Lightning talk on orientation-field models from CHiMaD Phase Field Workshop XVI.

1. A phase field model for grain growth:
capturing all five degrees of freedom of the grain
boundary energy
Phil Staublin1,
James A. Warren2, Peter W. Voorhees1
1. Materials Science & Engineering, Northwestern University, USA
2. Materials Science and Engineering Division, NIST, USA
Phase Field Workshop XVI, 20 March 2024

2. 2D Orientation-field Model
2
Order
Parameter, 𝜙
Orientation
Field, 𝜃
R. Kobayashi, J.A. Warren, W.C. Carter. Physica D: Nonlinear Phenomena 119(3-4), 415 (1998)
J.A. Warren, R. Kobayashi, A.E. Lobkovsky, W.C. Carter. Acta Materialia 51(20), 6035 (2003)
H. Henry, J. Mellenthin, M. Plapp. Phys. Rev. B 86, 054117 (2012)
𝐹 𝜙, 𝜃 =
1
2
∇𝜙 2
+ 𝐻𝑉 𝜙 + 𝜇2
𝑎 𝒏, 𝜃 2
𝑔 𝜙 ∇𝜃 2
𝑑𝑉
Gradient energy and single well potential
Energy from defect presence (𝜙 < 1)
Coupling to orientation
Energy from changes in orientation (𝜃)
Anisotropy coefficient
Energy depends on boundary plane normal
Materials parameters: 𝐻, 𝜇, 𝑔(𝜙)
Diffuse Grain
Boundary

3. How do we define the gradient of orientation in 3D?
3
𝒖𝐴 = 𝑂𝐴 𝑂𝐵
−1
𝒖𝐵 = 𝑂𝐴𝑂𝐵
−1
𝒖𝐵 = 𝑂𝐴𝐵𝒖𝐵
Misorientation between grains A and B: 𝑂𝐴𝐵
Infinitesimal misorientation at point 𝑥: Δ𝑂 = 𝑂−1 𝑂 + 𝑑𝑂 = 𝑂−1𝑂 + 𝑂−1𝑑𝑂 = 𝐼 + Ω
𝒖𝐴 = 𝑂𝐴𝒖 𝒖𝐵 = 𝑂𝐵𝒖
𝑂 𝒙 + 𝑑𝒙 = 𝑂 𝒙 + 𝑑𝑂
𝒖𝑥 = 𝑂 𝒙 𝒖

4. Infinitesimal rotation vectors: local misorientation
4
Consider a chance in rotation 𝑂 along spatial direction 𝑥:
The misorientation between 𝑂 and 𝑂 + 𝑑𝑂 is:
The rotation axis is: In terms of quaternions:
𝑂 𝒙 + 𝑑𝒙 = 𝑂 𝒙 + 𝑑𝑂
Δ𝑂𝑖𝑗 = 𝛿𝑖𝑗 + 𝜖𝑖𝑗𝑘𝜛𝑘
Δ𝑂 = 𝑂−1
𝑂 + 𝑑𝑂 = 𝑂−1
𝑂 + 𝑂−1
𝑑𝑂 = 𝐼 + Ω
𝒖 =
Δ𝑂32 − Δ𝑂23
Δ𝑂13 − Δ𝑂31
Δ𝑂21 − Δ𝑂12
=
−𝑑𝑂𝑥 − 𝑑𝑂𝑥
−𝑑𝑂𝑦 − 𝑑𝑂𝑦
−𝑑𝑂𝑧 − 𝑑𝑂𝑧
=
−𝜛1 − 𝜛1
−𝜛2 − 𝜛2
−𝜛3 − 𝜛3
= −2𝜛𝑖 𝜛𝑖 = −2𝜅𝑖𝛼𝛽𝑆𝛼𝛾𝑞𝛾
𝜕𝑞𝛽
𝜕𝑥
𝑆𝛼𝛽 =
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
where 𝜅𝛼𝛽𝛾 = 𝛿𝛼𝛽𝛿𝛾0 + 𝛿𝛼𝛾𝛿𝛽0 − 𝛿𝛼0𝛿𝛽𝛾 + 𝜖0𝛼𝛽𝛾

5. The “Misorientation-field Model”
5
𝜛𝑖𝑗 = −2𝜅𝑖𝛼𝛽𝑆𝛼𝛾𝑞𝛾
𝜕𝑞𝛽
𝜕𝑥𝑗
Crystal-frame grain boundary plane normal:
𝑛𝑖 = 𝜅𝑖𝛼𝛽𝜅𝛼𝛾𝑗𝑞𝛽𝑆𝛾𝜇𝑞𝜇
𝜕𝑗𝜙
𝜕𝑘𝜙𝜕𝑘𝜙 1/2
Crystal-frame infinitesimal rotation vector:
𝑓 𝜙, 𝒒, ∇𝜙, ∇𝒒 =
1
2
∇𝜙 2
+ 𝐻𝑉 𝜙 + 𝜇2
𝑔 𝜙 𝜒 𝜛 𝒒, ∇𝒒 , 𝑛 𝒒, ∇𝜙
Free energy density:
Choose a function of misorientation and inclination
𝛿𝐹
𝛿𝜙
=
𝜕𝑓
𝜕𝜙
− 𝜕𝑖 ⋅
𝜕𝑓
𝜕 𝜕𝑖𝜙
= −𝜕𝑖 ⋅ 𝜕𝑖𝜙 + 𝜇2
𝑔 𝜙
𝜕𝜒
𝜕𝑛𝑗
𝜕𝑛𝑗
𝜕 𝜕𝑖𝜙
+
𝜕𝑉
𝜕𝜙
+ 𝜇2
𝜕𝑔
𝜕𝜙
𝜒 𝜛 𝑞, ∇𝑞 , 𝑛 𝑞, ∇𝜙
𝛿𝐹
𝛿𝑞𝛼
=
𝜕𝑓
𝜕𝑞𝛼
− 𝜕𝑖 ⋅
𝜕𝑓
𝜕 𝜕𝑖𝑞𝛼
= 𝜇2
𝑔 𝜙
𝜕𝜒
𝜕𝜛𝑖𝑗
′
𝜕𝜛𝑖𝑗
′
𝜕𝑞𝛼
+ 𝑔 𝜙
𝜕𝜒
𝜕𝑛𝑖
𝜕𝑛𝑖
𝜕𝑞𝛼
− 𝜕𝑖 ⋅ 𝑔 𝜙
𝜕𝜒
𝜕𝜛𝑗𝑘
′
𝜕𝜛𝑗𝑘
′
𝜕 𝜕𝑖𝑞𝛼

6. Misorientation-dependent part
6
Proportional to the
misorientation angle
Coefficient
(Fitting parameter)
Components of the normalized
misorientation vector
𝜒(𝜛′) =
𝑘=1
3
𝑖=1
3
𝑎𝑘 𝜛𝑖𝑘
′
𝜛𝑖𝑘
′
𝑎𝑘 = 1 + 𝑐4
𝑖=1
3
𝜛𝑖𝑘
′ 4
𝑗=1
3
𝜛𝑗𝑘
′
𝜛𝑗𝑘
′ 2
𝑔 𝜙 𝜛𝑘
′ 2
100
𝑔 𝜙 𝜛𝑘
′ 2
111
≈ 0.5
𝑎𝑘 (c4 = 5)
𝑐4

7. Inclination-dependent part
7
𝜒 𝑛, 𝜛′ = 𝑏 𝑛 𝜒∗(𝜛′) = 𝑏(𝑛)
𝑘=1
3
𝑖=1
3
𝑎𝑘 𝜛𝑖𝑘
′
𝜛𝑖𝑘
′
𝑏 𝑛 = 1 + 𝜖4
𝑖𝑛𝑐𝑙
−
3
5
+
𝑖=1
3
𝑛𝑖
4
+ 𝜖6
𝑖𝑛𝑐𝑙
−
17
7
+ 66𝑛1
2
𝑛2
2
𝑛3
2
+
𝑖=1
3
𝑛𝑖
4
𝜕𝑏 𝑛
𝜕𝑛𝑖
= 4 𝜖4
𝑖𝑛𝑐𝑙
+ 𝜖6
𝑖𝑛𝑐𝑙
𝑛𝑖
3
+ 132𝜖6
𝑖𝑛𝑐𝑙
𝑛𝑖
𝑗=1,𝑗≠𝑖
3
𝑛𝑗
2 𝜕𝜒
𝜕𝑛𝑖
=
𝜕𝑏
𝜕𝑛𝑖
𝜒∗ 𝜛′
𝜕𝜒
𝜕𝜛𝑖𝑗
′ = 𝑏 𝑛
𝜕𝜒∗
𝜕𝜛𝑖𝑗
′
Cubic anisotropy
coefficient for inclination
Higher order terms are possible

8. Symmetry: “brute-force” method
8
𝑞𝑟
= 𝑆𝑞𝑏
For Each
Symmetry
Operation 𝑆
𝐿 =
𝛼=0
4
𝑞𝛼
𝑎 − 𝑞𝛼
𝑟 2 If
𝐿 < 𝐿𝑚𝑖𝑛
Set 𝐿𝑚𝑖𝑛 = 𝐿
Set 𝑞𝑚𝑖𝑛
𝑟
= 𝑞𝑟
∇𝑞 = 𝑞𝑎 − 𝑞𝑚𝑖𝑛
𝑟
True
False
Compute:
∇𝑞 = 𝑞𝑎
− 𝑞𝑏
Start
Done
Every quaternion gradient is computed using the
following iterative procedure:
Tamas Pusztai et al. in Europhysics Lett. 71 p131 (2005):

9. Grain boundary energy vs. misorientation
9
V.V. Bulatov, B.W. Reed, M. Kumar. Acta Materialia 64, 161-175 (2014).

10. Inclination dependence: 𝜸-plots
10
Phase field 𝛾𝐺𝐵
For [001] tilt boundaries
• Anisotropy varies with misorientation
(increases in this case)
• Axes of highest and lowest energy change
with misorientation
• Limited to modest anisotropy without
regularization

11. Triple junction velocities match theory
11
𝑣 =
2𝑀𝐺𝐵𝛾𝐺𝐵𝜃𝑇𝐽
𝐿
= 0.103304 = 2.58 𝑔𝑝/𝑡𝑖𝑚𝑒

12. Acknowledgements
This research was supported by the National Institute for
Standards and Technology through the Center for
Hierarchical Materials Design at Northwestern University.
Co-Authors:
James A. Warren (NIST)
Peter W. Voorhees (Northwestern)
12