Phase Interactions¶

In this section the user defines the phase interaction data and the grain boundary properties. Phase interactions can be defined for all pair-wise combinations of the phases which have been included in the Phases section.

In the standard input sequence, the user is requested to specify all phase interaction data in a fixed order, starting with the phase pair 0/1. Phase interactions which are not used are switched off using the keyword no_phase_interaction. No further input is required in this case. The standard input sequence is recommended for less experienced users and for low numbers of phases.

Example 1¶

No phase interaction between phases 0 and 1 and phase interaction between phase 1 and 1

...
# Phase Interactions
# ==================
#
# 0 (MATRIX)  /  1 (PHASE_1)
# --------------------------
# Simulation of interaction between 0 (MATRIX)  and  1 (PHASE_1) ?
# Options: phase_interaction  no_phase_interaction
#  [ standard | particle_pinning[_temperature] | solute_drag ]
#   | [ no_junction_force | junction_force ]
no_phase_interaction
#
# 1 (PHASE_1)  /  1 (PHASE_1)
# ---------------------------
# Simulation of interaction between 1 (PHASE_1)  and  1 (PHASE_1) ?
# Options: phase_interaction  no_phase_interaction  identical phases nb
#  [ standard | particle_pinning[_temperature] | solute_drag ]
#   | [ no_junction_force | junction_force ]
phase_interaction
...

If the number of phases involved in the simulation is high, the input file may get quite long and unreadable, even if most of the theoretically possible phase interactions are switched off. For this case, a terse mode is available, in which only those phase interactions which are needed can be defined in an arbitrary order. The other interactions are automatically switched off. Input is switching to terse mode if the line for the first phase interaction starts with two integers, corresponding to the numbers of the interacting phases. For technical reasons, the section then has to be finished by the keyword end_phase_interactions, if the terse mode is used.

Example 2¶

Phase interaction data with phase interaction in terse mode

...
# Phase Interactions
# ==================
#
# 0 (LIQUID)  /  1 (FCC_L12)
# --------------------------
# Simulation of interaction between 0 (LIQUID)  and  1 (FCC_L12) ?
# Options: phase_interaction  no_phase_interaction
#  [ standard | particle_pinning[_temperature] | solute_drag ]
#   | [ redistribution_control ] or [ no_junction_force | junction_force ]
0 1 phase_interaction
...
end_phase_interactions
...

Enabling phase interactions means that one of the phases may grow or shrink on the expense of the other. On the other hand, if a phase interaction is switched off, no movement of the corresponding interfaces is possible.

This also concerns the initialisation of the interface: if such an interface is created via the initial grain setting, the interface will stay sharp even if an initialisation is requested (see here). Nevertheless, in case of concentration coupling, there can be diffusion through switched off interfaces. The partition coefficients which are necessary for diffusion through interfaces are accessed from the other phase interactions using a constant approximation. If e.g. interactions are defined for phases 0/1 and 0/2, a simplified description can be derived for the ½ interface. This description is stored for each interface cell in the moment of creation (as initial structure or from moving triple junctions) and kept constant during the further simulation steps.

Important: If too many phase interactions are switched off, no calculation of the missing partition coefficients may be possible!

Switching off phase interactions can greatly reduce the complexity of a simulation, especially if many phases are included. In solidification, e.g., it is in most cases wise to discard all solid-solid interactions, if a continuation of the simulation (with heat treatment etc.) is not intended. In case of solid state reactions, only the interactions of the precipitations with the matrix and not the interactions between different types of precipitates should be included. If two contacting phases have a common stoichiometric component without solubility range (see Concentration Solver - Stoichiometry and more), the interaction between these two phases must be switched off!

When the interaction of one phase with itself is enabled, solid-state interactions between grains of the same phase will be activated. In this case, no chemical driving force is included and the movement will be controlled only by curvature (and by stored energies in case of recrystallisation).

As alternative to phase_interaction and no_phase_interaction, the keyword identical, followed by the phase numbers corresponding to another interaction (two integers), can be used for abbreviation purposes, if a phase interaction with exactly the same parameters has already been defined previously.

Phase Interaction options¶

The option phase_interaction can be followed by optional keywords which selects special interaction models:

• standard: this is the default interaction

• particle_pinning: enables a mesoscale model for particle pinning. If this keyword is selected, an effective mobility model is used which includes the pinning effect of not explicitly resolved particles. As further input, a critical pinning force, which can by defined either by a pinning pressure $[J/cm^3]$ or $[MPa]$ (default) or a pinning curvature $[\mu m^{-1}]$, and a minimal mobility will be requested later in this section.

• particle_pinning_temperature: enables the mesoscale particle pinning model with temperature dependent input parameters read from an ASCII file.

• solute_drag: the solute drag model is based on a two-step hysteresis description of the mobility for the free and loaded interface, as a function of the interface velocity. If this model is selected, a critical transition velocity, a transition range (for smooth transition) and a drag factor has to be specified.

• redistribution_control: this keyword activates special models which determine the redistribution behaviour. Currently, models for para-equilibrium, for nple as well as for antitrapping are available. These models are further specified in Phase Diagrams). Note that the keyword is also used to activate the automatic mobility correction in the thin-interface-limit for diffusion controlled phase interactions.

Driving Force options¶

In the next line, the driving force options have to be specified. Except for the local RX model, where these options have been specified always, they have been specified only in the case of interaction between different phases. All DeltaG options have to be written in one line, consisting of concatenated pairs of a keyword and the corresponding value (see Example 3).

The keyword avg is used to define an averaging of the driving force across the interface. Averaging prevents spreading of the interface if a strong concentration gradient is causing opposite driving forces on both sides of the interface. The user can specify a value between 0 (no averaging) and 1 (maximum averaging).

Averaging of the driving force helps stabilising the interface in case of diffusion controlled transformations, if the resolution is not high enough (i.e., the diffusion length is not big compared to the interface thickness). The recommended value is 0.5, the default value is 0.

The keyword max specifies the maximum driving force allowed (value above which the driving force will be cut-off). This value is useful to shrug off some temporary problems during initial transients or to reduce the impact of numerical fluctuations. The value should be chosen high enough in order not to limit kinetics during normal growth.

Important: If a too small value is chosen for the maximum allowed driving force, the movement of the interface can be drastically slowed down!

The smooth keyword has only effect if averaging is specified: The gradient direction along which averaging of the driving force is performed, is randomly rotated with the specified maximum value in degrees. Depending on other circumstances, this option may help to reduce the effect of grid anisotropy on the growth morphology. A typical value is 45, default is 0.

offset can be used to create an extra driving force shift. It can be used to correct small deviations in thermodynamic databases, to add an implicit curvature, to describe martensite as destabilized ferrite, or just to make an interface move in case of uncoupled phase-field.

Example 3¶

Phase interaction with driving force averaging, temperature dependent surface energy, constant mobility of the phase boundary and isotropic phase interaction between liquid and solid**

...
# Phase Interactions
# ==================
#
# 0 (LIQUID)  /  1 (FCC_L12)
# --------------------------
# Simulation of interaction between 0 (LIQUID)  and  1 (FCC_L12) ?
# Options: phase_interaction  no_phase_interaction
#  [ standard | particle_pinning[_temperature] | solute_drag ]
#   | [ redistribution_control ] or [ no_junction_force | junction_force ]
0 1 phase_interaction redistribution_control
# 'DeltaG' options:  default
# avg ...[] max ...[J/cm^3] smooth ...[Deg] noise ...[J/cm^3] offset ...[J/cm^3]
avg 0.55 max 100
# Type of interfacial energy definition between 0 (LIQUID) and  1 (FCC_L12) ?
# Options:  constant  temp_dependent
temp_dependent
# File for surface energy coefficient between 0 (LIQUID) and  1 (FCC_L12)?
C:\Desktop\Info.txt
# Type of mobility definition between LIQUID and FCC_L12?
# Options:  constant  temp_dependent  dg_dependent
constant
# Kinetic coefficient mu between LIQUID and FCC_L12 [cm**4/(Js)] ?
...
# Is interaction isotropic?
# Optionen: isotropic  anisotropic [harmonic_expansion]
isotropic
...

Interface Energy and Mobility¶

The interface energy, which scales the effect of curvature, can be given either as a constant value (keyword constant) or defined as temperature_dependent. Temperature dependent values can be read from an ASCII file where the first column represents temperature and the second the interfacial energy. Linear interpolation with temperature is performed on the tabular data. Interface energies have to be specified in $J/cm^2$.

One of the most important phase interaction parameter is the interface mobility which defines the interface velocity for a given driving force or curvature. The mobility may be defined as constant, temperature dependent or driving force dependent. In the both latter cases, the user will be asked to provide the name of a text file where the kinetic coefficient (second column) is given in tabulated form as a function of temperature or driving force (first column). Otherwise, just a constant value in $cm^4/Js$ is required. In case of some special models (solute drag, particle pinning), additional model parameters have to be included here (see above). As second optional parameter a minimum mobility can be defined which limits reduction of the mobility by automatic time stepping (minimal time step), anisotropy or special models like particle pinning. If the interface motion is controlled by solute diffusion, it is generally required to correct the effect of the finite interfacial width. This can either be done manually by calibration or automatically by selecting the keyword phase-interaction with the option redistribution-control and the further specification mob_corr in the phase-diagram section. Note that in case of complete diffusion-control, the exact value of the kinetic coefficients does not play any role, but has to be defined high enough not to slow down the interface motion.

Misorientation¶

The crystallographic interaction of two anisotropic phases is split into two factors: first, the misorientation function which considers the grain orientations relative to each other and second, the anisotropy function which considers the relative orientation of the boundary plane (given by the interface normal vector). If the keyword misorientation is set, the reduction of the mobility and the interfacial energy can be specified as function of the misorientation angle or by a constant factor (see Example 4). Note that the misorientation model accounts for the defined crystallographic phase symmetry and always calculates the lowest misorientation angle from all equivalent rotations. The keyword low_angle_limit allows specification of the transition angle (in degree) between low angle and high angle boundaries. Per default, this angle is set to $15°$. The additional option special_orient allows definition of special misorientation relations, such as twin and twist boundaries. It can be used in combination with the keyword aniso_special_orient (instead of anisotropic) to define an anisotropy function which only applies to interface with a special orientation relation, while all other interfaces are treated as isotropic.

The reduction of the interfacial energy as function of the misorientation angle $\theta$ is governed by the implemented Read-Shockley law¹:

where $\gamma_{HAB}$ and $\theta_{HAB}$ are the interfacial energy of high angle grain boundary, specified above, and of the transition angle respectively.

Note that for misorientation angles below $0.5°$, $\gamma(\theta) = 0.1 \gamma_{HAB}$.

Figure 1¶

Reduction of the interfacial energy and mobility as function of the misorientation angle between two cubic grains.

The reduction of the interfacial mobility as function of the misorientation angle between cubic grains is described by the sigmoidal law suggested by Humphreys[@Humphreys1997] (see Figure 1):

where $m_{HAB}$ is the interfacial mobility of a high angle grain boundary. B and n are two model parameters, which can be fitted if experimental data are available: Per default, $B=5$ and $n=4$. In order to avoid small mobilities at very small misorientation angles, the Humphreys law is cut below $1.5°$:

Example 4¶

Reduction of the surface energy and of the mobility of a low angle misorientation between cubic grains, according to the Read-Shockley and Humphreys model respectively.

...
# Phase interaction data
# ==================
#
# 1 (PHASE_1)  /  1 (PHASE_1)
# ---------------------------
# Simulation of interaction between 1 (PHASE_1)  and  1 (PHASE_1) ?
# Options: phase_interaction  no_phase_interaction  identical phases nb
#  [ standard | particle_pinning[_temperature] | solute_drag ]
#   | [ no_junction_force | junction_force ]
phase_interaction
...
# Shall misorientation be considered?
# Options:   misorientation    no_misorientation
#           [low_angle_limit (degrees)] default:15    [special_orient (nb)]
misorientation low_angle_limit 15
# Input of the misorientation coefficients:
# Modification of interfacial energy for low angle boundaries
# Options:  factor   read-shockley
Read-Shockley
# Modification of the mobility for low angle boundaries
# Options:  factor   humphreys [min_reduction + parameters B and N
#                         (default: min_reduction=0. B=5.0 N=4.0)]
humphreys 0.01 5.0 4.0
...

Anisotropy¶

If at least one of the two interacting phases has been defined as not isotropic, the pairwise interfacial properties may be defined as anisotropic, i.e. as function of the local interface normal. Two alternative model versions can be chosen to describe weak non-faceted anisotropy: The default standard model offers predefined anisotropy functions. The specific choice of the anisotropy function depends on the crystal symmetry of the anisotropic phase, previously defined by the user. The user can specify anisotropy coefficients $\delta_{\sigma*}$ and $\delta_{\mu}$ in the range of $\delta \in [0,1]$. A typical value for $\delta_{\mu}$ in cubic metal systems is $0.05$. The more complex harmonic_expansion model allows to formulation of anisotropy functions by combination of spherical harmonics in accordance with the selected crystal symmetry. The implemented harmonic functions are listed in Table 4 of Topics:Phase Interaction. In case of faceted anisotropy, individual sets of anisotropy coefficients $\delta_{\sigma*}$ and $\delta_{\mu}$ can be defined for each facet type. Note that the value 1 here corresponds to isotropic growth and the strength of anisotropy increases with decreasing coefficient, see Topics/Phase Interaction/Faceted anisotropy.

Phase Diagram¶

An enabled phase interaction requires a phase diagram if the phases are not identical.

In case of uncoupled or temperature coupled simulations (phase or temperature in the Flags section), the pair-wise thermodynamic data are described only by the equilibrium temperature and the entropy of fusion (see Example 5). For concentration coupling, however, thermodynamic data are much more complex. The rest of this chapter deals with this case.

Example 5¶

Definition of equilibrium temperature and entropy of fusion

...
# Phase Interactions
# ==================
# 0 (MATRIX)  /  1 (PHASE_1)
# --------------------------
phase_interaction
...
# Equilibrium temperature between MATRIX and PHASE_1 ? [K]
800.00
# Entropy of fusion between MATRIX and PHASE_1 ? [J/(cm**3 K)]
0.50000
...

If Thermo‑Calc™ coupling (via its TQ interface) is NOT to be used, the user has to provide a linear phase diagram description. There are two formats which can be selected either linear or linearTQ. The first corresponds to the normal linear description which consists of the temperature at the reference point, the entropy of fusion and the reference concentrations and slopes for all elements in both phases. The matrix component is omitted in this description.

Example 6¶

Phase diagram input data with no coupling to a thermodynamic database

...
# Which phase diagram is to be used?
# Options:       linear        linearTQ
linear
# Temperature of reference point? [K]
350.0000
# Entropy of fusion between MATRIX and PHASE_1 ? [J/(cm**3 K)]
0.8
# Input of the concentrations at reference points
# Reference point 1: Concentration of component Comp_1 in phase MATRIX ? [at%]
10.
# Reference point 2: Concentration of component Comp_1 in phase PHASE_1 ? [at%]
5.000
# Input of the slopes at reference points
# Slope m = dT/dC at reference point 1, component Comp_1 ? [K/at%]
-4.000000
# Slope m = dT/dC at reference point 2, component Comp_1 ? [K/at%]
-8.0000
...

The second format is identical to the internal extrapolation which MICRESS® uses in case of Thermo-Calc™ coupling and thus exactly corresponds to the output format of initial linearisations in the .log output file. In addition to the normal linearisation parameters (keyword linear), this format includes a driving force at the reference point and an explicit temperature dependence of the reference concentration. Moreover, the transformation entropy is replaced by the temperature derivatives of the driving force with fixed composition in phase 1 or phase 2 (dSf+, dSf-).The advantage of the linearTQ input format is that linearisation parameter obtained in Thermo-Calc™ coupled simulations (found in the .log or TabLin output file) can directly be used as linear phase diagram description in a subsequent simulation. Another important application is the use for eutectic systems with demixing solid-solid phase interactions: The slopes of the phase diagram lines for the interaction of the two solid eutectic phases typically have opposite signs. This reflects their immiscibility which is enhanced with lowering temperature. This demixing effect can be compensated for by using the linearTQ phase diagram description with different signs for, dSf+ and dSf.

Note: The .log output of the initial linearisation parameters for a phase interaction additionally contains information on the transformation enthalpy which is not part of the phase diagram description. For sake of simplicity, during input of the linearisation parameters in the linearTQ format, a dummy variable is read for the enthalpy. Thus, the output block in the .log file can be directly transferred to the driving file by copy and paste without removing the transformation enthalpy. If data are taken from other sources (e.g. the .TabLin file), a dummy value has to be included in the corresponding place.

A phase diagram description in the linearTQ format can NOT be exactly translated into a normal linear phase diagram description. If in a linearTQ phase diagram description dSf+ and dSf- have opposite sign, the slopes of ALL elements in the two phases must also have opposite signs.

Then, the thermodynamic description of the phase interaction has to be given: for each interaction, data can be taken from the specified database, from a linearised description (linear) or according to the linearTQ format (see above).

In case of database, an optional keyword in the same line with database can be used to specify a more global or local phase diagram description: In case of local (default), each interface cell which exists for the given phase interaction bears its own local linearized description obtained and updated from the database according to the specified updating conditions. In the global cases, only one description is obtained and updated for each corresponding reference region. The global option are generally less accurate, especially for spatially extended interfaces. In turn, it is computationally much more efficient and reduces numerical noise. The global options are:

• global: Each interface between two specific grains will get its own data set, which strongly reduced the effort of updating. In order to, this spatial scope can be narrowed by

• globalF: Similar to **global, but increased locality and exactness by narrowing the spatial scope to each interconnected segment (fragment) of the interface region for a specific grain pair.

• globalG: Uses one set of linearisation data for all interfaces between two phases, so that it comes close to a linearized phase diagram where the data are automatically calculated and updated based on the average interface composition and temperature. This mode can be regarded as a fully global description.

• globalGF: Defines common linearisation data for interconnected segments (fragments) of all phase interfaces which thus may span several grain interfaces belonging to the same phase pair.

In terms of locality, the following order can be defined:

local > globalF > global > globalGF > globalG

In multi-phase regions (triple junctions), this procedure may lead to local inconsistencies between the different phase interfaces, because they use linearisation data sets which may be based on concentration averages on different spatial regions (as the integration area for each interface also includes dual interface regions). If necessary, this problem can be avoided using the option database_consistent (see Database) which separates the triple point areas from the dual interface regions both in terms of the range of averaging of interface compositions as in terms of the range of validity of the linearisation data.

A further additional keyword (second or third in the row, depending on whether the global/local keyword is used) allows controlling the composition of which phase is used as start value for the iteration of the quasi-equilibrium. The options are start_value_1 and start_value_2, which choose the first or second phase of the phase pair for delivering the start compositions. Default is the use of the phase with higher fraction if no stoichiometric conditions are set. This option has to be considered as an absolute expert option.

As hidden expert option, the keyword init_fraction must appear in an extra line following the thermodynamic description of the phase-interaction. This option allows specifying the phase fractions which are applied when initializing the thermodynamic interaction for the given phase pair. This initialization performed either at the beginning of the simulation or when nucleation for a phase which would establish this interaction is checked first time. The initial set of linearization data is written in the .log output. Given the fact that this initialization is crucial for a successfully proceeding simulation, the explicit specification of the phase fractions can be important. Without this option, a fraction of phMin is used for nucleating phases or 0.5 for phases already existing in the initial microstructure. After the keyword init_fraction, the fraction of the second phase (e.g. phase 2 in the interaction 0/2) can be specified.

Also, for each phase interaction, the relinearisation mode must be defined. A maximum temperature deviation for updating the thermodynamic description can be specified using the option automatic. Setting a high value speeds up the calculation by limiting the number of calls to the TQ-library. However, this is achieved at the expense of precision. Alternatively, an updating time interval can be specified for this phase interaction using the keyword manual or reading time-dependent from file using from_file. This definition of a relinearisation mode per interface can be helpful if one phase interaction, e.g. for a specific intermetallic phase, needs more care.

Example 7¶

Specifying relinearisation option for an interface

...
#
# Input of the phase diagram of phase 0 and phase 1:
# ----------------------------------------------------------------------------
# Which phase diagram is to be used?
# Options:  database [local|global[F]|globalG[F]][start_value_{1|2}]  linear  linearTQ
database local
# Relinearisation mode for interface 0 / 1
# Options:  automatic   manual   from_file   none
manual 0.001
#
# Input of the phase diagram of phase 0 and phase 2:
# ----------------------------------------------------------------------------
# Which phase diagram is to be used?
# Options:  database [local|global[F]|globalG[F]][start_value_{1|2}]  linear  linearTQ
database globalF start_value_2
# Fraction for initialization of phase 2
init_fraction 0.01
# Relinearisation mode for interface 0 / 2
# Options:  automatic   manual   from_file   none
from_file
# File name for the time-dependent relinearisation interval
relinInterval0_1
...