In the phase-field dislocation model, each phase corresponds to a "state of slip". Individual dislocations are thus phase boundaries whose configurations are determined by minimizing the total system energy, which in its most general form contains the elastic energy, the lattice energy, the gradient energy, and the external energy.
There are many phase-field dislocation models, e.g., the phase-field microelasticity model and the microscopic phase-field model. Below you will find a brief, historical overview of the phase-field dislocation dynamics (PFDD) model. In the current context, the name "PFDD" refers to a specific model first developed by Marisol Koslowski and later advanced by her students, postdocs, and colleagues. Note that (i) early PFDD papers did not use the name "PFDD", but they were retrospectively considered to use the PFDD model, and (ii) some papers used the name "PFDD" for their models, but they were authored/co-authored by other groups, e.g., Pi et al. Papers in the second category are not included in this overview.
Here is the very first PFDD paper:
in which (i) the isotropic elasticity was assumed, (ii) the elastic tensor was assumed the same everywhere, (iii) the lattice energy was based on a simple 1D function, (iv) the lattice energy was assumed the same everywhere, (v) the gradient energy was omitted, (vi) the external energy involved the stress-controlled loading only, (vii) the model was for FCC crystals only, and (viii) no numerical grid was used. These eight issues were addressed in subsequent years.
As mentioned, the first PFDD paper did not use the name PFDD. The first paper that used the name PFDD was
Anisotropic elasticity was first implemented into PFDD in 2019:
Later that year, details in the formulation of anisotropic elasticity were presented:
Elastic heterogeneity was introduced to PFDD, for a void in 2013:
and for bi-phase materials in 2016:
Note that the two papers above did not address issues (i), (iii), or (v). In 2022, issues (i), (ii), and (iii) were addressed in the same model:
Also in 2022, the PFDD model was extended to multi-phase materials:
in which the issues (i), (iii), and (v) were also addressed.
In 2011, the lattice energy formulation was advanced such that it was based on a 2D function that approximates the generalized stacking fault energy (GSFE) surface in FCC crystals:
Then in 2019, the lattice energy was advanced again such that it was based on the GSFE surface that was entirely informed by atomic-level calculations without using any function:
In the paper above, effects of the two GSFE surfaces were studied and it was recommended that the one not using a function is more desirable.
The GSFE surface is used only for dissociated dislocations, e.g., those in FCC crystals and on basal planes in HCP crystals. For non-dissociated dislocations, e.g., those in BCC crystals and on some planes in HCP crystals, GSFE curves should be used. In early PFDD models, the GSFE curve was approximated by a 1D function. That changed in 2020:
in which the GSFE curve was informed entirely by atomic-level calculations without using any function.
In materials such as multi-principal element alloys, the GSFE is not spatially uniform. As a result, the lattice energy cannot be the same everywhere. The feature of spatially varying lattice energy was added to PFDD in 2019:
In 2022, a method was developed to generate lattice structures with given short-range order:
The spatial variation in the lattice energy inevitably generated a large amount of data, leading to difficulties in relating the inputs and outputs analytically yet providing an opportunity to use machine learning:
In 2020, the feature of character angle-dependent lattice energy was added to PFDD, for BCC metals:
The rational was that the Peierls stress differs greatly between edge and screw dislocations in BCC metals.
In 2021, the lattice energy was extended to be a function of the applied shear stress and the angle between the maximum resolved shear stress plane and the glide plane:
The purpose was to capture the non-Schmid effects in BCC metals.
In 2019, the gradient energy was added to the total energy in the PFDD model:
Originally the gradient energy was intended only for dissociated dislocations, e.g., those in FCC crystals and on basal planes in HCP crystals. Later, it was found that inlcuding the gradient energy may result in a match between the PFDD-based dislocation core width and the atomistic-based one in BCC metals:
Note that including the gradient energy in the total energy is not necessarily desirable in all FCC crystals. In some FCC crystals, it may be better not to include it. An analysis of edge and screw dislocations in eight FCC crystals was conducted in:
In any case, if one decided to include the gradient energy, the formulation would ask for the values of two independent gradient energy coefficients per slip plane in an FCC crystal. In Xu et al. MSMSE (2019), the two coefficients were different. However, they may have the same value. The uniform coefficient was first used in the paper below, shortly after the preceding MSMSE paper was published:
The two types of gradient energy coefficients, i.e., non-uniform and uniform, were compared later in:
which recommended that the non-uniform coefficients be used whenever possible.
In 2015, the strain-controlled loading was implemented into PFDD:
Another line of advancement is to extend PFDD from FCC crystals to other types of crystals. Three advancements were made in 2020. First, PFDD was extended to {110} slips in BCC crystals:
Soon after, PFDD was extended to {112} and {123} slips in BCC crystals:
Then in 2022, PFDD was extended to {134} slips in BCC crystals:
PFDD was also extended to HCP crystals, for slips on basal, prismatic-I, and pyramidal-II planes:
As of March 2023, the PFDD model had not been used to study slips on pyramidal-I planes in HCP crystals, but doing so should be straightforward.
In 2004, 2D orthogonal numerical grids were introduced to PFDD:
Then in 2011, 3D orthogonal numerical grids were used:
which studied a 3D problem. In the above paper (and many others that were not referenced here), slips were not confined to pre-defined slip planes in a 3D grid.
The first paper in which all slips were confined to pre-defined slip planes in PFDD was published in 2019:
Later, effects of the confinement were studied in:
which recommended that the confinement be used whenever possible.
One may wonder: wouldn't the confinement reduce a 3D model to a 2D model? This is indeed the case for a 2D problem, i.e., when only one slip plane is involved in the system. However, the 3D model, even confined, can be used for a 3D problem, i.e., when multiple slip planes, either parallel or non-parallel, are involved. Clearly, the 3D model is advantageous to the 2D model which cannot simulate a 3D problem.
In 2021, 3D non-orthogonal numerical grids were developed for FCC and BCC crystals:
To sum up, it is always a good idea to use feature (i) in any PFDD work. Feature (iii) is highly recommended unless in the case of intensive high-throughput computing. Whether other features are used depends on the specific slip systems, material systems, and/or loading conditions.
The PFDD code was parallelized in 2011:
and was accelerated by GPU in 2018:
For a more detailed report of the progress in the PFDD method up to March 2022, please read: