Modelling of

Microstructural Evolutions in

Structural Materials

Yoshiyuki　SaitoDepartment of Ｅｌｅｃｔｒｏｎｉｃ　and Ｐｈｏｔｏｎｉｃ　Ｓｙｓｔｅｍｓ

Waseda University

Contents１．Introduction　　　Present State 　　　Target ２．Computational Models based on Thermodynamics　　and Phase Transformation Theory３．Computer Simulation(Topics) 　　　grain growth, spinodal decomposition　　　　4. Future plan 3-D multiscale, Nanostructural materials

ProblemsModelling of microstructural evolutionds

in structural steels　・Synergetic effects grain growth, recrystallization, second

phase precipitation, spinodal decomposition, γtoα phase transformation, 　・Multiscale　　several nm to several handeredsµm　・Multicomponent system

Phase separation

0

300

600

900

1200

0 0.25 0.50 0.75 1.00

Temp., K

Cr, at.frac

binodalspinodal

Grain growth

Modelling of microstructural evolution in structural steel

• 1st stage(1985-1995) Integrated model based on thermodynamics and phase transformation theory• 2nd stage(1995-2005) MC, PF for Grain growth, Spinodal decomposition

etc.• 3rd stage(2005- ) Multiscale 3-D simulation for microstructural evolution in structural steel.

Computational Models based on

Thermodynamicsand

Phase Transformation Theory

1．Theories of Microstructural Evolution　　　　Nucleation, Growth and Interface Migration2. Integrated Models for Predicting Microstructural

Evolution in HSLA steels Grain growth, Precipitation, Phase

Transformation

Modelling of microstructural evolution in structural steel

• 1st stage(1985-1995) Integrated model based on thermodynamics and phase transformation theory• 2nd stage(1995-2005) MC, PF for Grain growth, Spinodal decomposition

etc.• 3rd stage(2005- ) Multiscale 3-D simulation for microstructural evolution in structural steel.

Computer Simulation

• Discrete　model 　　　Monte Carlo method• Continuum　model　　　　Phase field model

Grain Growth

Classification of simulation models

Model

Soap frothPolycrystal

Direct Simulation Statistical

Deterministic Probabilistic

Equation of motion for velocity of boundary

・Vertex model

・tracking of boundaries

Monte Carlo Non Monte Carlo

・Cellular Automata(*)

Phase-field

Simulation of Grain Growth by

：the local free energy density：the gradient energy coefficients

： total number of orientation field variable

:Onsager’s phenomenological parameters(mobility of a interface)

Since the orientation field variables are non-conserved order parameter, the temporal evolutions of these parameters are written as follows

We can write the total free energy of an inhomogeneous system in terms of all the orientation field variables and their gradients as:

The　Phase　field　model

Simulation of grain growth

0

0.025

0.050

0.075

0.100

0 1.25 2.50 3.75 5.00

Frequency

Normalized Grain Size,R/Rav

100200300400

0

0.0375

0.0750

0.1125

0.1500

0 1.25 2.50 3.75 5.00

Frequency

Normalized Grain Size,R/Rav

100180200220300400

-Isotropic Grain Growth

Variation in the scaled grain size distribution function

・3D-simulation

・2D cross section of 3D simulation

System size：3203

0

0.025

0.050

0.075

0.100

0 1.25 2.50 3.75 5.00

Frequency

Normalized Grain Size,R/Rav

100200300400

0

0.0375

0.0750

0.1125

0.1500

0 1.25 2.50 3.75 5.00

Frequency

Normalized Grain Size,R/Rav

100180200220300400

-Isotropic Grain Growth

Variation in the scaled grain size distribution function

・3D-simulation

・2D cross section of 3D simulation

System size：3203

0

200

400

600

800

1000

0 100 200 300 400 500

<Rv>^2

Simulation time

0

0.05

0.10

0.15

0.20

0 0.625 1.250 1.875 2.500

Frequency

Normalized Grain Size, R/Rav

t=50t=100t=200t=300t=400

0

3

6

9

12

0 5 10 15 20 25 30 35 40

Frequency, %

Number of faces per grain, Nf

t=50t=100t=200t=300t=400

0

3.1

6.2

9.3

12.4

15.5

0 100 200 300 400 500

average number of faces

simulation time

10.0

12.5

15.0

17.5

20.0

0 0.0625 0.1250 0.1875 0.2500

m(Nf)

1/Nf

Simulation of Grain Growth by

The　Metropolis　Algorithm

Importance　Sampling--- Markov process

A configuration {s'} is generated from the knowledge of a configuration {s}.

W({s},{s'}): the conditional probability to select {s'} starting from {s}.The equilibrium distribution:

Peq({s})=exp[-E({s})/kBT]/Z.

E: internal energy function, kB : the Boltzmann constant, T: temperature , 　Z : canonical partition function.

The　procedure　of　the　simualtion

• (1). A grain number from 1 to the system size, N, is assigned to each lattice point. • (2). A number corresponds to an orientation is randomly assigned to each grain.• (3). The evolution of microstructure is tracked by the change of orientations on

each lattice• (a). One lattice site is selected at random .• (b). If the lattice site belongs to grain boundary, then a new orientation is

generated.• (c).If one of nearest neighbor lattices has the same orientation as the newly

selected grain orientation, a re-orientation trial is attempted.• (d). The change in energy, ΔE, associated with the change of grain orientation is

calculated. The re-orientation trial is accepted if ΔE is less than or equal to zero. If the value ΔE is greater than zero, the re-orientation is accepted with probability,

• W = exp(-ΔE/kBT).

The change of interfacial energy accompanying re-orientation

Condition of the simulation

• System size: 1283

• The number of orientation: 16 or 32• 3-dimensional• J/KBT=0.75

0

52.254

104.508

156.762

209.016

0 2,000 4,000 6,000 8,000 10,000

Area

Time,MCS

0

1.25

2.50

3.75

5.00

0 0.625 1.250 1.875 2.500

frequency

R/Rav

400 MCS500 MCS750 MCS1000 MCS

0

2.5

5.0

7.5

10.0

0 10 20 30 40

Frequency, %

Number of faces per grain

400 MCS500 MCS750 MCS1000 MCS

10.0

12.5

15.0

17.5

20.0

0 0.0625 0.1250 0.1875 0.2500

m(Nf)

1/Nf

Phase Field vs. Monte Carlo

0

1.25

2.50

3.75

5.00

0 0.625 1.250 1.875 2.500

f(Rv/<Rv>)

Rv/<Rv>

MCMPFM Hillert-3D

0

2.5

5.0

7.5

10.0

0 10 20 30 40

frequency, %

number of faces per grain

MCMPFM

0

5

10

15

20

0 0.0625 0.1250 0.1875 0.2500

m(Nf)

1/Nf

PFMMC

Phase Field vs. Monte Carlo

Problems related to interface migration

Effect of anisotropy on grain growth Abnormal grain growthEffect of second phaseTexture controlSolute drag effectRecrystallizationPhase transforamtion

Spinodal Decomposition

• Background• 　　Duplex stainless steel

• ・high strength, ・good weldability, ・high resistance to stress corrosion cracking 　　→　　Chemical reactor

•

• Problem: 475°C embrittlement　

• 　　Hardening due to thermomechanical instability of ferrite phase in duplex stainless steels

• 　　　Prediction of long time stability of

• chemical reactor

0

300

600

900

1200

0 0.25 0.50 0.75 1.00

Temp., K

Cr, at.frac

binodalspinodal

Fe-Cr binary alloy

Second order differential parameter1D

3D

2D

Second order differential parameter1D

3D

2D

0

300

600

900

1200

0 0.25 0.50 0.75 1.00

Temp., K

Cr, at.frac

binodalspinodal

Fe-Cr binary alloy

Chemical free energy

f0=Ωx(1-x) +RT[xln(x)+(1-x)ln(1-x)]}

Phase boundary

Spinodal

Proto type simulation(Fe-40wt.%Cr ally）８５０K

0

0.225

0.450

0.675

0.900

0 6.25 12.50 18.75 25.00

Cr, at.frac.

Distance, nm

00.25hr0.5hr1hr2hr5hr10hr20hr50hr100hr200hr

　Fe-40％Cr　binary　alloy

Simulation of Phase Separation

Cahn-Hillirad equation for Fe-X-Y ternary alloys

Cahn-Hillirad equation for Fe-X-Y ternary alloys

Condition　

Mobility, M：

Energy gradient coefficient

L：

Aging temperature T：

Composition：

K：

Condition　

Mobility, M：

Energy gradient coefficient

L：

Aging temperature T： 800KComposition：

Fe-40at.%Cr-5at.%Mo Fe-40at.%Mo-5at.%Cr

K：

Diffusion coeeficient D：

Interaction parameter Ω：

Diffusion coeeficient D：

Interaction parameter Ω：

1D simulationFe40Cr5Mo(800K)

1D simulationFe40Cr5Mo(800K)

Cr Mo

100時間後

Cr Mo

16 nm

Fe40Cr5Mo 2D simulation

Cr Mo

100時間後

0.83 0.11

0.09 0.01

Cr Mo

16 nm

Fe40Cr5Mo 2D simulation

Cr Mo

1000時間後

Cr Mo

16 nm

Fe40Cr5Mo 2D

Cr Mo

1000時間後

0.82

0.01

0.12

0.09

Cr Mo

16 nm

Fe40Cr5Mo 2D

Cr Mo

1000時間後

0.82

0.01

0.12

0.09

Cr Mo

16 nm

Fe40Cr5Mo 2D

Isosurface of Cr Isosurface of Mo

1000時間後

0.82Cr Mo

0.11Fe40Cr5Mo3D simulation

Isosurface of Cr Isosurface of Mo

1000時間後

0.82Cr Mo

0.65

0.11

0.06

Fe40Cr5Mo3D simulation

Cr Mo

MoCr

3D1000時間後 Fe40Cr5Mo

0.82

0.09

0.11

0.01

Cr Mo

MoCr

16 nm

3D1000時間後 Fe40Cr5Mo

0.82

0.09

0.11

0.01

Cr Mo

MoCr

16 nm

3D1000時間後 Fe40Cr5Mo

Monte Carlo　Simulation

Lenard –Jones Parameter

：equilibrium atomic spacing

Interatomic potential at equilibrium atomic spacing

Lenard-Jones potentila

Normalized Lenard-Jones parameter

Condition

Fe-40at.%Cr-5at.%Mo 750K800K850K

Fe-40at.%Mo-5at.%Cr

62.0 KJ/mol

0.9537 1.2296 0.9809 1.3001 1.6390

Monte Carlo　Simulation

Separation of peaks of Mo

Behavior of element Y along the trajectory of peak top of element X

Behavior of element Y along the trajectory of peak top of element X

Regular solution model

Modelling of microstructural evolution in structural steel

• 1st stage(1985-1995) Integrated model based on thermodynamics and phase transformation theory• 2nd stage(1995-2005) MC, PF for Grain growth, Spinodal decomposition

etc.• 3rd stage(2005- ) Multiscale 3-D simulation for microstructural evolution in structural steel.

Microscopic model　Thermodynamics model 　　Kinetic model　　→Prediction of microstructure　　　　　＋FEM　→ Prediction of mechanical propertiedMesoscopic model　Phase field model　Monte Carlo methodNeural networkMathematical method

Dynamics and Microstructure Evolution in Metals during and after the process of Severe Plastic Deformation

　　　　　Tetsuya Ohashi (Kitami Institute of Technology),　　　　　Mitsutoshi Kuroda　(Yamagata University), 　　　　　Yoshiyuki Saito (Waseda University), 　　　　　Tadanobu Inoue (National Institute of Material Science)

　　Objectives of this study group are to clarify underlying mechanisms for the development of fine-grained microstructures and to understand the mechanical response of metal polycrystals with high-density lattice defects by using numerical techniques of macro- and meso-mechanics. In the macroscopic analyses, strain histories in materials processed by ARB or ECAP are quantitatively evaluated. The crystal plasticity analyses are performed to understand the characteristics and particularities of dislocation accumulations under different deformation modes. Macroscopic mechanical responses of nano-structured metals are also predicted by crystal plasticity analyses. Thermodynamic stability of ultrafine grained structures is discussed with phase field simulations.

Crystal plasticity analyses of dislocation

accumulation

Phase field simulation of microstructural

evolutions

Macroscopic analyses of giant straining process

item ２０００ ２００５２０１０

２０１５ needs

Precipitation

Dynamic recrystallization

Dynamic phase

transformation

Fracture mechanics

Microstructure vs

properties

Modelling of microstructure during TMCP

Mechanical properties

Corrosive p@roperties

Modelling　Equilibrium state　　Calphad　Dynamics　　Microstructural evolution　　Properties

Modelling ofMechanical propertiesMagnetic properties、Corrosive properties

Microstructure vs. mechanical propertiesMD,MC、PFM、FEM、neural networkMechanical properties, magnetic properties

Ab-initio,・PFM・FEM

Database, Knowledge base　Thermodynamic DB、Diffusion coefficient DBの拡充、Properties DB

Database

Ab-initioMD,MCPFMFEMEmpirical

ThermodynamicsDBDiffusion Coeffi. DBMicrostructure DBProperties DB

Computed phase diagram

Microstructural evolution

Computational mechanics　deformation, fracture

Multiscale 3D-modelMicrostructure Phase field+MC and FEMProperties FEM Neural network