Mellenthin, Jesper (2007) Phase field modelling of polycritaline materials. PhD thesis PMC, PMC, EP/X p.150.

Alternative Locations: http://www.imprimerie.polytechnique.fr/Theses/Files/Mellenthin.pdf

Abstract

The phase-field method has become in recent years the method
of choice to model microstructural pattern formation during
solidification. For monocrystals, quantitative agreement
with experiments and analytical solutions has been obtained.
The modeling of polycrystals, which consist of many grains
of the same thermodynamic phase, but different orientations
of the crystalline lattice, is far less advanced. Two types
of models have been proposed: multi-phase-field models use
a separate phase field for each grain, and orientation-field
models use a small number of fields, but have non-analytical
terms in their free energy functional.

This work examines various aspects of phase-field modeling
of polycrystals and is divided in three parts. In the first,
a new possibility of describing the local orientation is
explored, using a tensorial order parameter which represents
automatically the local symmetry of the system. This approach is
tested by developing a phase-field model for the nematic-isotropic
phase transition in liquid crystals. The model is applied to
simulate the directional ''solidification'' of a liquid crystal.
The effect of the coupling between nematic orientation and the
interface shape is investigated. The simulation results for the stability
of a planar interface agree well with a generalized stability
analysis, which takes into account a new anchoring condition
at the interface: the nematic orientation at the interface is
the result of the interplay between bulk deformation and interface
anisotropy. The shape and stability of well-developed cells is
also influenced by this effect.
Numerically, the use of a tensorial
order parameter simplifies the treatment of the symmetries in the
system significantly, while the equations of motions become considerably
more complicated.

In the second part, grain boundaries are investigated on a smaller
length scale, using a phase field crystal model, where elastic
properties and dislocations appear naturally. With this model,
the local order in interfaces is examined and the stability of
liquid films between two solid grains is studied below the
melting point. This situation can be described by an interaction
potential between the two solid-liquid interfaces, which is
extracted numerically. The results are compared with a
phenomenological model which is found to hold for high-angle
grain boundaries, where the dislocations overlap. For low-angle
grain boundaries, premelting around dislocation as well as
a symmetry breaking (dislocations form pairs) is observed. As a
result, the interaction potential becomes nonmonotonous, and
consists of a long-range attraction and a short-range repulsion.

In the third part, a new phase-field model is developed using an
angle variable to describe the crystalline orientation. Contrary
to the already existing models, the free energy is constructed
without a term proportional to the modulus of the gradient of
the orientation field. Instead, the standard squared gradient
is used, but it is coupled to the phase field with a singular
coupling function. Various benchmark simulations are carried
out to test the model. It is found that it presents several
artifacts such as spurious grain rotation and interface motion;
however, these effects are extremely small, such that the
model yields satisfactory results unless the undercooling is
very small. Finally, the observed problems are analyzed and
ways of obtaining a better description of the dynamics of the
angle field are discussed.

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