Grain boundary (GB) related phenomena are responsible for many of the properties in crystalline materials from plasticity to electrical conductivity. An important phenomenon which happens in crystalline materials during plastic deformation or at high temperature is GB motion. Due to more sophisticated nature of GB in comparison with grain interior, GB motion attracted a lot of effort and research in the past few years. Studying materials under different conditions, e.g. high temperature, and at small scales, e.g. nanometers, is extremely difficult using available experimental techniques, thus computer simulations are of importance to illustrate mechanisms at microscales. Although thickness of GB barely reaches 1 nm, the surface of GBs even between two grains are of micron scale. Therefor atomistic simulations with huge computational costs are not efficient to study large network of grains. Recently a continuum based simulation method has been developed to study the evolution in systems with distinct phases via minimizing the free energy. In this method, the material is treated as a continua and different phases/interphases have same/different energies. By minimizing the system via minimizing the energy of interphases the system reaches the ground state. Some examples of using phase field modeling include solidification, phase separation in alloys and grain growth in crystalline materials. In this work first, the basis of phase field modeling is explained, then governing equations are derived based on continuum mechanics for a bicrystal sample (a sample consisting of two grains and a GB between them) and a simple code is developed based on the formulation to show GB motion of a bicrystal sample. Using the results of this work, the mobility of different GBs can be computed easily.

1. Phase Field Modelling of
Grain Boundary Motion
Mohammad Aramfard
23 April 2018

2. Why Nanocrystalline?
Hall-Petch relationship:
𝜎 𝑦 = 𝜎0 +
𝐾
𝑑
2
Yield strength
Grain size NC realm

3. Volume Fraction of GB
High volume fraction of GBs and triple junctions (TJs)
makes them important in nanocrystalline metals.
3Palumbo et al. 1990

4. Stress-Induced Grain Growth: Experiments
4
Pt thin film under
tension. JA Sharon et
al., Scripta Mater, 2011
Al thin film under tension.
DS Gianola et al., Acta Mater, 2006.
G Gottstein et al., Acta Mater, 2002.

5. GB Motion: Bicrystal Experiment
5Gorkaya et al., Acta Mater, 2009.
Al bicrystal specimen under tension, GB: symmetric <100> with misorientation angle of 32o
𝑣 = 𝑀 𝐺𝐵Δ𝑃

6. Molecular Dynamics (MD) vs. Phase Field Modelling (PFM)
MD: in the order of
nanometer and
nanosecond.
Phase Field Modelling: in
the order of meter and
seconds
6

7. Mathematical Background of PFM
• Stefan problem (solidification):
•
𝜕𝑇
𝜕𝑡
= 𝛻. 𝛼𝛻𝑇
• 𝜌𝐿 𝑓 𝑉𝑛 = 𝑘 𝑠 𝛻𝑇. 𝑛𝑖𝑛𝑡
𝑆
− 𝑘 𝐿 𝛻𝑇. 𝑛𝑖𝑛𝑡
𝐿
• 𝑇𝑖𝑛𝑡 = 𝑇 𝑚 −
𝛾𝑇 𝑚
𝐿 𝑓
𝜅 −
𝑉𝑛
𝜇
7Lesar R., Intro to Comp Mat Sci, 2013.
Provatas N. Elder K., PFM in Mat Sci and Eng, 2010.

8. Mathematical Background of PFM
• Sharp interface problems:
• Case dependent
• Numerically difficult
• It needs one more equation to locate the
interface.
8
Lesar R., Intro to Comp Mat Sci, 2013.
Provatas N. Elder K., PFM in Mat Sci and Eng, 2010.
Gomez H. Zee K., Comp PFM, 2015.
Solution:
introducing phase equation

9. Mathematical Background of PFM
𝜙 𝑥, 𝑡 = tanh
𝑑 𝑥 − 𝑥𝑖𝑛𝑡
2𝜖
Phase variable or order parameter.

10. A-C and C-H Equations
• Allen-Cahn: mass of each phase is not
conserved (solidification).
• Cahn-Hilliard: mass of each phase is
conserved (phase separation).
Gomez H. Zee K., Comp PFM, 2015.

11. A-C Equation
• Constitutive equation (free energy)
• Ψ = Ψ 𝜙, 𝛻𝜙
• Free energy dissipation
•
𝑑
𝑑𝑡 𝒱
Ψ 𝜙, 𝛻𝜙 𝑑 𝑥 = 𝒲 𝒱 − 𝒟 𝒱 ; 𝒟 𝒱 ≥ 0
• Mass balance for a mixture
•
𝜕𝜙
𝜕𝑡
= −𝑚 𝜙 𝜇 ; 𝜇 = chemical potential
• Ψ = 𝐺 𝜙 +
𝜖2
2
𝛻𝜙 2
• 𝜇 =
𝛿
𝛿𝜙 𝒱
Ψ 𝜙, 𝛻𝜙 𝑑 𝑥 = 𝐺′ 𝜙
− 𝜖2
Δ𝜙
•
𝜕𝜙
𝜕𝑡
= −𝑚 𝜙 𝐺′ 𝜙 − 𝜖2Δ𝜙
𝐺 𝜙
Gomez H. Zee K., Comp PFM, 2015.

12. Governing Equations
• Constitutive equation (free energy), 1-D
• ℱ 𝜙 𝑥, 𝑡 = Ω
𝐺 𝜙1, 𝜙2, … , 𝜙 𝑝 + 𝑖=1
𝑝 𝜖 𝑖
2
𝜕𝜙 𝑖
𝜕𝑥
2
𝑑𝑥
• 𝐺 𝜙1, 𝜙2, … , 𝜙 𝑝 = −
𝛼
2 𝑖=1
𝑝
𝜙𝑖
2
+
𝛽
4 𝑖=1
𝑝
𝜙𝑖
2 2
+ 𝛾 −
𝛽
2
𝑖=1
𝑝
𝑖≠𝑗
𝑝
𝜙𝑖
2
𝜙𝑗
2
•
𝜕𝜙 𝑖
𝜕𝑡
= −𝐿𝑖
𝛿ℱ
𝛿𝜙 𝑖
= −𝐿𝑖
𝜕𝐺
𝜕𝜙 𝑖
+ 𝐿𝑖 𝜖𝑖Δ𝜙𝑖
• Assumptions:
• 𝜙1 𝑥, 0 = tanh
𝑑
2𝜖
; 𝜙2 𝑥, 0 = 0
• Numerical implementation:
• Finite difference
Tikare V. et al., Acta Mater, 1999.

13. Results
• Comparison between
disturbed sample and
normal sample
• Grain Boundary Motion
• Stable results
• With some parameters: non-
physical results

14. Comparison with 3-D
Tikare V. et al., Acta Mater, 1999.
• Polycrystalline sample.
• FEM

15. Summary and Future Works
• Summary:
• Capability of PFM to model GBM under driving force.
• Flexibility of studying different behaviors using the governing equation.
• Future works:
• Optimizing the model to match experiments/MD.
• Including more features of real materials (Phase-Field Crystal).
• Using the approach to model bulk 3-D models for designing.