Theoretical and Applied Phase-Field: Glimpses of the activities in India

1. Theoretical and Applied Phase-Field: Glimpses of
the activities in India
Abhik Choudhury
Assistant Professor
Department of Materials Engineering
Indian Institute of Science, Bangalore, India

2. Outline
Brief introduction of the phase-field groups in India and their
activities (15 mins)
Particular focus problem: Predicting equilibrium shapes of
precipitates under the influence of coherency stresses using the
phase-field method

3. Phase-field groups in India
Indian Institute of Technology, Bombay (M.P. Gururajan)
Indian Institute of Technology, Kanpur (Rajdip Mukherjee)
Indian Institute of Technology, Hyderabad (Saswata Bhattacharyya)
Indian Institute of Technology, Madras (Gandham Phanikumar)
Indian Institute of Science, Bangalore (T.A. Abinandanan, Abhik
Choudhury)
As an introduction, I will highlight some of the recent works in the
individual groups

4. T.A. Abinandanan (Indian Institute of Science)
Solid-state phase
transformations
Grain-growth and interfacial
instabilities
Influence of elastic stresses
on phase transformations
Growth and coarsening in
multi-component systems
Spinodal decomposition

5. Tricontinuous microstructures in a ternary system
T. Shukutani, T. Myojo, H. Nakanishi, T. Norisuye, and Q. Tran-Cong-Miyata. Tricontinuous morphology of ternary polymer blends
driven by photopolymerization: Reaction and phase separation kinetics. Macromolecules, 47(13):4380–4386, 2014.

6. T. Shukutani, T. Myojo, H. Nakanishi, T. Norisuye, and Q. Tran-Cong-Miyata. Tricontinuous morphology of ternary polymer blends
driven by photopolymerization: Reaction and phase separation kinetics. Macromolecules, 47(13):4380–4386, 2014.

7. Cocontinuous microstructures in 2D/3D

8. Cahn-Hilliard Model

10. Findings
1. Spinodal decomposition in three-component alloys can yield
tricontinuous microstructures.
2. Equal (and near-equal) volume fractions of the three phases
produce tricontinuous microstructures under a variety of conditions.
3. This understanding allows us to explain the microstructures we have
observed in ternary alloys, and demonstrate that tricontinuous
microstructures may be produced even in alloys with volume
fractions as low as 25%

11. Saswata Bhattacharyya (Indian Institute of
Technology, Hyderabad)

17. M.P.Gururajan,Indian Institute of Technology Bombay,
Computational Materials Engineering Group

18. Introduction
To develop phase field models to study microstructural evolution:
induced by phase transformation and deformation
Develop in-house codes in C (and, CUDA) to implement the phase
field models
Examples: Interfacial energy and attachment kinetics anisotropy,
elastic stress effects and phase field dislocation dynamics
Funding: We acknowledge DST, GoI, Tata Steel, and GE
Computational facilities: We thank DST-FIST, Space-Time: IITB, and
C-DAC, Pune

19. Precipitate morphological evolution
From Master’s thesis of Mr. Abhinav Soni: based on extended
Cahn-Hilliard model with sixth rank tensor terms
Figures show the effect of hexagonal anisotropy in both interfacial
energy and attachment kinetics
Relevant to morphological evolution of graphite in graphitic cast iron

20. Spinodal decomposition
From unpublished work of Mr. Sagar Girimath, intern: based on
extended Cahn-Hilliard model with sixth rank tensor terms
Figures show the effect of pronounced hexagonal anisotropy in
interfacial energy on morphological evolution during spinodal
decomposition
See Nani and Gururajan (Phil. Mag., 2014) and Arijit Roy et al (Phil.
Mag., 2017) for details of the formulation

21. Phase Field Dislocation Dynamics
Phase field dislocation dynamics code under development by Mr.
Rahul Chigurupati (Master’s student): in collaboration with Mr. Arjun
Varma, PhD student and Prof. Prita Pant
Figure shows a dislocation loop on the (111) plane with no applied
stress

22. Gandham Phanikumar, Indian Institute of Technology,
Madras
Applied phase-field methods
ICME based approach to
materials science problems
Multi-component solidification
Levitation experiments,
controlled solidification setups
for determining stresses
during solidification
Additive manufacturing

26. Abhik Choudhury, Indian Institute of Science,
Bangalore
Multi-phase multi-component
solidification
Multi-component growth and
coarsening
Multi-physics problems,
coupling of electric, thermal,
mechanical effects->
corrosion, electromigration,
precipitate growth
Interfacial instabilities
Multi-scaling
DS experiments

27. Predicting three-phase microstructures
Simulation along (Path II)

28. Three-phase microstructures in a realistic system
L. Ratke and A. Dennstedt. DLR
Koln
Coupling with databases according to Choudhury et al. Current Opinion in Solid-State and Materials Science, Vol. 287, 2015

29. Other possible attractors

30. Statistical characterization of structures
2-point spatial correlations with principal component analysis (in collaboration with
Prof. Surya Kalidindi (Acta Materialia 110 (2016) 131–141))

31. Modeling of electronic performance of polymer organic
photovoltaic blends

32. Electromigration/Thermomigration
in collaboration with Praveen Kumar(IISc, Bangalore)
(a) (clockwise) Representative
SEM micrographs showing (i)
cathode side before
electromigration test, (ii) anode
side before electromigration test,
(iii) near to the cathode side after
electromigration test, and (iv)
near to the anode side after
electromigration tests. The
vertical arrows in (iii) and (iv)
indicate the initial position of the
Cu film. The metal film was Cu
and the interlayer between Cu
and the Si substrate was W.
(b) Phase field model prediction
showing effect of imposition of an
electric field from right to left and
a temperature gradient from (i)
right to left and (ii) left to right on
the mass transport after various
instances. Asymmetric mass
transport at the anode and the
cathode is clearly observed.
Experimental results shown in (a)
and phase field modelling results
shown in (b) are mutually
consistent.
(i) (ii)
0 500 1000 1500 2000 2500
0
50
100
150
200
250
300
350
400
Chakraborty et al. Acta Materialia Vol. 153(2018)

33. Predicting the equilibrium precipitate shape morphologies under the
influence of coherency stresses

34. Equilibrium of coherent precipitates
Minimization of the sum of interfacial and elastic energies for a given
volume
Shapes minimizing the elastic and interfacial energies are not the
same
Shape becomes an unknown

35. Prediction of shape, shape bifurcations
In the presence of coherency equilibrium shapes of precipitates
depends on the size

36. Sharp-interface solutions: Green’s functions based

37. Sharp-interface solutions: Green’s functions based

38. Sharp-interface solutions: FEM Based methods

39. Sharp-interface solutions: FEM based methods

40. Local thermodynamic equilibrium at the interface
(Su and Voorhees, Thomson and Voorhees)
ρα
− ρβ
M = W (E − E0) |β
α − T.E|β
α − s.ξ (1)
M is the constant diffusion potential in the system, ξ is the
Cahn-Hoffmann vector.
(Eshelby, Schmidt and Gross, Leo and Sekerka)
nT
· P|β
α · n − γκ + λ = 0 (2)
P is the Eshelby momentum tensor WI − uT · σ.

41. Free energy functional
Free energy functional
F(φ) =
V
γWa2
(n) | φ|2
+
1
W
ω (φ) dV
+
V
fel (u, φ) dV + λβ
V
h (φ) dV
Interpolation function
h (φ) = φ2
(3 − 2φ) (3)
Obstacle potential
ω (φ) =
16
π2
γφ (1 − φ) φ ∈ [0, 1],
= ∞ otherwise.

42. Volume-preserved Allen-Cahn dynamics
Allen-Cahn dynamics for evolution of order parameter
τW
∂φ
∂t
= −
δF
δφ
τW
∂φ
∂t
= 2γW · a (n)
∂a (n)
∂ φ
| φ|2
+ a (n) φ
−
16
π2
γ
W
(1 − 2φ) −
∂fel (u, φ)
∂φ
− λβh (φ)
Mechanical equilibrium
ρ
d2u
dt2
+ b
du
dt
= · σ · σ = 0
Lagrange multiplier λβ
λβ = V rhsα
V h (φ)
,
Volume preserved Allen-Cahn; SIAM Vol.18(8)1347–1381

43. Interpolation of the elastic energy density
Khachaturyan type Interpolation
fel(u, φ) =
1
2
Cijkl(φ)( ij − ∗
ij(φ))( kl − ∗
kl(φ)),
ij =
1
2
∂ui
∂xj
+
∂uj
∂xi
Cijkl(φ) = Cα
ijklφ + Cβ
ijkl(1 − φ),
∗
ij(φ) = ∗α
ij φ + ∗β
ij (1 − φ).

44. Tensorial Interpolation: Basic Idea
Transform coordinates into n, t (normal and tangential)
Interpolate different elements of the stiffness matrix such that the
following conditions are satisfied
From the continuity of tractions and tangential displacements
σα
nn = σβ
nn = σnn
σα
nt = σβ
nt = σnt
α
tt = β
tt = tt
The other components get interpolated across the interface
σtt = σα
tt h (φ) + σβ
tt h (1 − φ)
nt = α
nt h (φ) + β
nt h (1 − φ)
nn = α
nnh (φ) + β
nnh (1 − φ)
Scheider et al. Computational Mechanics 55(2015)887–901, Durga et al. Mod. Sim. Mater. Engg. 21(2013), Bhadak et al. 2018,
Met. Trans. A

45. Some feel for the interpolation conditions
-0.01
-0.005
0
0.005
0.01
0.015
0 0.5 1
εnt,εtt,εnn
Normalized distance
εnt
εtt
εnn
φ=0.5
-2
-1
0
1
2
0 0.5 1
σnt,σtt,σnn
Normalized distance
σnt
σtt
σnn
φ=0.5

46. Interface width convergence
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ρ(Normalizedaspectratio)
W/R (interface width/precipitate radius)
Tensorial interpolation
Khachaturyan interpolation
λβ = γκ + (ωβ − ωα) − − > Sharp Interface problem
ω = fel − σ0
nn nn − 2σ0
nt nt

47. Parameter initialization
C44 = µ, C12 = 2ν C44
1−2ν , C11 = C12 + 2C44
Az
.
Poison ratio(ν) = 0.3, Shear modulus(µ)µmat = 125,
Zener anisotropy parameter(Az : 0.3 − 3.0), Inhomogenity ratio(δ),
Interfacial energy(γ) = 0.15
Eigenstrain (misfit strain) ∗:dilatational or tetragonal
∗
=
∗
xx 0
0 ∗
yy

48. Results: Equilibrium shapes of the precipitate
Case I: Isotropic elastic energy:
(Az = 1.0, ∗
xx = ∗
yy = 0.01)(dilatational misfit)
Case II: Cubic anisotropy in elastic energy:
(Az = 1.0, ∗
xx = ∗
yy = 0.01)(dilatational misfit)
Case IIIA: Cubic anisotropy in elastic energy:
(Az = 1.0, ∗
xx = ∗
yy )(tetragonal misfit: same sign)
Case IIIB: Cubic anisotropy in elastic energy:
(Az = 1.0, ∗
xx = ∗
yy )(tetragonal misfit: opposite sign)
Case IV: Anisotropy in both the energies:
Figure: Schematic

49. Isotropic elastic energy with dilatational misfit
L = Rµmat
∗2
γ , ρ =
c − a
c + a
0 500 1000 1500 2000 2500
0
50
100
150
200
250
300
350
400
Precipitate size(R)=30(L=2.5)
0 500 1000 1500 2000 2500
0
50
100
150
200
250
300
350
400
Precipitate size(R)=60(L=5.0)

50. Isotropic elastic energy
Az = 1.0, ∗
xx = ∗
yy = 0.01, L = Rµmat
∗2
γ
, ρ = (c − a)/(c + a)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 1 2 3 4 5 6 7 8
ρ(Normalizedaspectratio)
L (Characteristic length)
Johnson-Cahn
Phase field
FEM
(a)
(b)
Figures (a)Bifurcarion diagram, (b) equilibrium shapes of
the precipitate

51. Elastic energy: Cubic anisotropy and dialatational
misfit
Precipitate size(R) = 25(L = 2.08) and 55(L = 4.58), ρ =
c − a
c + a
0 500 1000 1500 2000 2500
0
50
100
150
200
250
300
350
400
0 500 1000 1500 2000 2500
0
50
100
150
200
250
300
350
400

52. Elastic energy: Cubic anisotropy and dialatational
misfit
(Az = 1.0, ∗
xx = ∗
yy = 0.01) L = Rµmat
∗2
γ
, ρ =
c − a
c + a
(a)
(b)

53. Elastic energy: Cubic anisotropy and tetragonal misfit
Az < 1.0, ∗
xx > ∗
yy , ∗
xx / ∗
yy (t) = +2.0
∗ =
0.01 0
0 0.005
, ρ = N
i=1
Xi Yi
NV
0 500 1000 1500 2000 2500
0
50
100
150
200
250
300
350
400

54. Elastic energy: Cubic anisotropy and tetragonal misfit
Az < 1.0, ∗
xx > ∗
yy , ∗
xx / ∗
yy (t) = +2.0
∗ =
0.01 0
0 0.005
, ρ = N
i=1
Xi Yi
NV
(a) (b)
Figures (a)Bifurcation diagram, (b) Comparison of results
from FEM and PF simulations and equilibrium shapes of
precipitate as function of size

55. Elastic energy: Cubic anisotropy and tetragonal misfit
(Az < 1.0, ∗
xx > ∗
yy , ∗
xx / ∗
yy (t) = +2.0)
0.55
0.555
0.56
0.565
0.57
0.575
3.45 3.5 3.55 3.6 3.65 3.7 3.75
(Totalenergy)
L (Charateristic length)
twisted diamond shape
ellipse-like shape
(a)
(b)
Figures (a)Elastic energy as function of precipitate size,
(b) equilibrium shapes of the precipitate with same size:
stable (dotted line) and metastable (thick line)
equilibrium

56. Elastic energy: Cubic anisotropy and tetragonal
misfit(opposite sign), (Az = 1.0, ∗
xx/ ∗
yy (t) = −2.0)
(a)
(b)
Figures (a)Equilibrium morphologies of precipitate with
Az = 0.3 and (b) Az = 2.0

57. Competition between anisotropies in interfacial and
elastic energies
Az = 1.0 Az = 3.0 Az = 0.3
(a) (b) (c)
Equilibrium shapes of the precipitate with (dotted line)
and without (continuous line) interfacial anisotropy

58. Competition between anisotropies in interfacial and
elastic energies (Shape factor(η) = R100/R110)
(a)
(b)
Figures (a)Shape factor (η) as function of Az at different
strengths of interfacial anisotropies and (b) Equilibrium
morphologies of precipitate at Az = 3.0, ε = 0 − 0.04

59. Comparison with experiments
Experiments
precipitate morphologies in
Ni-Al-Mo alloy (Fahrmann et
al. 1995)
Simulations
Equilibrium shapes of
precipitate at Az = 3.0

60. Comparison with experiments continued...
Experiments
Mg-stabilized zirconia
microstructure (Lanteri et
al. 1986)
Simulations
Equilibrium shapes of the
precipitate at Az = 0.3, t = +2.0

61. Conclusion
Presented a phase-field model for the determination of equilibrium
morphologies of precipitate under coherency stresses.
Predicted the symmetry breaking transitions and bifurcation
diagrams which occur for the shapes of the precipitates as a
function of size for various combinations of misfits as well as
anisotropies in the elastic constants.
Find excellent agreement between predicted results and respective
analytical and FEM solutions.
Combined influence of elastic and interfacial energy anisotropy both
above and below bifurcation.

62. Thank You...