PanEvolution tables

Table 1:  Summary of Equations for Nucleation Models
Name Operation Examples
J

The transient nucleation rate

Nν

and

Potential nucleation sites

Z

Zeldovich factor

β*

Atomic attachment rate

τ

Incubation time

ΔG*

Nucleation barrier energy

R*

Critical nucleation radius

ΔGvol

ΔGV+ΔGS

Volume energy change during nucleation.ΔGV is the chemical driving force per volume and ΔGS is the elastic strain energy

ΔGS

Elastic strain energy. The volume misfit ∆ and particle aspect ratio AR are given in kdb file

σeff

Effective interfacial energy

σαβ

The estimated interfacial energy based on GBB method

Deff

Effective diffusivity for multi-component alloys

 

Table 2:  Model Options for precipitation in kdb
Name Options Description

model

KWN, Fast-Acting(FA)

Refer to Figure 1 in Precipitation Nucleation

morphology

Sphere; Cylinder

Refer to Figure 1 in Precipitation Growth. The aspect ratio AR and shape factors are set to be 1 automatically when “Sphere” is selected

nucleation

Modified_Homogeneous; Grain_Boundary; Grain_Edge; Grain_Corner; Dislocation

Refer to Table 1. Both homogeneous and heterogeneous nucleation can be considered by “Modified_Homogeneous”. In this case, the values of Nν and ∆G* must be manually adjusted through Nf, ∆Gv and ∆GS as discussed in Table 4.

growth

Simplified;

SFFK;

SFFK_Shape_Evolution

A constant value of The aspect ratio AR can be assigned for “Simplified” and “SFFK”. Choose “SFFK_Shape_Evolution” for shape evolution, which means AR varies during particle growth

phase_name

 

Each “PrecipitatePhase” has a “name” and “phase_name”. “phase_name” must be consistent with the name in tdb/pdb. The “phase_name” tag can be empty if “name” and “phase_name” are same.

 

Table 3:  Model Options for grain growth in kdb
Name Options Description

model

Grain_OneSize, Grain_MultiSize

Refer to Figure 1 in Precipitation Nucleation

morphology

Sphere

Only spherical approximation is considered

nucleation

N/A

Recystallization process is not included

growth

Grain_Simplified

The simplified grain growth model is implemented as shown in Equations Eq. 1 and Eq. 4. User’s grain growth model can be defined in KDB (see example Section User-defined Growth Rate Model for Grain Coarsening)

 

Table 4:  Kinetic model parameters for precipitation in kdb

Name

Unit

Description

Equation

Molar_Volume

m3/mole

Molar volume of matrix or precipitate phase

<Parameter type="Molar_Volume" value="6E-6" description="Molar Volume" />

Vm in Eq. 2 and Vain Eq. 3

Grain_Size

m

The grain size of the matrix phase

<Parameter type="Grain_Size" value="1e-4" description="Grain size, default value = 1e-4m" />

D in Eq. 12

Dislocation_Density

m-2

The dislocation density in the matrix phase

<Parameter type="Dislocation_Density" value="1e13" description = "Dislocation density, Default value  =1.0e12/m^-2" />

ρi in Eq. 10

Grain_Aspect_Ratio

N/A

The aspect ratio for the matrix grain

<Parameter type="Grain_Aspect_Ratio" value="1.0" description="grain aspect ratio, default value = 1.0" />

A in Eq. 12

Contact_Angle

degree

Contact angle of nucleus on grain boundary, default value = 90 degree

in Eq. 8

Aspect_Ratio

N/A

The aspect ratio of the precipitate phase. The value of ARis evolving if “SFFK_Shape_Evolution” is chosen as growth model.

<Parameter type="A_R" value="1" description ="Initial aspect ratio" />

AR in Eq. 6

Interfacial_Energy

J/m2

Interfacial energy

<Parameter type="Interfacial_Energy" value="0.2" description ="Interfacial Energy" />

User keyword “IFE_CAC(*)” to get the calculated interfacial energy:

<Parameter type="Interfacial_Energy" value="IFE_CALC(*)" description="Interfacial Energy" />

in Eq. 2 and Eq. 13

Interfacial_Energy_L

J/m2

Interfacial energy in L direction

<Parameter type="Interfacial_Energy_L" value="0.05" description="Interfacial Energy in L direction" />

Used in “SFFK_Shape_Evolution” model

Antiphase_Boundary_Energy

J/m2

Antiphase boundary energy

in Eq. 2 and Eq. 4

Atomic_Spacing

m

Usually use lattice constant

<Parameter type="Atomic_Spacing" value="7.6E-10" description="Atomic Spacing" />

a in Eq. 4

Nucleation_Site_Parameter

N/A

Homogeneous: choose a value close to solute concentration;

Heterogeneous: choose a value close to nucleation density when “Modified_Homogeneous” option is chosen for nucleation model. Otherwise, use the model automatically estimate the nucleation density and default value of 1.0 can be used. Such an example is given in Section Another Example for Heterogeneous Nucleation

Nf in Eq. 7 and Eq. 11

Driving_Force_Factor

N/A

A factor adjusting chemical driving force obtained by thermodynamic calculation

A pre-factor applied to ΔGV in Eq. 2

Strain_Energy

 

The elastic strain energy per volume of precipitate offsetting the calculated value by equation Eq. 5.

 

Volume_Misfit

N/A

The volume misfit

Δ in Eq. 6

Kinetic_Parameter_Factor

N/A

A factor adjusting kinetic parameter obtained by thermodynamic and mobility calculation

A pre-factor applied to adjust K in equation Eq. 1

Effective_Diffusivity_Factor

N/A

A factor adjusting effective diffusivity for nucleation obtained by mobility calculation

A pre-factor applied to adjust Deff in Eq. 4

Steady_State_Nucleation_Rate

N/A

0: transient nucleation rate;

1: steady state nucleation rate;

in Eq. 1

 

Table 5:  Kinetic model parameters for grain growth in kdb

Name

Unit

Description

Equation

Interfacial_Energy

J/m2

High angle grain boundary energy

γ in equations Eq. 3, Eq. 4 and Eq. 5

Grain_Boundary_Width

m

Grain boundary width. A suggested value is equal to twice the atomic radius.

δ in Eq. 2

Grain_Boundary_Mobility_Factor

N/A

A pre-factor to adjust grain boundary mobility.

A2 in Eq. 2

Zener_Drag_Force_Factor

N/A

A factor for Zener drag force, default is 0 for not considering Zener pinning effect.

b in Eq. 4

 

Table 6:  User-defined models for grain growth in kdb

Name

Unit

Description

Equation

Grain_Boundary_Mobility

m/(s·J/mol)

User-defined grain boundary mobility. It can be an expression, for example: 1e-18/2.86e-10

<VariableTable name="Variables replacing built-in variables">
Parameter type="Grain_Boundary_Mobility" value="KP(@Grain)/2.86e-10" description="Grain_Boundary_Mobility" />
</VariableTable>

KP(@Fcc) ( in Eq. 2 is the effective atomic mobility for multi-component alloys calculated from mobility database.

Zener_Drag_Force

J/mol

User-defined drag force for Zener pinning effect. It can be an expression, for example: -1.5*7.1E-6*0.75*0.15/(0.3 * 1e-6), which is "-factor*Vm*Sigma*fv/size"

 

Growth_Rate

m/s

User-defined grain growth rate. It can be an expression, for example: KP(@Grain)/2.86e-10*2*0.75*7.1e-6/s(@Grain) with KP(@Grain) is the effective mobility and s(@Grain) is the mean grain size

 

 

Table 7:  Mechanical model parameters defined in kdb

Name

Unit

Description

Equation

Shear_Modulus

Pa

The shear modulus of the matrix phase

μ in Eq. 6 and Eq. 4

Burgers_Vector

m

The Burgers vector of the matrix phase

b in Eq. 4

Taylor_Factor

N/A

The Taylor factor of the matrix phase

M in Eq. 5

Solution_Strengthening_Factor

N/A

scaling factor of alloying element for solution strengthening

aj in Eq. 11

Strength_Parameter

N/A

Strengthening parameter due to precipitation hardening

kP in Eq. 7

Shearing_Critical_Radius

m

Critical radius shifting from shearing to looping mechanism

RC in Eq. 7

Intrinsic_Strength

MPa

The baseline contribution including lattice resistance, work-hardening and grain boundaries hardening.

s0 in Eq. 10

Hardness_Factor

N/A

The yield strength in MPa can be converted to hardness in VPN based on Eq. 11

A in Eq. 12

Hardness_Constant

VPN

The yield strength in MPa can be converted to hardness in VPN based on Eq. 11

B in Eq. 12

 

Table 8:  Model parameters for RX and Dislocation Density in kdb

Name

 Unit

Description

 Equation

f_WH

N/A

Work hardening coefficient for dislocation density evolution model

fw in Eq. 2

f_DRV

N/A

Dynamic recovery coefficient for dislocation density evolution model

fv in Eq. 3

f_SRV

N/A

Static recovery coefficient for dislocation density evolution model

 

f_RX

N/A

Recrystallization coefficient for dislocation density evolution model

fx in Eq. 4

Avrami_Exponent

N/A

Exponent for JMAK

n in Eq. 1

time_half

s

The time for 50% RX for JMAK

t0.5 in Eq. 1

Nucleation_Site_Parameter

N/A

Parameter for potential nucleation sites for the Fast-Acting nucleation model

Nf in Eq. 4

Kinetic_Parameter_Factor

N/A

Kinetic parameter factor for the Fast-Acting growth model, which is a pre-factor to adjust the effective self-diffusivity Deff

Pre-factor of Deff in Eq. 7
Table 9:  User-Defined variables for RX and Dislocation Density in kdb

Name

Description

Dislocation_Density_Rate

User-defined dislocation density rate to replace the built-in K-M model defined by Eq. 1

Nucleation_Rate

User-defined nucleation rate to replace the built-in model defined by Eq. 5

Nucleation_Barrier_Energy

User-defined Nucleation barrier energy to replace the built-in model defined by Eq. 3

Critical_Grain_Size_RX

User-defined Critical nucleus size for RX to replace the built-in model defined by Eq. 2

Dislocation_Mean_Free_Path

User-defined Dislocation mean free path for critical density to replace the built-in model defined by Eq. 1

Growth_Rate

User-defined Growth rate to replace the built-in RX growth model defined by Eq. 6

 

Table 10:  Symbol and syntax for retrieving system quantities

Name

Unit (SI)

Comments

time

second

Time

T

K

Temperature

vft

 

Total Transformed Volume Fraction: where  is the transformed volume fraction of phase

x(comp), w(comp)

 

Overall alloy composition

 

Table 11:  Symbol and syntax for retrieving quantities of precipitate phases or grain

Name

Unit (SI)

Comments

s(@phase)

m

Average size/radius of equivalent sphere particles or grain

D(@phase)

m

Diameter of cylinder

L(@phase)

m

Length/Height of cylinder

A_R(@phase)

m

Aspect ratio of cylinder

nd(@phase)

#m-3

Number density

nr(@phase)

m-3sec-1

Nucleation rate

vf(@phase)

 

Volume fraction of specified phase

x(comp@phase), w(comp@ phase)

 

Instant composition of the matrix or precipitate phases

IFE_CALC(@phase)

J/m2

Model calculated interfacial energy

dgm(@phase)

J/mole

Nucleation driving force of phase(s)

vf_range(@phase,lb,ub)

 

The volume fraction for different particle groups defined by a size range [lb, ub] such as primary, secondary and tertiary in Ni-based super alloys, for example vf_range(@L12_FCC,0.5e-8, 0.5e-7)

s_range(@phase,lb,ub)

 

average size for different particle groups defined by a size range [lb, ub], for example s_range(@L12_FCC,0.5e-8, 0.5e-7)

Grain_Size

m

Mean grain size for the deformed matrix grains and the recrystallized grains

dislocation_density(@*)

1/m2

Dislocation density in the deformed matrix grains or the recrystallized grains. Example: dislocation_density(@Grain)

 

Table 12:  Symbol and syntax for retrieving quantities of particle size distribution (PSD)

Name

Unit

Comments

time

s

The PSDs are saved for the user-specified times; the PSD for the last time step is automatically saved. Using time = t to get the PSD for time “t”.

psd_id

N/A

The PSD consists of a certain number of cells (size classes); psd_id gets the cell id.

psd_s(@phase)

m

The characteristic size of a precipitate phase or grain for each cell.

psd_nd(@phase)

#m-3

The number density of a precipitate phase or grain for each cell.

psd_gr(@phase)

m/sec

The growth rate of a precipitate phase or grain for each cell.

psd_ns(@phase)

 

Normalized size of the cell

psd_nnd(@phase)

 

Normalized number density of the cell

psd_df(@phase)

 

The distribution function: with being the cell width

psd_cvf(@phase)

 

Cumulative volume fraction of phase(s). Example: psd_cvf(@L12_FCC).

 

Table 13:  Symbol and syntax for retrieving mechanical properties

Name

Unit (SI)

Comments

sigma_y

MPa

Overall yield strength. Example: sigma_y

Hv

vpn

Overall microhardness. Example: hv

sigma_i

MPa

Intrinsic yield strength. Example: sigma_i.

sigma_ss

MPa

Yield strength due to solution strengthening. Example: sigma_ss.

sigma_p(@*)

MPa

Yield strength due to precipitation hardening. Example: sigma_p(@Mg5Si6).

sigma_m(@Grain)

MPa

Yield strength due to dislocation interactions in grain structure.

 

Table 14:  Constants of mathematics and physics

Name

Comments

_K

Boltzmann constant

_PI

Archimedes' constant.

_R

Molar gas constant.

_NA

Avogadro constant.

_E

Natural Logarithmic Base.