Using an Explicit Nucleation Model in PRISIMS-PF to Predict Precipate Microstrucutures

Center for PRedictive Integrated
Structural Materials Science
Using an Explicit Nucleation Model
in PRISMS-PF to Predict Precipitate
Microstructures
Stephen DeWitt
Dept. of Materials Science and Engineering
University of Michigan
Phase Field Workshop VII

Part 1:
Overview of PRISMS-PF and new features
Part 2:
Discussion on explicit nucleation models for phase field
simulations and implementation in PRISMS-PF
Part 3:
2D simulations of nucleation, growth, and coarsening in
an Mg-Nd alloy

Lead Developer:
Stephen DeWitt1
PRISMS-PFAn Open-Source Phase Field Modeling Framework
Associated PRISMS Faculty:
Katsuyo Thornton1
Primary Contributors:
Shiva Rudraraju2*, Larry Aagesen1**,
David Montiel1, and Beck Andrews1
1. Dept. of Materials Science and Engineering, University of
Michigan
2. Dept. of Mechanical Engineering, University of Michigan
* (Now at the University of Wisconsin) ** (Now at Idaho National Laboratory)

What’s hard about writing a phase
field code?
1. Wide diversity of models and coupled physics
makes code reuse difficult
No real “typical”
governing equations
Large variety of
formulations and
terms

What’s hard about writing a phase
field code?
2. Simulating large, physically representative
systems is computationally intensive
Simulations often take days on
10s-100s of cores
Simulations are often done in 2D
for tractability
Physical fidelity requires strong
numerical performance
nersc.gov

Four Principles Guiding PRISMS-PF
Development
1. Its computational performance, including
parallel scalability, should meet or exceed that
of typical phase field codes
2. It should accommodate a wide variety of phase
field models and applications
3. The interface for creating or modifying
governing equations should be simple, quick,
and separate from the numerics
4. It should be open source with a permissive
license so it is available to everyone and
advances can be shared by the community

User-Friendly:
Simple interface to solve
an arbitrary number of
coupled PDEs
Detailed online user guide
24 applications to get you
started
Simple Docker-based
installation
High-Performance:
Ideal scaling for >1,000
processors
Improved performance
over finite difference
(approx. 10x w/o adaptive
meshing, 100+ x w/
adaptive meshing)
An Open Source, Finite Element,
General Purpose Phase-Field Platform
(github.com/prisms-center/phaseField)
Advanced Capabilities:
Matrix-free finite element
approach
High-order elements
Hybrid parallelization:
MPI/Threads/Vectorization
Adaptive meshing
Explicit nucleus placement
Grain-remapping

PRISMS-PF Performance:
Matrix-Free Approach
• Leverages the matrix-free finite element capabilities of the deal.II library
• Cell-based strategy eliminates the need for a sparse matrix
– Operator instead applied element-by-element
– Unit cell operations same for all cells permitting vectorization
• Sum factorization
– Restructuring of the operation as the tensor product of 1D operations
• Gauss-Lobatto elements
– Uses the same points for support and quadrature
– Diagonal “mass matrix” enables explicit time stepping without “mass
lumping” or solving a system of equations
– Well-conditioned for high order elements
• Together these:
• Reduce memory bandwidth which often limits sparse matrix operations
• May reduce floating point operations (large reductions at high order)
Kronbichler and Kormann, Computers & Fluids, 63 (2012)

PRISMS-PF Performance:
Adaptive Meshing
Adaptive octree meshes
using the p4est library
Depending on the geometry
can massively decrease the
size of a calculation

Performance vs. Finite Difference
• PRISMS-PF and custom finite
difference code
– Written in Fortran with MPI
parallelization
– Second order central differencing
– Explicit time stepping (forward
Euler)
• 2 growing particles in 3D
– Coupled Cahn-Hilliard/Allen-Cahn
– Closely related to a number of
problems of physical interest
(precipitation, solidification, etc.)

Speed-up vs. Finite Difference at 4
Error Levels

Performance on PFHub BM3
• Simulation with cubic elements and adaptive meshing
• h = 1/384, Δt = 0.005
• Result is well converged
– Decreasing h by 1/2 and Δt by 1/10, leads to 1% change in the velocity

Comparison to Other Uploads
Framework Tip Velocity
at t = 1500
Core-h Required Memory
(GB)
PRISMS-PF 8.6×10-4 0.30 0.29
MOOSE (#2) 8.1×10-4 0.51 36.3
MOOSE (#1) 5.5×10-4 241 6.7
FreeFem++ 7.4×10-4 100 0.098
FiPy 1.2×10-3 74 1.9
PRISMS-PF

New Features:
Grain Remapping
• Each grain is identified
using a recursive flood fill
• Simplified representation
of each grain is stored
– Currently a circle/sphere
– A persistent grain ID is saved
for each grain
• Modified greedy coloring
algorithm to determine the
new home for a grain
marked for transfer
Order Parameter 1 Order Parameter 2
Too close
Marked for transfer
Permann, Tonks, Fromm, Gaston, Comp. Mater. Sci. (2016)

New Features:
Grain Remapping
• Each grain is identified
using a recursive flood fill
• Simplified representation
of each grain is stored
– Currently a circle/sphere
– A persistent grain ID is saved
for each grain
• Modified greedy coloring
algorithm to determine the
new home for a grain
marked for transfer Initial microstructure
imported from
Permann, Tonks, Fromm, Gaston, Comp. Mater. Sci. (2016)

New Features:
Materials Commons Integration
Command line tool automatically parses the PRISMS-PF app files and
uploads the metadata and simulation results to Materials Commons

New Features:
Website and User Manual
PRISMS-PF has a brand new website, including
an online manual replacing the PDF file
https://prisms-center.github.io/phaseField/

Other New(ish) Features:
• Explicit nucleus placement (more on this later)
• Hybrid Newton/Picard nonlinear solver
• Checkpoint/restart
• Postprocessing
• Docker-based installation
• Automatic integration testing with Travis CI

Summary
• Flexible, and easy-to-use framework for phase field
modeling
• Highly competitive performance
– 1-2 orders of magnitude faster than finite difference
– Fastest published time for BM3
• Advanced features such as nucleation and grain remapping
• Talk to me if you want to know more
https://prisms-center.github.io/phaseField/

Noise-Based Nucleation vs Explicit
Nucleation
Noised-based nucleation
• Noise is added to the solution,
which lead to super-critical
nuclei forming
• Pro: Mimics reality
• Pro: Easily accounts for all
coupled physics and
heterogeneities
• Con: Intractable for physical
noise scales
• Con: Artificially large noise can
degrade the solution
• Con: No easy connection
between artificially large noise
and nucleation rates
Explicit nucleation
• Super-critical nuclei are
algorithmically placed in the
domain
• Pro: Flexible, based on choice
of nucleation model
• Pro: Works on a mesoscale
length/time scale
• Pro: General nucleation
behavior is know a priori
• Con: Need a functional form
for the nucleation rate and
parameters

Explicit Nucleation: General Approach
1. Pick a form for the nucleation rate, j(c,η,t,…)
2. During a phase field simulation, periodically sample
subvolumes
3. In each subvolume, compare a random number
[0,1] to the following probability (from a Poisson
distribution)
4. If the random number is greater than p, add a
nucleus in the subvolume
Simmons, Chen, Wang, Scripta Materialia (2000)

How to Choose the Nucleation Rate?
• Need a model for the nucleation rate
• One common choice is classical nucleation
theory
– Although something else could be used

Classical Nucleation Theory (I)
• Continuum model usually for a spherical
nucleus
• Assume that the total energy change is broken
into volumetric energy and interfacial energy
• Assume the volumetric term can be split into
chemical and elastic terms
Balluffi, Allen, Carter, Kinetics of Materials (2005)

Classical Nucleation Theory (II)
• The volumetric term is
negative (favors nucleation,
the interfacial term is positive
(disfavors nucleation)
• The interfacial term dominates
for small nuclei, the
volumetric term dominates for
large nuclei
• Leads to a critical nucleus size Aaronson, Enomoto, and Lee, Mechanisms of
Diffusional Phase Transformations in Metals and
Alloys (2010)

Classical Nucleation Theory (III)
• Critical size is at:
• Which can be used to calculate the critical
energy and radius

Classical Nucleation Theory (IV)
• Allow clusters to grow or shrink by adding or
removing atoms (thermally activated)
• Assume an equilibrium distribution of nuclei
(with a constraint on the max size)
• Can calculate the rate of formation of stable
nuclei
Zeldovich factor
(Accounts for super-critical nuclei
that become sub-critical)
Density of nucleation
sites
Attachment rate

Classical Nucleation Theory (V)
• Assumes that the distribution of clusters is at
equilibrium
• It takes time for the clusters to grow
• Add an incubation time
Incubation time

Some Problems
1. How do we get the energies?
2. What if the clusters aren’t spherical?
3. What about heterogeneous nucleation?
These problems are magnified by the
exponential, small uncertainties can blow up

One Solution:
Throw Out (Much of) the Detail
Following the general approach from (Simmons, 2000)
Exploiting that, for dilute
solutions without strain energy
Where ρ1, ρ2, and τ can
be treated a free
parameters

How to Place the Nucleus?
1. Modify the composition field (Simmons, 2000)
2. Modify the nonconserved order parameter field
(Jokisaari, 2016)
Jokisaari, Permann, Thornton, Comp. Mater. Sci. (2016)
Move solute from the matrix into a cluster, leaving a depletion zone
Need to be careful not to introduce discontinuities, which cause instability
Add a nucleus to the order parameter
Don’t need to worry about conservation
Allow the composition field to naturally evolve
May want to “hold” the nucleus, as the composition field evolves

Implementation of Nucleation in
PRISMS-PF
• Both explicit nucleation and (naïve) noise-based nucleation
supported
– Our applications focus on order-parameter only explicit
nucleation
– Explicit nucleation requires some work in the background
• Available in any app
• Modular approach, works with:
– Grain remapping
– Multiple phases
– Couple time-independent PDEs (elasticity, electrostatics, etc.)
– Adaptive meshing
– Any other PRISMS-PF feature

Explicit Nucleation in PRISMS-PF
• User-defined expression for the nucleation
rate
• Nuclei are ellipses/ellipsoids with arbitrary
rotation
• Nucleus size/rotation and rate can be set
independently for different phases

Explicit Nucleation Procedure
At an user-specified frequency:
For the mesh held by each processor:
For every element:
Determine average variable values (c, n, etc.)
For each nucleating order parameter:
Calculate local nucleation probability
Compare with random number
If p > random number:
Check if the new nucleus overlaps with existing ones
Add it to the local list of new nuclei
Communicate local lists to build a global list
Eliminate overlapping nuclei
Refine the mesh near the new nuclei
Place the nuclei via a source term in the governing equations

Example: Grain Boundary Nucleation
David Montiel (UM)

Part 1:
Overview of PRISMS-PF and new features
Part 2:
Discussion on explicit nucleation models for phase field
simulations and implementation in PRISMS-PF
Part 3:
2D simulations of nucleation, growth, and coarsening
in an Mg-Nd alloy

Background: Mg-RE Precipitates
• Mg alloys of interest for
lightweighting structural
components
• Mg-RE alloys have
precipitates with habit
planes perpendicular to
the basal plane
• Effectively block
dislocation motion
Plots courtesy E. Deda, A. Githens (UM)
Schematics: J.F. Nie, Met. Trans. A, 43A,
3891 (2012)

Precipitation Sequence in Mg-Nd
Early work in the PRISMS Center examined the precipitation sequence:
[1] Natarajan, et al., Acta Materialia, 108 (2016)
[2] Solomon, et al., Scripta Materialia, 128 (2017)
[3] Saito and Hiraga, Materials Transactions, 52 (2011)
[1] [1] [3]
Primary phases that lead to precipitation strengthening

Previous Work: β’’’ Precipitates
Phase field modeling used to examine morphology and the relationship
between the misfit strain and the composition
Inputs from first principles/stat. mech.
Predicted equilibrium shapes
Quantitative comparisons to experiments
Identification of interactions that change
the morphology
DeWitt, et al., Acta Materialia, 136 (2017)

β1 Precipitates in Mg-Nd Alloys
• Now, we are building off of this success to
examine β1 precipitates
• Simulations of entire β1 microstructures during
aging
– β’’’ work showed the importance of interactions on
morphology
– Synthetic microstructures can be passed to
dislocation dynamics to predict strengthening
– Aging simulations get us closer to the capability to
optimize aging conditions for the desired
microstructure

Long-Term Vision
Fourier Space DFT
(VASP)
Statistical Mechanics
(CASM)
Phase Field
(PRISMS-PF)
Experiments
(HAADF-STEM,
APT)
Number
Density,
Volume
Fraction,
Dimensions
Finite Temp.
Energies
0K Energies
Elastic Constants, Transformation Strains
Enumerate
Configurations
Inform problem space
Dislocation Dynamics
(LLNL-ParaDIS)
Synthetic Precipitate
Microstructures
Experiments
(Tensile Tests)
Yield
Strength
(Validation)

Shorter-Term Approach
(This Work)
Fourier Space DFT
(VASP)
Phase Field
(PRISMS-PF)
Experiments
(HAADF-STEM,
APT)
Number
Density,
Volume
Fraction,
Dimensions
Elastic Constants, Transformation Strains
Inform problem space
Note: Work in progress

Nucleation Model for β1 Precipitates
• Use the 2D version of the simple classical nucleation
theory model for the nucleation rate [1]
• In principle could be parameterized via DFT, but in practice
it is too sensitive to error
• Instead, ρ1, ρ2, and τ are fit from experiments
– Need the number density and supersaturation during aging
[1] Simmons, Shen, and Wang, Scripta Materialia, 43 (2000)

[experimental results redacted]

Parameterizing the Nucleation Model
First, fit the composition using a hyperbolic tangent
Next, fit the number density using the nucleation model
Inhibit nucleation
before 1h

Phase Field Simulation of Nucleation,
Growth, and Coarsening
• Use this parameterized nucleation model for 2D phase field simulations
– 2D for computational tractability
– Longest precipitate axis perpendicular to the simulated plane
• 3 order parameters for 6 orientation variants
– Orientation variants with the same transformation strain share an order
parameter
– Random number decides the nucleus orientation for one variant or the other
• KKS model + linear elasticity
• Model inputs
– Homogenous free energies: Parabolas with the experimental equilibrium
compositions
– Interfacial energy: Isotropic, 50 mJ/m2 (estimate from Liu, 2014)
– Stress-free transformation strains: From DFT (Liu, 2014)
– Elastic constants: Homogenous, anisotropic, from DFT (Ji, 2014)
– Diffusion constant: From experiment
Liu et al., Acta Materialia, 77 (2014)
Ji et al., Acta Materialia, 76 (2014)PRISMS-PFAn Open-Source Phase Field Modeling Framework

Simulated Microstructure Evolution
• Precipitate shape is
qualitatively similar to
experiments
• All 6 variants present
• Largest precipitate is
approx. 0.5 μm at 6h
– Consistent with
experiments
• Fine precipitates
nucleating and
dissolving at the end
1.1μm

Simulated Microstructure Evolution
• Before 4 h, results are as expected
• After 4 h, new stable nucleus
formation stops prematurely
• Nucleation is still happening, but the
nuclei dissolve back into the matrix
• Average composition has a sigmoidal
shape, as expected
• Depletion of the matrix is delayed
compared to expectations
• Occurs even where the number
density is consistent with the
analytic model

Simulation vs. Experiment
Microstructure Comparison (4 h)
Simulation (4 periodic copies) TEM
Images are on the same scale
Diad/triad structures are much more common experimentally
Typical length of precipitates are similar, in the 200 nm range
Distribution in precipitate sizes wider in the simulation
5 0 0 n m

What’s going on?
Was this a typical
result? (yes)
What about
increasing the
diffusivity?

Why does stable nucleation end early?
• Mismatch between the nucleation model and the phase field model
• Nucleation events are still happening (nuclei stable in the nucleation
model)
• Nuclei no longer stable in the phase field simulation
• This possibility a consequence of the fitting strategy
• Possibilities:
• Interfacial energy is too high
• Experimental uncertainty (composition spatial variation, foil thickness)

Possible improvements
More experiments
- Understand what happens
between 1-4 h
Including strain energy in the
nucleation rate
- The characteristic triad
structures are favorable due to
strain energy [2]
- Should cause preferential
nucleation of particular variants
at precipitate tips
[1] Paliwal, et al., Scripta Materialia, 108 (2015)
[2] Liu, et al., Scientific Reports, 5 (2015)

On the path to quantitative phase field modeling
of precipitation during aging
Current strategy is valid for examining the effect
of average composition on the microstructure
Next large step to include temperature
dependence for validation and to eventually
optimize aging conditions
The microstructures from phase field can be
passed to dislocation dynamics to predict strength
Concluding Thoughts

Acknowledgements
Funding:
US DOE, Office of Science, Basic Energy Sciences
Award DE-SC0008637
PRISMS-PF Collaborators:
Shiva Rudraraju (now U. Wisconsin), Larry Aagesen (now INL),
David Montiel, Beck Andrews, Katsuyo Thornton
Experimental Collaborators
Qianying Shi, Zhihua Huang, John Allison

Questions?

Acknowledgements
Computational Resources:
Funding:
US DOE, Office of Science, Basic Energy Sciences
Award DE-SC0008637
deal.II Developers:

Using an Explicit Nucleation Model in PRISIMS-PF to Predict Precipate Microstrucutures

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