A typical way to address the sub-µ gamma' microstructures in Ni-base superalloys would be to model solidification and heat treatments on the scale of the dendrites or grains. Then, picking the local composition in one point of interest, one would set up a simulation on a smaller scale and simulate phenomenae related to gamma' growth and morphology (see here).

However, there are cases where you need to take the effect of gamma' precipitates into account while you are still on the scale of the dendrites or dendrite arms. Then, the difficulty is that for reasons of the local thermodynamic consistency gamma' cannot be neglected, but the precipitates also cannot be displayed properly at that scale.

From a principle point of view, there could be 3 approaches to the problem:

**a) Effective Phase Approach**

If it were possible to describe 2-phase mixtures of gamma and gamma' as an effective phase, one could just use this phase description instead of gamma in the phase-field simulation. However, theoretical analysis shows that this is possible only for binary systems or those which can be approximated by a pseudo-binary system (see e.g.

*J. Rudnizki, B. Böttger, U. Prahl, and W. Bleck, "Phase-Field Modeling of Austenite Formation from a Ferrite plus Pearlite Microstructure during Annealing of Cold-Rolled Dual-Phase Steel", METALLURGICAL AND MATERIALS TRANSACTIONS A 42A (2011) 2516*).

**b) Diffuse Phase Approach**

If it is possible to represent the two-phase mixture as a diffuse two-phase region in MICRESS, the thermodynamic consistency problem of an effective phase can be overcome (because there is an extra degree of freedom given by the local phase fractions). This approach can be used in MICRESS for eutectic or eutectoid reactions where a eutectic front, which consists of the two phases, is moving ("unresolved" model, see e.g. here). However, this model is not useful for precipitiation of one solid phase in another one.

**c) "Big Particle" Approach**

What remains for multicomponent systems (after rejecting a and b) is to accept that - if morphology anyway cannot be correctly represented - the size of the particles also is not relevant. This means, if we accept that the particles, which can be represented at the given resolution, are by a factor of maybe 10 too big, but still cover the whole areas such as to establish the correct local equilibria everywhere, then we can live with that (see e.g.

*B. Böttger, M. Apel, B.Laux, S. Piegert, "Detached Melt Nucleation during Diffusion Brazing of a Technical Ni-based Superalloy: A Phase-Field Study", 2015 IOP Conf. Ser.: Mater. Sci. Eng. 84 012031, http://dx.doi.org/10.1088/1757-899X/84/1/012031*).

While the "Big Particle" approach sounds quite simple at first glance, there are some important difficulties linked to it:

- the particles must be quite regulary distributed in order to fill the whole space (i.e. practically all space should be covered by interface between gamma and gamma'. To obtain that, using "bulk restrictive" and a suitable nucleation distance is essential.

- the number of particles is very high. Thus, it is not possible that each one represents an own grain number. Instead, particles must be "categorized" or already set as several "new sets" in conjunction with the nucleation option "add_to_grain".

- there should be no coalescence which creates few big particles instead of many small ones. Adjacent particls should not have the same grain number, and the gamma'-gamma' interface energy has to be chosen big enough.

- in any case, a lot of interface regions are created which reduce performance. Timestepping and relinearisation schemes need to be optimized to reduce this effect.

Bernd