phase field modeling of microstructure evolution in ... · phase field modeling of microstructure...

Phase field modeling of Microstructure Evolution in Zirconium base alloys

Gargi Choudhuri, S.Chakraborty, B.K.Shah, D S i t G K DD. Srivastava, G.K.Dey

Bhabha Atomic Research CentreMumbai, India- 400085,

17th ASTM International Symposium on Zirconium in Nuclear Industry, Feb 3-7, 2013, Hyderabad, India

OUTLINE

Morphological variation in Zr-Nb microstructure

Phase Field Model

Model Development

- Construction of Free Energy Functional

- Anisotropy in Interfacial Energy

- Model Parameters

Results & Discussion

Conclusions

Motivation

Generation of desired microstructure with required texture in commercial Zr-Nb alloys has been achievedthrough empirical formulation and traditional trial and error method. Due to cost effectiveness of

)800862oC

Computational method over traditional trial and error method, alloy development and designing of desiredmicrostructure through modeling and simulation route is gaining ground day by day.

Ms(')

pera

ture

(o C)800

400

18.5610oC I+II

I (Zr )+ I(Zr) ’ ( )

Ms() Tem

p400

25'

‐hcp’ hcp

I (Zr) I (Zr) + II (Nb)

Zrwt % Nb

252015105'+ metastable ‐ hcp

‐ bcc‐ hexagonal

• The phase transformation and microstructural evolution in Zr-Nb alloys are complex.• Depending upon composition of the alloy, soaking temperature and cooling rate

β Zr transforms to Allotrimorph alphaAllotrimorph alphaWidmanstatten alpha (parallel plate/ basket weave morphology)Martensitic microstructure (lath/plate morphology)Omega phase

Hydride formation

o

Zr rich corner of Zr-Nb phase diagramMicrostructure: Distribution of phase as finespherical precipitates within the equiaxed matrix

Ms(')

ratu

re (o C

)800862oC

18.5610oC I+II

spherical precipitates within the equiaxed matrixgrains.

• Average precipitate size

Ms() Tem

per

400

25

• Average precipitate size~ 30 nm

• precipitate volume fraction~ 3 %

Zrwt % Nb

252015105'+ metastable

transformation

b t hi t i lbcc stereographic triangle where the superimposition of the variants could be seen along with the reflections in [110], [210], [311] and [211] zones

8000C - 30 mins + Q

[311] and [211] zones.

S. Neogy, K. V. Mani krishnaD. Srivastava and G. K. Dey , Phil. Mag. 2011

S. Neogy*, K.V. Mani Krishna, D. Srivastava and G.K. DeyPhilosophical MagazineVol. 91, No. 35, 21 December 2011, 4447–4464

Zr-2.5 Nb alloy Gas quenched from + β

50C/sec10C/sec

25C/sec

martensiteWidmanstatten

10C/sec

Different morphologies

Saibaba.et al., J. of ASTM Int., June 2011, 8, Issue 6

Micrograph showing the microstructure of the Zr‐7Nb alloy after isothermally transforming at 823 K for 15 min Misfit dislocations of the α /β interface can be clearly seenat 823 K for 15 min. Misfit dislocations of the α /β‐interface can be clearly seen.

G.K.Dey et.al Journal of Nuclear Materials 224 (1995) 146-157

(a) Bright‐field and (b) dark‐field micrograph showing the formation of the plate‐shaped internally twinned a‐phase at the grain boundaryinternally twinned a phase at the grain boundary.

G.K.Dey et.al Journal of Nuclear Materials 224 (1995) 146-157R. Tewari, D. Srivastava, G.K. Dey, J.K. Chakravarty, S. Banerjee,

Journal of Nuclear Materials 383 (2008) 153–171G. K. Dey and S. Banerjee, Journal of Nuclear Materials 125 (1984) 219

S. Neogy*, K.V. Mani Krishna, D. Srivastava and G.K. DeyPhilosophical MagazineVol. 91, No. 35, 21 December 2011, 4447–4464

OUTLINE


Phase Field Model

Model Development



- Model Parameters


Conclusions

Phase field method (PFM) grows out of the work of Allen Cahn and

Phase field model :

Phase field method (PFM) , grows out of the work of Allen , Cahn and Hilliard has been used here to model phase boundary motion during β to phase transformation.

Diffuse InterfacePhase field model :Computational approach to modeling and predicting meso-scale morphological & microstructure evolution

Diffuse Interface

& microstructure evolution.

Entire microstructure is represented Interface thi k δcontinuously by a non conservative

phase field variable, φ,(order parameter/crystal structure) am

eter

thickness δ

where φ=1,φ=0 (at precipitate & matrix/at two phases)

& rder

par

&0<φ<1 represent the interface region.

It is a Diffuse Interface ConceptDistance

Or

The evolution of microstructure with time is assumed to be

proportional to the variation of the free energy functional with

Allen-Cahn Equation:

p p gy

respect to the phase field variables:

Allen Cahn Equation:

Wh G t t l f f th i t tWhere G = total free energy of the microstructure,

Mφ= Order parameter Mobility that can be related to interface mobility (M)related to interface mobility (M)

Allen S. M. , Cahn J. W., Acta Metall., 1979, 27, 1085– 1095

Conservative Phase field variables :

Concentration / mole fraction (c)Concentration / mole fraction (c)

Cahn-Hilliard Diffusion Equation :

Where L ̋ related to the Diffusional mobility of M Nb through

OUTLINE


Phase Field Model

Model Development



- Model Parameters


Conclusions

The total free energy functional of the microstructure (G):

Homogeneous free energy density for

phases with no gradient

Energy associated with local gradient

Vm = Molar volume (assumed to be constant for both the phases )

εφ = the gradient energy coefficient for order parameter

Temperature (T K) is taken as constant in both phases

d h id h d i--- due to the rapid heat conduction

Cahn JW, Hilliard JE, J. Chem. Phys., 1958, 28, 258–267

Construction of Homogeneous free energy density

g(c,φ,T)= Interpolation function + Double-well function

I t l ti f ti + D bl ll f tiInterpolation function + Double-well function

free energy expressions of the

coexisting phases (α&β)

Weight function p(φ) = 1in β= 0 in α

• zero at both the phases &• zero at both the phases & maximum value at φ=.5Homogeneous free energy

expressions

& w - can be adjusted to fit interfacial energy

Thermodynamic analysis of stable phases in Zr-Nb system and calculation of phase diagram" by Armando Fernandez Guillermet and SGTE database for pure element by AT Dinsdale .

(ε φ ) Gradient free energy Coefficient of φ:.

Interface thickness is a balance between two opposing effects.Interface thickness is a balance between two opposing effects.

1. The interface tends to be sharp to minimize the volume of material

where 0<φ<1.

2. The interface tends to be diffuse to reduce the energy associated with the gradient of φ, g φ,

3. For pure material interface thickness ( δ) is related with ε and w by the expression ,

4. Similarly interfacial energy (σ) is related to them as4. Similarly interfacial energy (σ) is related to them as

5. Combining the above two expression, w becomes , w= 3*σ/δ

• In the present model εφ and w are assumed as independent of temperature and composition.

i i f i l ( )

This orientation relationship results in a coherent or semi-coherent interface of very low energy.

Anisotropy in Interfacial Energy (σ)

It is the free energy associated with the compositional and/orIt is the free energy associated with the compositional and/or structural in-homogeneities present at interfaces.

The extent of anisotropy in interfacial energy determines the morphology of the phase.morphology of the phase.

• In case of Widmanstatten morphology,

Incoherent InterfaceHi h t i t f i l

Coherent /Semi-coherent interface • Highest interfacial energy,

max. interface thickness• High Mobility

interface• Lowest interfacial energy &

min. interface thickness• Low mobility

This anisotropy in interfacial energy can be introduced through Anisotropy function:

According to McFadden et al Diffuse interface thickness &According to McFadden et al. Diffuse interface thickness & interfacial energy follow the same anisotropy function

σ = 0 3J/m2σ o = 0.3J/m2

(typical value for incoherent phase interfaces)

δ o = 5 nm

interfaces)

Where is the extent of anisotropy,Interfacial energy of in-coherent interface (ic)

f i l f h i f (Interfacial energy of coherent interface (c

Loginova I, Årgen J, Amberg G, Acta Mater, 2004,52 (13), 4055–4063.

In the final form, the phase field equations become,

* C

* A* * B BD

A, B and D specify the operating points for widmanstatten plates &

C for planar growth respectively.

Molar Volume Vm (m3/mol) 1.4060e-5

Simulation parameters for Phase Field Model

m

Incoherent Interface

thickness

δ0 (nm) 5

Interfacial Energy of

incoherent interface

σ0(J/m2

) 0.3

Nb diffusion in α Zr D hcp 6 6*10-10*exp( 15851 4/T)Nb diffusion in α-Zr DNbhcp

(m2/sec)

6.6*10 10*exp(-15851.4/T)

Nb diffusion in β-Zr DNbbcc (m2/sec) 9*10-9*(T/1136)^18.1*

exp(-(25100+35.5*(T-

1136))/(1.98*T))

Initial state: Homogeneous β with a very thin layer of having Nb concentration determined from phase diagram.

OUTLINE


Phase Field Model

Model Development



- Model Parameters


Conclusions

Mullins- Sekerka(M-S) model : Interface instability

They described the conditions for the onset of a perturbed interface & thescale of such a perturbation for both liquid-solid & solid-solid phasetransformations.transformations.

According to Townsend & Kirkaldy,Widmanstätten plate spacing = f( M–S type instability)

AB

λ

In this PF simulations M-S instability theory is assumed.

A perturbed interface with wavelength λ, A & B are the highest & lowest point of the interface.

The nucleation event can be introduced in simulations in two ways:-Implicit event: Adding suitably amplified noise term in source term of the equation.q-- Explicit event is free from this shortcoming.

For studying growth of single widmanstatten lath equilibrated protrusion was made in the planar surface of allotrimorph .

Phase field Simulations performed using single protusion at different temperatures :

Widmanstatten Lath Formation at operating point B &Dp g p

Distribution of φ during lath formation Concentration profile (c) during lath formation for Zr-2.5Nb

•When >20, widmanstatten plate , pgrows otherwise initial perturbation decays

• Movement of planar interface is restricted due to solute accumulation and growth of tip leads to lath formation.

Concentration profile (c) during lath formation for Zr-1Nb

Multiple Lath Formation:

C t ti fil ( l f ti f Nb) Concentration profile(mole fraction of Nb) during growth of multiple lath from allotrimorph α at same temperature.

1 Interaction of diffusion field of neighbouring protrusions1. Interaction of diffusion field of neighbouring protrusions change the morphology ( width )of the growing phase.

2. With increasing time inter lath location becomes rich in solute gcontent as the diffusion field of neighboring laths overlapped and prevents further widening of each lath.

ff f ( i f i f i l f i h i f hEffect of ( ratio of interfacial energy of incoherent interface to the coherent interface)

2.0x10-5

2.1x10-5

)

4.0

1.7x10-5

1.8x10-5

1.9x10-5

2.0x10

ning

rate

(m/s

ec)

2.5

3.0

3.5

g ra

te(

m/s

ec)

1.3x10-5

1.4x10-5

1.5x10-5

1.6x10-5

Plat

e Le

ngth

en

1.5

2.0

Plat

e W

iden

ing

The lengthening rate of single plate increases linearly with value where as widening rate decreases

0.00 0.01 0.02 0.03 0.04 0.05 0.06

1/(1+0.00 0.01 0.02 0.03 0.04 0.05

1.0

1/(1+

where as widening rate decreases.

In case of multiple lath the phase fields of neighboring plates interact hindering the growth of plates in width direction.

a Effect of Temperature on morphologyb

Effect of Temperature on morphology

Lower temperature (850K) (Plate width less)

Higher Temperature (890K) (wider Plate Width)(Plate width less) (wider Plate Width)

Classical Diffusional Planar Growth at operating pt. C (1054K) at low d liundercooling

• Protrusion decays, planar interface grows• Allotrimorph

More or less uniform distribution of concentration field (mole fraction of Nb) across the entire interface leads to planar growth (low undercooling).

Planar Growth at operating pt. C (1054K) at low undercooling

Planar interface

Distribution of concentration profile (mole fraction of Nb) p ( )during development of allotrimorphs from the protrusions of

grain boundary at 1054K,

The black line denotes the initial position of the interface.The black line denotes the initial position of the interface.

Effect of Initial protrusion size

Growth of lath also dependent on initialGrowth of lath also dependent on initial

protrusion size.

Wider protrusions grow fast.

Due to overlapping of diffusion field certain

protrusions may not grow at all.

OUTLINE


Phase Field Model

Model Development



- Model Parameters


Conclusions

1. The growth of widmanstatten plates in Zr-2.5 Nb alloy has beeng p y

modeled taking thermodynamic and kinetic data of Zr-Nb system as input.

2. The effect of temperature and parameter during growth of single and

multiple side plates has been evaluated.

3. The lengthening rate of single plate increases linearly with value where

as widening rate decreasesas widening rate decreases.

4. In case of multiple laths the phase fields of neighboring plates interact

hindering the growth of plates in width direction resulting in different

aspect ratio compared to single lath.

5. The side plates grow in a range of temperature. Higher temperature

favours formation of wider side plate.

6. At very high temperature with low under cooling classical diffusional

planar growth is observed rather than widmanstatten growth. Asplanar growth is observed rather than widmanstatten growth. As

temperature is lowered, the movement of planar surface is restricted and

widmanstatten growth is favored.

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