code/vacancy_generation.f90 index b2a979482..e0de863ad 100644 --- a/code/vacancy_generation.f90 +++ b/code/vacancy_generation.f90 @@ -29,15 +29,23 @@ module vacancy_generation vacancy_generation_freq, & vacancy_generation_formationEnergy, & vacancy_generation_diffusionEnergy, & + vacancy_generation_diffusionCoeff0, & !< the temperature-independent pre-exponential of diffusion coefficient D_0 vacancy_generation_stressCoeff, & vacancy_generation_jogHeight, & !< the height of jogs in Burgers vectors vacancy_generation_jogSeparation, & !< the jog seperation vacancy_generation_nLatticeSites, & !< the number of lattice sites per unit volume 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 :: & - kB = 1.38e-23_pReal !< Boltzmann constant in J/Kelvin + kB = 1.38e-23_pReal !< Boltzmann constant in J/Kelvin enum, bind(c) enumerator :: undefined_ID, & @@ -138,9 +146,16 @@ subroutine vacancy_generation_init(fileUnit) allocate(vacancy_generation_jogHeight(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_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(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) phase = 0_pInt @@ -175,33 +190,51 @@ subroutine vacancy_generation_init(fileUnit) IO_lc(IO_stringValue(line,positions,2_pInt)) end select - case ('atol_vacancyGeneration') + case ('atol_vacancygeneration') vacancy_generation_aTol(instance) = IO_floatValue(line,positions,2_pInt) case ('vacancy_frequency') 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) - 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) - 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) - case ('vacancy_jogHeight') + case ('vacancy_jogheight') 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) - case ('vacancy_nLatticeSites') + case ('vacancy_nlatticesites') 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) + 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 endif; endif enddo parsingFile @@ -213,11 +246,11 @@ subroutine vacancy_generation_init(fileUnit) !-------------------------------------------------------------------------------------------------- ! 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_nLatticeSites(instance)/ & vacancy_generation_burgersVec(instance)/ & - vacancy_generation_burgersVec(instance) + vacancy_generation_burgersVec(instance) !-------------------------------------------------------------------------------------------------- ! Determine size of postResults array @@ -313,7 +346,8 @@ subroutine vacancy_generation_dotState(nSlip, accumulatedSlip, Tstar_v, Temperat vacancyState use math, only: & math_Mandel6to33, & - math_trace33 + math_trace33, & + pi implicit none integer(pInt), intent(in) :: & @@ -326,23 +360,57 @@ subroutine vacancy_generation_dotState(nSlip, accumulatedSlip, Tstar_v, Temperat real(pReal), intent(in), dimension(6) :: & Tstar_v !< 2nd Piola Kirchhoff stress tensor (Mandel) real(pReal), intent(in) :: & - Temperature !< 2nd Piola Kirchhoff stress tensor (Mandel) + Temperature !< 2nd Piola Kirchhoff stress tensor (Mandel) real(pReal) :: & - pressure !< 2nd Piola Kirchhoff stress tensor (Mandel) + pressure !< 2nd Piola Kirchhoff stress tensor (Mandel) integer(pInt) :: & 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) constituent = mappingConstitutive(1,ipc,ip,el) instance = phase_vacancyInstance(phase) 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) = & vacancy_generation_freq(instance)* & exp(-(vacancy_generation_formationEnergy(instance) - vacancy_generation_stressCoeff(instance)*pressure)/ & (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 !--------------------------------------------------------------------------------------------------