# Phase¶

## Anisotropic phase definition and crystallographic symmetry¶

The growth of a solid phase may strongly be affected by its underlying atomic lattice structure. Especially the interfacial mobility and the interfacial energy are commonly functions of orientation. In order to consider crystallographic aspects of a solid phase it first has to be defined as non-isotropic, i.e. as either anisotropic or faceted (faceted_a, faceted_b, faceted_c/faceted, antifaceted). This is the basic requirement to define crystallographic orientations and anistropic phase-interactions.

Moreover, a crystallographic symmetry can be defined for this phase by selecting one of the following options: cubic, hexagonal, tetragonal or orthorhombic as illustrated in Table 1. The most common type of anisotropy in metals is the cubic anisotropy applicable to fcc and bcc phases and also to diamond structures. The hexagonal symmetry may be used e.g. for simulation of Mg or Zn alloys. Note that in MICRESS®, the hexagonal symmetry is defined with the basal direction $c$ parallel to $y$ such that the six-fold basal plane corresponds to the $x$-$z$ plane which is used as standard in 2-D simulations). Tetragonal symmetry and orthorhombic symmetry are both distorted cubic structures defined with the $c$ axis parallel to the $z$ axis of the grain coordinate system. If none of these symmetry types are applicable simply select none. The definition of the crystallographic symmetry is required for the following tasks:

• to reduce grain orientations to equivalent orientations in the fundamental zone,
• to calculate the disorientation between neighboring grains, i.e. the equivalent misorientation with smallest angle,
• to determine the anisotropy functions for the interfacial energy and the interfacial mobility in case of weak anisotropic phase interactions.
###### Table 1¶

Crystallographic symmetries implemented in MICRESS®

cubic tetragonal orthorhombic hexagonal
$a \ne c$ $a \ne b \ne c$

## Faceted growth¶

In addition to the standard weak anisotropy model, MICRESS® offers special anisotropy models for faceted growth. These models can be selected by defining a solid phase as either faceted_a, faceted_b or antifaceted.and by specifying individual normal vectors of the facets in the local coordinate system. Additionally, a parameter kappa, which defines the sharpness of the anisotropy function has to be specified. Facets with equal properties can be grouped into a facet type. To treat phases with more than one facet type an amended model version faceted_b is recommended which better selects the facet type with lower energy. An alternative model to simulate plate-like growth is given by the antifaceted model which in contrast to the faceted model describes fastest growth in direction of the facet vectors.

In the phase interaction input, a static and a kinetic anisotropy coefficient have to be specified for each facet type, if one of the two interacting phases is faceted. The anisotropy of the interfacial energy is implicitly taken into account by the anisotropy of the interface stiffness which allows a direct matching to the modified Gibbs-Thomson equation for anisotropic interfaces, which reads: $\nu = \mu \left( \Delta G - \sigma^* \kappa \right)$ with $\nu$ being the velocity in normal direction, $\mu$ the kinetic coefficient, $\Delta G$ the driving force, $\kappa$ the curvature and $\sigma^*$ the interface stiffness.

## Definition of crystallographic orientations¶

Crystallographic orientations have to be defined for each grain of an anisotropic. A grain orientation is generally defined as relative rotation between the local grain coordinate-system and the global reference coordinate-system. In 2-D simulations, orientations can simply be described by a single angle, but the description of orientations in 3-D is complex and no common standard exists. Therefore, MICRESS® offers various ways to formulate 3-D orientations.

### Euler angles¶

According to the Euler theorem, any two independent orthonormal coordinate frames can be related by a sequence of three rotations about coordinate axes, where no two successive rotations may be about the same axis. Be careful when importing Euler angles, because there are $12$ different representations! MICRESS® uses the $ZXZ'$ convention, i.e. the Euler angles $\phi$, $\theta$, $\psi$.

• Rotation 1 ($\phi$): (anticlockwise) about the $[001]$ axis; $Z1$.
• Rotation 2 ($\theta$): (anticlockwise) about the (rotated) $[100]$ axis; $X$.
• Rotation 3 ($\psi$): (anticlockwise) about the rotated $[001]$ axis; $Z2$.

### Angle/Axis¶

According to the definition, any two independent orthonormal coordinate frames can be related by a single rotation with an angle $\phi$ about a specific axis $(a_z, a_y, a_x)$.

### Miller indices¶

They are the reciprocals of the fractional intercepts which between the planes and the crystallographic $x$, $y$, $z$ axes of the three non-parallel edges of the cubic unit cell

Note that if a crystallographic symmetry has been defined, MICRESS® may replace a user defined orientation by an equivalent orientation in the fundamental zone. In case of crystal symmetry, there exist mathematically distinct orientations which are physically indistinguishable. In 3-D, there exist $24$ equivalent orientations for cubic symmetry, $12$ for hexagonal symmetry, $8$ for tetragonal and $4$ equivalent orientations for orthorhombic symmetry. For a unique representation of orientations, the full orientation space can be reduced to a fundamental zone. In MICRESS®, the fundament orientation is defined as the crystallographically equivalent orientation with smallest rotation angle $\phi$ and $a_z > a_y > a_x$, with $(a_z, a_y, a_x)$ being the components of the rotation axis.

###### Table 2¶

The orientation spectrum is reduced to the fundamental zone

Symmetry none cubic
Fundamental zone $0°-360°$ $0°-90°$
Orientation $\alpha$ $30°$ $30°$
Orientation $\beta$ $80°$ $80°$
Orientation $\gamma$ $120°$ $30°$
Misorientation $\alpha \beta$ $30°/80° \rightarrow 50°$ $30°/80° \rightarrow 40°$
Misorientation $\alpha \gamma$ $30°/120° \rightarrow 90°$ $30°/30° \rightarrow 0°$
Misorientation $\beta \gamma$ $80°/120° \rightarrow 40°$ $80°/30° \rightarrow 40°$

## Categorization¶

The keyword categorize is used in the MICRESS® driving file with two meanings:

merging several grains into one

concentrating continuous orientations into a few distinct orientations

The main reason why categorization has been introduced in MICRESS® is the performance increase which can be achieved by reducing the number of effective grains. As soon as the number of grains exceeds $\sim 1000$, the performance of MICRESS® decreases noticeably due to the administrative effort of the interface list structure (Figure 1). At the present stage, it gets very hard to get beyond $\sim 10000$ grains due to the high memory usage. In many simulation cases, the number of grains or precipitates can get very high, while the individuality of each single grain is of minor importance, as e.g. in the case of precipitation of a secondary phase. Then, a huge performance gain can be obtained if several grains can be categorized in the sense that they are treated as one single grain. One prerequisite is that they have identical properties, i.e. the phase, orientation, recrystallisation energy etc. Furthermore, if they have been nucleated as small grains, they need to have already reached full size, otherwise they still have their individual analytical or stabilized curvature and hence cannot be categorized.

###### Figure 1¶

Correlation between categorization and CPU time

The categorization option is controlled in the following way:

• Specifying categorization in the phase data allows switching on/off this option for the given phase. All grains of this phase will be checked during runtime with respect to the possibility of assigning them a common grain number.
• If categorization is activated for a given phase which has been defined as anisotropic, one is prompted to switch on/off categorisation for each grain or seed type within this phase in both the initial grain setting and the nucleation input. This categorize flag allows the user to arrange the random nuclei orientations in categories, e.g. $10°$, $20°$, $30°$, etc. The number after the categorize keyword is the number of categories, if no number is given, the default value of $36$ (for $36$ categories with an interval of $10$ degrees in 2-D) is used. During simulation, grains of the same orientation category can be categorized` to the same grain number. In the case of isotropic phases, the number of categories is meaningless, because all grains can be treated as if they had the same orientation.

The effects of categorization are the following:

• The grains lose their identity, they appear with the same grain number in the korn output.
• The grains will not form an interface between each other if they touch afterwards due to further growth.
• Categorization will not apply to nuclei that already touch each other, that means that no existing interfaces will be removed.
• The grain statistics output (mean radius etc.) will be meaningless or at least difficult to interpret.
• If global TC coupling is used (MICRESS® 5.404 and above), the approximation of one set of linearisation parameters per grain may be worse if this grain exists in several places.
• The orientation distribution may get less smooth if one throws orientations into categories.
• The results should not be affected apart from the choice of discrete grain orientations, the special case of global TC-coupling and the grain-related statistic outputs.