Rapid Solidification

Transcription

1 La métallurgie en fabrication additive novembre Arts et Métiers ParisTech Rapid Solidification Michel Rappaz Emeritus Professor - Ecole Polytechnique Fédérale de Lausanne MRC-Consulting, Lausanne Switzerland 1

2 Table of contents 1. Introduction 2. What is rapid solidification? 3. Mullins-Sekerka analysis Absolute stability 4. Departure from equilibrium Solute trapping Attachment kinetics 5. A few results 6. Conclusion 2

3 A few references Fundamentals of Solidification, Chap. 7 W. Kurz and D. Fisher (Trans Tech Pub, 1998) Solidification, Chap. 2 J. Dantzig and M. Rappaz (PPUR, 2009) Rapid Solidification W. J. Boettinger (notes, Zermatt seminar, 1988) Banded Solidification Microstructures W. Kurz and R. Trivedi, Met. Mater. Trans. 27A (1996) 625 Rapid solidification processing and microstructure formation W. Kurz and R. Trivedi, Mater. Sc. Engng A 179/A I80 (1994) 46 Microstructure and Phase Selection in Laser Treatment of Materials W. Kurz and R. Trivedi, Trans. ASME 114 (1992) 450 3

4 1. Introduction In several solidification (welding) processes, a very fast solidification speed of the interface is expected Atomisation tower Droplet 4

5 1. Introduction In several solidification (welding) processes, a very fast solidification speed of the interface is expected Planar flow casting Ribbon 5

6 1. Introduction In several solidification (welding) processes, a very fast solidification speed of the interface is expected Heat source Liquid pool Weld trace Melting Solidification Spot welding / flash melting Continuous welding 6

7 1. Introduction Heat flow will determine the solidification speed. For a pure substance (without a mushy zone): A small example: flash melting of 1 kw (effective) for 10 ms on pure Al V melt 1 mm 3 The thermal diffusion layer is also on the order of 1 mm. Assuming no thermal gradient in the liquid, the solidification speed is on the order of: 7

8 1. Introduction Warning: for an alloy, the heat balance conditions the speed of the isotherms, not necessarily that of the solid-liquid interface! A small example: recalescence during atomisation, but with an average grain size of 100 µm 8

9 2. What is rapid solidification? To answer this question, we have to understand the various departures from equilibrium at a local scale (microstructure): Full chemical equilibrium No chemical potential gradients, i.e., locally uniform composition of phases solid liquid 9

10 2. What is rapid solidification? To answer this question, we have to understand the various departures from equilibrium at a local scale (microstructure): Only the interface is at local equilibrium including curvature effects, i.e., the phase diagram is corrected for the Gibbs-Thomson effect 10

11 2. What is rapid solidification? To answer this question, we have to understand the various departures from equilibrium at a local scale (microstructure): Metastable local interfacial equilibrium The stable phase cannot nucleate (or grow fast enough) The metastable phase diagram can still be still be used for the interface Tempering of triacylglycerols (cocoa butter) Ph. Rousset, PhD EPFL 11

12 2. What is rapid solidification? To answer this question, we have to understand the various departures from equilibrium at a local scale (microstructure): Interfacial non-equilibrium The phase diagram does no longer apply for the interface The chemical potentials are no longer equal at the interface J. C. Baker and J. W. Cahn in Solidification, ASM, Metals 12 Park, OH, p. 23, 1971.

13 3. Mullins-Sekerka analysis Steady-state solution of a planar front (growth at the solidus temperature) solid liquid undercooled zone Constitutional undercooling (and instabilities) occur when: 13

14 3. Mullins-Sekerka analysis In order to analyse the development of instabilities, Mullins and Sekerka developed a 1 st -order analysis by perturbing the interface The solute profile in the liquid is approximated by: solid liquid From the diffusion equation, one deduces: 14

15 3. Mullins-Sekerka analysis The temperature at the interface T(z*) must be equal to the liquidus temperature, taking into account the curvature effect This allows to determine A Finally, from the solute balance at the interface, one has 15

16 3. Mullins-Sekerka analysis From this equation, we can take the terms independent of S (0 th -order) and those linear in S (1 st -order). One gets: O th -order The terms cancel, showing that the 0 th -order solution is indeed satisfied 1 st -order After some rearrangement and introducing the value of A, one obtains the important result: 16

17 3. Mullins-Sekerka analysis Analyses of a few situations: Small velocity (v*/d l << ω): constitutional criterion for a stable planar front Marginal stability mode ω i : smallest wavelength marginally stable Absolute stability: as m l G C >> G at high velocity and the last term becomes dominant, one can show that there is no unstable mode when: 17

18 3. Mullins-Sekerka analysis In summary, below v C or above v a, the planar front is stable. In between, there is a range of unstable modes. Thermal length Log λ i [m] Solute length Capillary length Log v* [m/s] 18

19 3. Mullins-Sekerka analysis At low speed, the planar front/cells/dendrites transition has been studied extensively. A full theory of dendrite tips kinetics (including anisotropy) has been developed for 100 dendrites (solvability). The so-called marginal stability criterion gives the following trend for the tip temperature. plane front cells dendrites plane front? 19

20 4. Departure from equilibrium Mullins-Sekerka analysis is still based on the assumption that the interface is at equilibrium. However, close to or before the absolute stability limit is reached at high velocity, departure from interfacial equilibrium can occur. Solute trapping: for alloys, solute elements may not have enough time to diffuse across the solidliquid interface low speed high speed 20

21 4. Departure from equilibrium In order to go from equilibrium to partition-free solidification, several models have been developed, that of Aziz et al being the most widely used 21

22 4. Departure from equilibrium In order to go from equilibrium to partition-free solidification, several models have been developed, that of Aziz et al being the most widely used 22

23 At very high velocity, when solidification is partition-less, another contribution can finally contribute to the interface kinetics Attachment kinetics: 4. Departure from equilibrium This contributes significantly for metals only at very high speed. When all these contributions are put together, one has the following scheme: 23

24 5. A few results Band structures observed in longitudinal sections of Al Fe specimens after surface remelting at high speed with a CO2 laser Al 4 wt%fe v laser = 2 m/s Al 0.5 wt%fe v laser = 9 m/s M. Gremaud PhD thesis #885, EPFL (1990) 24

25 Bands made of α-al + θ prec. α-al + θ Al 2 Cu 5. A few results Microstructures in laser remelted surfaces of eutectic Al Cu alloys Al 33 wt%cu α-al + θ Al 2 Cu M. Zimmermann PhD thesis #899, EPFL (1990) 25

26 5. A few results Ondulated eutectic structure in Al 33wt%Cu alloys Al 33 wt%cu v* = 0.42 m/s v laser = 1 m/s M. Zimmermann PhD thesis #899, EPFL (1990) 26

27 5. A few results Microstructure map of Al Cu alloys as a function of the nominal composition and velocity of the interface 27

28 6. Conclusion The term Rapid solidification is ambiguous and can encompass: Formation of an unexpected (metastable) phase due to nucleation and/or growth kinetics True departure from equilibrium of the solid-liquid interface (solute trapping, attachment kinetics) Attachment kinetics (for a pure metal) occurs at very high speed (a fraction of the sound velocity) Solute trapping becomes important when: With The velocity of the process (e.g., traverse speed of the laser) IS NOT that of the solid-liquid interface: angle of columnar front, equiaxed grains Observations at high speed are required before any conclusion (and simulation)! 28

29 Et si tout n était pas très clair pour vous COURS DE SOLIDIFICATION Villars, Suisse 29 mai 3 juin 2016 Calcom ESI EPFL solidification.course@esi-group.com 29