Phase field modeling of Microstructure Evolution in Zirconium base alloys

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1 Phase field modeling of Microstructure Evolution in Zirconium base alloys Gargi Choudhuri, S.Chakraborty, B.K.Shah, D. Si Srivastava, GKD G.K.Dey Bhabha Atomic Research Centre Mumbai, India th ASTM International Symposium on Zirconium in Nuclear Industry, Feb 3-7, 2013, Hyderabad, India

2 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

3 Motivation Generation of desired microstructure with required texture in commercial Zr-Nb alloys has been achieved through empirical formulation and traditional trial and error method. Due to cost effectiveness of Computational method over traditional trial and error method, alloy development and designing of desired microstructure through modeling and simulation route is gaining ground day by day. Zr 862 o C 610 o C M s ( ') M s ( ) 18.5 I + II Temp perature ( o C) 25 '+ ' hcp metastable wt % Nb 25 I ( Zr )+ I ( Zr ) I ( Zr ) I ( Zr ) + II ( Nb ) 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 alpha Allotrimorph alpha Widmanstatten alpha (parallel plate/ basket weave morphology) Martensitic microstructure (lath/plate morphology) Omega phase Hydride formation

Zr rich corner of Zr-Nb phase diagram o 862

5 400 M s ( ) 25 '+ metastable Zr 5 10 15 20

Microstructure: Distribution of phase as

$volume fraction ~ 3 % transformation bcc$ superimposition of the variants could be

4 Zr rich corner of Zr-Nb phase diagram o 862 o C o C I + II M s ( ') M s ( ) 25 '+ metastable Zr wt % Nb Temper rature ( o C) Microstructure: Distribution of phase as fine spherical precipitates within the equiaxed matrix grains. Average precipitate size ~ 30 nm precipitate volume fraction ~ 3 % transformation bcc stereographic triangle where the superimposition of the variants could be seen along with the reflections in [110], [210], [311] and [211] zones C - 30 mins + Q S. Neogy*, S. Neogy, K.V. Mani V. Mani Krishna, krishna D. Srivastava and G.K. D. Dey Srivastava and G. K. Dey, Philosophical Magazine Phil. Mag Vol. 91, No. 35, 21 December 2011,

5 Zr-2.5 Nb alloy Gas quenched from + β 50 C/sec 25 C/sec 10 C/sec Widmanstatten martensite 10 C/sec Different morphologies Saibaba.et al., J. of ASTM Int., June 2011, 8, Issue 6

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 seen. G.K.Dey et.al Journal of Nuclear Materials 224 (1995)

7 (a) Bright field and (b) dark field micrograph showing the formation of the plate shaped internally twinned a phase at the grain boundary. G.K.Dey et.al Journal of Nuclear Materials 224 (1995) R. Tewari, D. Srivastava, G.K. Dey, J.K. Chakravarty, S. Banerjee, Journal of Nuclear Materials 383 (2008) G. K. Dey and S. Banerjee, Journal of Nuclear Materials 125 (1984) 219

8 S. Neogy*, K.V. Mani Krishna, D. Srivastava and G.K. Dey Philosophical Magazine Vol. 91, No. 35, 21 December 2011,

9 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

10 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. Phase field model : Diffuse Interface Computational approach to modeling and predicting meso-scale morphological & microstructure evolution. Entire microstructure is represented continuously by a non conservative phase field variable, φ, (order parameter/crystal structure) where φ=1,φ=0 (at precipitate & matrix/at two phases) & 0<φ<1 represent the interface region. ameter Or rder par It is a Diffuse Interface Concept Interface thickness δ Distance

11 The evolution of microstructure with time is assumed to be proportional p to the variation of the free energy functional with respect to the phase field variables: Allen-Cahn Equation: Where G = total t free energy of the microstructure, t M φ = Order parameter Mobility that can be related to interface mobility (M) Allen S. M., Cahn J. W., Acta Metall., 1979, 27,

12 Conservative Phase field variables : Concentration / mole fraction (c) Cahn-Hilliard Diffusion Equation : Where L related to the Diffusional mobility of M Nb through

13 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

the phases ) ε φ = the gradient energy coefficient for order parameter Temperature (T K) is taken as

14 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 ) ε φ = the gradient energy coefficient for order parameter Temperature (T K) is taken as constant in both phases --- due to the rapid heat conduction Cahn JW, Hilliard JE, J. Chem. Phys., 1958, 28,

energy expressions & zero at both the phases & maximum value at φ=.

15 Construction of Homogeneous free energy density g(c,φ,t)= Interpolation function + Double-well function Interpolation ti function + Double-well ll function free energy expressions of the coexisting phases (α&β) Weight function p(φ) = 1in β = 0 in α Homogeneous free energy expressions & zero at both the phases & maximum value at φ=.5 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

The interface tends to be sharp to minimize the volume of material where

The interface tends to be diffuse to reduce the energy associated with

For pure material interface thickness ( δ) is related with ε and w by

16 . (ε φ ) Gradient free energy Coefficient of φ: 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, 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 as 5. Combining the above two expression, w becomes, w= 3*σ/δ In the present model ε φ and w are assumed as independent of temperature and composition.

17 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/or structural in-homogeneities present at interfaces. The extent of anisotropy in interfacial energy determines the morphology of the phase. In case of Widmanstatten morphology, Coherent /Semi-coherent interface Lowest interfacial energy & min. interface thickness Low mobility Incoherent Interface Highest interfacial i energy, max. interface thickness High Mobility This anisotropy in interfacial energy can be introduced through Anisotropy function:

18 According to McFadden et al. Diffuse interface thickness & interfacial energy follow the same anisotropy function σ =03J/m o 0.3J/m 2 (typical value for incoherent phase interfaces) δ o = 5 nm Where is the extent of anisotropy, Interfacial energy of in-coherent interface ( ic) Interfacial energy of coherent interface ( c Loginova I, Årgen J, Amberg G, Acta Mater, 2004,52 (13),

19 In the final form, the phase field equations become, * C * A * * B D A, B and D specify the operating points for widmanstatten plates & C for planar growth respectively.

20 Simulation parameters for Phase Field Model Molar Volume V m (m 3 /mol) e-5 Incoherent Interface thickness Interfacial Energy of incoherent interface m δ 0 (nm) 5 σ 0(J/m 2 ) 0.3 Nb diffusion in α-zr D hcp 6*10-10 Nb 6.6*10 *exp( /t) (m 2 /sec) Nb diffusion in β-zr D Nb bcc (m 2 /sec) 9*10-9 *(T/1136)^18.1* exp(-( *(t- 1136))/(1.98*T)) Initial state: Homogeneous β with a very thin layer of having Nb concentration determined from phase diagram.

21 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

22 Mullins- Sekerka(M-S) model : Interface instability They described the conditions for the onset of a perturbed interface & the scale of such a perturbation for both liquid-solid & solid-solid phase transformations. According to Townsend & Kirkaldy, Widmanstätten plate spacing = f( M S type instability) A B λ A perturbed interface with wavelength λ, A & B are the highest & lowest point of the interface. In this PF simulations M-S instability theory is assumed. The nucleation event can be introduced in simulations in two ways: -Implicit event: Adding suitably amplified noise term in source term of the equation. -- Explicit event is free from this shortcoming. For studying growth of single widmanstatten lath equilibrated protrusion was made in the planar surface of allotrimorph.

23 Phase field Simulations performed using single protusion at different temperatures : Widmanstatten Lath Formation at operating point B &D Distribution of φ during lath formation Concentration profile (c) during lath formation for Zr-2.5Nb When >20,, widmanstatten plate grows otherwise initial perturbation decays Concentration profile (c) during lath formation for Zr-1Nb Movement of planar interface is restricted due to solute accumulation and growth of tip leads to lath formation.

$Multiple Lath Formation: Concentration ti profile(mole l fraction of Nb) during growth of multiple lath from$ Interaction of diffusion field of neighbouring protrusions change the morphology ( width )of the growing

Interaction of diffusion field of neighbouring protrusions change the morphology ( width )of the growing

24 Multiple Lath Formation: Concentration ti profile(mole l fraction of Nb) during growth of multiple lath from allotrimorph α at same temperature. 1. 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 g content as the diffusion field of neighboring laths overlapped and prevents further widening of each lath.

25 Effect of ( ratio of interfacial i energy of incoherent interface to the coherent interface) 2.1x Plate Lengthen ning rate (m/sec) 2.0x x x x x x x x /(1+ Plate Widening rate( m/sec) /(1+ The lengthening rate of single plate increases linearly with value 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.

26 b a Effect of Temperature on morphology Lower temperature (850K) (Plate width less) Higher Temperature (890K) (wider Plate Width) Classical Diffusional Planar Growth at operating pt. C (1054K) at low undercooling 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).

$C (1054K) at low undercooling Planar interface Distribution of concentration profile (mole fraction of Nb) during$ development of allotrimorphs from the protrusions of grain boundary at 1054K, The black line denotes the initial

development of allotrimorphs from the protrusions of grain boundary at 1054K, The black line denotes the initial

27 Planar Growth at operating pt. C (1054K) at low undercooling Planar interface Distribution of concentration profile (mole fraction of Nb) during development of allotrimorphs from the protrusions of grain boundary at 1054K, The black line denotes the initial position of the interface. Effect of Initial protrusion size Growth 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.

28 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

29 1. The growth of widmanstatten plates in Zr-2.5 Nb alloy has been 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 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. As temperature is lowered, the movement of planar surface is restricted and widmanstatten growth is favored.

30 THANK YOU

Part IV : Solid-Solid Phase Transformations I Module 2 : Cellular precipitation

Part IV : Solid-Solid Phase Transformations I Module 2 : Cellular precipitation 2. Cellular precipitation 2.1 Motivation Consider the microstructure (schematic) shown in Fig. 18. This is a typical microstructure