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Solidification modeling: from electromagnetic levitation to atomization processing

Ch.-A. Gandina*, D. Tourreta, b , T. Volkmannb, D. M. Herlachb,

A. Ilbagic, H. Heneinc

(a) MINES ParisTech, CEMEF UMR CNRS 7635, 06904 Sophia Antipolis, France;

(b) Deutsches Zentrum für Luft- und Raumfahrt, Institut für Materialphysik im Weltraum, 51147 Köln, Germany;

(c) University of Alberta, Chemical and Materials Engineering Department, Edmonton, Alberta, T6G 2G6, Canada;

(*) Corresponding author: Charles-Andre.Gandin@mines-paristech.fr

Abstract

A segregation model for multiple phase transformations is presented together with solidification experiments using electromagnetic levitation. Comparison is made between predictions and measurements. Thanks to coupling with thermodynamic equilibrium calculations, as well as the treatment of dendritic, peritectic and eutectic reactions, quantitative agreement is found for the cooling curves and the fraction of microstructures measured, thus providing a validation of the kinetics of phase transformations. Application is then possible for the impulse atomization process. It reveals the importance of the competition between the various microstructures formed from the melt and the role of the nucleation and growth kinetics.

Keywords

Solidification, Segregation, Dendritic, Peritectic, Eutectic, Electromagnetic Levitation, Atomization

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1. Introduction

Modeling of solidification during atomization is rarely compared with segregation measurements. The main reason is the difficulty to measure the thermal history of an individual droplet during its flight in the atomization tower. This data is yet required for calibration of model parameters such as the cooling rate prior and during solidification.

In the present contribution electromagnetic levitation is proposed as an experimental technique to study segregation taking place upon solidification of an almost spherical system. Because the volume of the sample is a few cubic millimeters and is fixed in space, it is possible to use a pyrometer to record its temperature evolution during solidification. Measurements are conducted on an aluminum-nickel (Al-Ni) alloy at a composition for which successive dendritic, peritectic and eutectic reactions are expected. Metallurgical characterizations are performed to produce distribution map of the volume fraction of the microstructures and the average dendrite arm spacing. Results from impulse atomization [1] are also presented for Al-Ni alloys, permitting to assess the role of cooling rate on solidification.

Volume averaged segregation models offer an alternative to simple analytical models (i.e. lever rule or Gulliver-Scheil [2,3]) and to numerical solution of the complete diffusion problem by direct tracking of the phase interfaces [4,5], heavily demanding on computing resources. With the averaging method [6-11], the physical quantities are averaged over a representative elementary volume and the conservation equations for averaged quantities are solved. This approach leads to fast simulations compared to direct tracking methods, while still accounting for microstructural features, such as dendrite arm spacing. It is presented hereafter into details and applied to electromagnetic levitation and impulse atomization.

The paper starts with a presentation of the experimental set-ups used to collect data during and after solidification of Al-Ni samples. The segregation model for multiple phase transformations is then summarized [12-15], together with its coupling with thermodynamic equilibrium calculations using the CalPhaD method [16-19]. The solidification of a spherical domain is simulated and the results are compared with the measured cooling curves and fraction of phases in a levitated sample. Finally, application to atomization is presented. Original interpretation is provided to explain the evolution of the phase fractions observed in atomized particles. A discussion on the calibration and the necessary data for application to the atomization process is also given.

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2. Electro-Magnetic Levitation

The Electro-Magnetic Levitation (EML) is the most suitable technique to levitate stable metallic samples and to achieve direct measurements in undercooled states [20, 21]. An Alternating Current (AC) flowing through a copper coil produces an Electro-Magnetic (EM) field generated in a conically shaped coil. A metallic sample placed within the EM field is levitated by the Lorentz force that compensates the gravitational force. It is also heated by the Eddy currents. In order to limit the contamination of the melt and the evaporation of light elements, the set-up is placed within an ultra high vacuum chamber, which is evacuated to a pressure of 10-8 mbar before backfilling with high purity inert gas (Helium 6.0). Illustration of the EML apparatus is provided in Figure 1(a).

The droplet is melted and stabilized several hundred Kelvin above its melting temperature during 10 to 20 seconds – which is useful as well to remove the possible residual oxides. Then, the power of the current in the coil is rapidly decreased to the minimum necessary for the levitation, while cooling gas (He) jets are opened. The current in the coil and the opening of the gas jet are then kept the same up to the complete solidification of the droplet, while the operating pressure is maintained using a monitoring vacuum pump.

During the experiment, the temperature of the droplet is measured with a digital infrared pyrometer (Impac IGA 10-LO). The solidification involves multiple phase transformations and the identity of the phases facing the pyrometer changes. A significant gap between a temperature calibrated regarding emissivity of a liquid or a solid was already highlighted [22]. Since the theoretical equilibrium temperatures of reactions are known from thermodynamic calculations, the emissivity of the sample can be calibrated regarding its melting curve. Upon heating, the melting of a solid comes with a change of slope in the temperature-time profile due to the heat of melting needed for the transformation and the change in heat capacities of the present phases. On the raw measured temperature, melting events of the solid phases occur at non-calibrated measured temperatures. From the theoretical equilibrium temperature, the actual temperature T then is calibrated using the Wien’s law.

The solidified samples are analyzed with the back scattered electron detector of a Scanning Electron Microscopy (SEM) to acquire grey scale digital images with a high magnification. Since the droplets usually exhibit an axis of symmetry corresponding to the axis of the coil, the structures observed in a median cross-section of the droplet are representative of its overall volume. The SEM images are post-processed with the software ImageJ [23]. Several images are captured, cropped and combined into a high definition image of the full cross-section. The grey level thresholds corresponding to the interfaces between structures are determined and the surface fraction of microstructures, representative of the

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whole volume, is estimated. Furthermore, if the microstructure is assumed isotropic at the scale of the droplet, the secondary Dendrite Arm Spacing (DAS), may be evaluated by measuring the spacing between intercepts with the boundaries defined by the eutectic microstructure with evenly spaced lines through the cross-section image.

(a) (b) Figure 1. Schematics of the apparatus for (a) Electro-Magnetic Levitation (EML) [14] and (b)

Impulse Atomization.

3. Impulse atomization

The Impulse Atomization (IA) technique takes advantage of the fact that a liquid jet emanating from a capillary can be destabilized when disturbed by a particular shock wave with a given frequency. The impulse unit includes a plunger made of refractory material attached to a pulsator which impulses the molten metal through a series of capillaries on the bottom of the crucible. The impulses generate discontinuous streams of molten metal, which in turn break up into droplets. The droplets fall in a static gas atmosphere and solidify as spherical particles as illustrated in Figure 1(b). [1]. Collected particles were sieved into different size ranges according to Metal Powder Industries Federation Standard 05 [24].

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No temperature could be measured upon atomization runs on in-flight particles. Similarly, due to the size of the particles that typically varies in the range [50-500] μm, the detailed analyses performed from SEM images could not be achieved for the particles. Quantitative metallurgical characterization thus requires advanced methods such as Transmission Electron Microscope (TEM), Scanning Electron Microscope (SEM), Electron Backscatter Diffraction (EBSD), Neutron Diffraction (ND) and X-ray Diffraction (XRD) together with Rietveld analyses. Phase fractions were determined as a function of the particle size, the alloy compositions and the gas atmosphere.

4. Modeling

In the present contribution, a volume averaged segregation model is coupled with thermodynamic CalPhaD calculations [16-19]. The model is applied to a one-dimensional spherical domains [7, 8] that represents a single EML processed droplet or a single IA particle. The domain radius R is schematized in Figure 2, with external area A and volume V. Several solid structures, numbered s1, s2,…, sn form in a liquid phase, denoted as l. Each solid is assumed to nucleate at the center of the domain and to propagate by radial growth. Several zones are defined, which follow the developing solids, and are numbered as (1), (2),…, (n). Several potential solidification steps may be considered. Figure 2 illustrates the specific case of three simultaneous reactions involving a dendritic structure s1, a peritectic structure s2 and a eutectic structure s3. Each structure sk grows with its corresponding zone (k) of radius R(k), whose velocity vsk is defined by the growth kinetics of solid sk. Each new zone produces new boundaries with other zones and new interfaces within the zone. Distinction is made between “interfaces”, which are real physical interfaces between two different phases within the same zone, e.g. s(1)

1 /l(1), and “boundaries”, which are model-defined frontiers splitting the same phase between two different zones, e.g. l(1)/l(0). The middle part of Figure 2 shows the existing interfaces (plain lines) and boundaries (dashed lines) where mass transfers are considered. In each part of a phase – defined as a given phase si or l in a given zone (n), s(n)

i or l(n) – and at each phase interface and boundary, diffusion fluxes are considered both in the radial growth direction and at the scale of the microstructure, i.e. between the secondary dendrite arms. As an exception, no diffusion is assumed within the final eutectic structure, which forms with the same composition as the remaining liquid it replaces. Specific surfaces and diffusion lengths describe respectively the microstructure geometry and the solute diffusion fluxes at interfaces and boundaries [13]. The bottom part of Figure 2 shows the representative secondary dendrite arm spacing for each zone. The white dotted rectangles spot the half spacing where plate-like geometry and composition profiles are assumed. Specific surfaces and diffusion lengths are given hereafter.

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General assumptions

The following assumptions are considered:

(i) application to binary alloys with nominal composition of the solute element w0 in weight fraction or x0 in atomic fraction;

(ii) equal and constant densities for all phases α, ρα=ρ; (iii) uniform domain temperature, T; (iv) no advection; (v) closed system with respect to mass exchange with its surrounding at boundary R; (vi) in phase α, of interdiffusion coefficient Dα and solute element concentration wα in

mass fraction, the diffusion flux of the solute element, jα, follows: jα = -Dα grad(ραwα) = - ρ Dα grad(wα) . (1)

Figure 2. Schematics of the one-dimensional model [14]. Possible solidification steps are schematized involving simultaneous (1) dendritic, (2) peritectic and (3) eutectic growths. On top are spotted the indices of the growing zones, their constituting phases, radii and growth

velocities. The middle schematic shows the existing “phases” si(m), interfaces (plain lines) and

boundaries (dashed lines). The lower part illustrates the secondary dendrite arm spacing, representative of the microstructure behavior over the zone.

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Mass conservations

Under the above assumptions, the averaged mass conservation equations over a phase α and over an interface/boundary α/β between phases α and β respectively write:

∂gα

∂t = Σα/β

⎝⎛

⎠⎞|

|Sα/β vα/β , (2)

vα/β + vβ/α = 0 , (3)

where gα is the volume fraction of α and vα/β is the average normal velocity of α/β. The specific surface of α/β, Sα/β, is defined from its area Aα/β as:

Sα/β= Aα/β/V . (4)

The average conservation equations for the solute mass over a phase α, and at an interface or boundary α/β are:

gα ∂∂t ⎝

⎛⎠⎞|

|<wα>α = Σα/β

⎣⎢⎡

⎦⎥⎤

Sα/β ⎝⎛

⎠⎞|

|wα/β - <wα>α

⎝⎜⎛

⎠⎟⎞

vα/β + Dα lα/β , (5)

⎝⎛

⎠⎞|

|wα/β - wβ/α vα/β +

Dα lα/β

⎝⎛

⎠⎞|

|wα/β - <wα>α +

Dβ lβ/α

⎝⎛

⎠⎞|

|wβ/α - <wβ>β = 0 , (6)

where <wα>α is the intrinsic average composition of a phase α. In the α-phase at α/β, the average composition over the interface is wα/β and the composition gradient ∂wα/∂nα/β is expressed via the length lα/β:

lα/β = - ⎝⎛

⎠⎞|

|wα/β - <wα>α / ∂wα

∂nα/β α/β

. (7)

Furthermore, through a boundary α(m)/α(n) splitting the same phase α into two adjacent zones (m) and (n), the composition is continuous, i.e. wα(m)/α(n)

=wα(n)/α(m). Thus, when written at

a boundary, the first term in the solute balance (6) vanishes and the boundary composition is expressed directly from average compositions and diffusion lengths in α(m) and α(m).

Specific surfaces

Assuming a plate like geometry in the half secondary dendrite arms spacing of size λ2 /2 (sketched in Figure 2) in zone (m), which volume fraction is g(m), the specific surface of α(m)/β(m) writes [13]:

S α(m)

/β(m)

= g(m) 2/λ2 . (8)

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In the propagation direction, boundary densities are expressed from the radii of the zones R(n). The total density of the boundary between zones (m) and (n), with n>m, is distributed to each phase α split into zones (m) and (n). The specific surface of a boundary thus writes:

S α(m)

/α(n)

= gα

(m)

g(m) 3 R (n) 2

R 3 . (9)

Diffusion lengths

Composition profiles are assumed in two directions in order to establish analytical expressions of the solute diffusion lengths at the interfaces and boundaries. Quadratic composition profiles are assumed in λ2 /2. If the phase α(m) is in contact with one only other phase β(m) in zone (m) (e.g. s(2)

1 or l(2) in Figure 2), the composition profile leads to [13]:

lα(m)

/β(m)

= 13

gα(m)

g(m) λ2

2 . (10)

If α(m) is surrounded by two other phases in zone (m) (e.g. s(2)2 in Figure 2), the

composition profile is a piecewise quadratic profile with a symmetry condition in the center. The diffusion length in α(m) at α(m)/β(m) then writes:

lα(m)

/β(m)

= 16

gα(m)

g(m) λ2

2 . (11)

As in the microstructure length scale, quadratic – or piecewise quadratic – profiles are assumed in the solid phases between the radii delimiting the zones. If the phase α(m) exists only in one zone (n) adjacent to (m) (e.g. s(2)

2 in Figure 2), and the radii delimiting zone (m) are R(m) and R(p) (e.g. m=2 and p=3 for zone (2) in Figure 2), the diffusion length at this boundary is:

lα(m)

/α(n)

= ( R(m)-R(p) ) / 3 . (12)

If phase α(m) exists in both zones neighboring (m) (e.g. s(2)1 in Figure 2, with α = s1 in

zone (m = 2), delimited by radii R(m) and R(p=3) at the boundary with s(n=1)1 or s(n=3)

1 ):

lα(m)

/α(n)

= ( R(m)-R(p) ) / 6 . (13)

In the liquid, the analytical expression of quasi-stationary diffusion profile in front of a spherical moving boundary with imposed boundary composition is established [25]. In a

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liquid zone, between boundaries of moving internal radius Ri and a fixed external radius Re, the expression of the liquid/liquid boundary solute diffusion length, l(m)/(n), writes:

l(m)/(n) = Ri

Pee3 - Pei

3 ⎣⎢⎡

⎦⎥⎤

⎝⎜⎛

⎠⎟⎞|

|Pee3- Pei

2- Pei- Iv( Pei )Pee3 - ⎝⎜

⎛⎠⎟⎞|

|Pee2- Pee- 1 - Iv(Pee )Pee

2 e - Pee

e - Pei Pei . (14)

where Pei=Ri vi / Dl and Pee=Re vi / D

l are the solute diffusion Peclet numbers in the liquid, with regard to the internal and external radius, Ri and Re respectively, and vi is the normal average velocity of the internal moving boundary. Expression (14) is equivalent to those derived in previous papers [12, 25], but written in a simpler way (avoiding, as much as possible the double use of radii and Peclet numbers).

Nucleation

The nucleation event of a solid phase si in an undercooled liquid l(i-1) of average composition <wl(i-1)

>l(i-1) takes place when the temperature becomes lower than the prescribed

nucleation temperature of the solid phase, T n si = Teq

si - ΔT n si:

( )||T ≤ T si

n ⇒ ( )||solid si nucleates in a new zone (i) at position r=0 (15)

where Teq si = Teq

si( )<wl(i-1)>l(i-1)|

| is the equilibrium temperature of the solid phase si facing the liquid phase l(i-1).

Heat balance

The volume averaged segregation models usually write a heat balance with global and constant latent enthalpy of fusion and heat capacity [7-13]. Here, the thermodynamic equilibrium calculations give direct access to enthalpies of phases. Thus, if the average enthalpy of a phase α, < Hα>α, depends only on the temperature T and on its average composition <wα>α, the global heat balance of the system writes:

d<H>dt = Σ

α ⎝⎜⎛

⎠⎟⎞|

|∂gα

∂t <Hα>α + gα ∂<Hα>α

∂T ∂T ∂t + gα

∂<Hα>α ∂<wα>α

∂<wα>α ∂t . (16)

In the right hand side of equation (16), the first term stands for the enthalpy of phase transformations. The second term stands for the change of heat capacity, ∂<Hα>α/∂T, and the third term is due to the variation of enthalpy with composition, ∂<Hα>α/∂<wα>α. The latter term is so far neglected in most of the segregation models. A convective external boundary condition is assumed with a heat transfer coefficient hext:

d<H>/dt = - A ρV hext ( T - Text ) . (17)

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Thermodynamics data

The simulations are coupled with thermodynamic equilibrium calculations. Using the CalPhaD method [17-19], phase diagrams are computed, as well as metastable equilibria, as shown in Figure 3. The main advantage of using such a coupling is the possibility to run systematic simulations on any alloy, without a priori knowledge of the solidification sequence. At the interfaces between phases α and β, thermodynamic equilibrium is assumed and compositions on both sides, wα/β and wβ/α, are computed from the temperature T. Knowing the temperature T and the average composition of a phase α, <wα>α, the equilibrium involving this phase in its state (T,<wα>α) is computed. The values of the average specific enthalpy <Hα>α and its partial derivatives with respect to temperature, ∂<Hα>α/∂T, and composition, ∂<Hα>α/∂<wα>α, are thus made available for direct use in the heat balance.

Figure 3. Aluminum-Nickel phase diagram with metastable extensions (dashed) computed with the thermodynamic database PBIN [14, 16-19].

Growth kinetics

The growth kinetics of the solid structures defines the velocities of the boundaries between zones. Here, theories proposed by Kurz et al. [26] and Jackson and Hunt [27] are respectively adopted for dendritic and eutectic growth. The peritectic reaction – i.e. the growth along the existing solid/liquid interface – is usually very fast compared to the peritectic transformation [28] – i.e. the thickening of the peritectic layer surrounding the

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primary dendritic solid. Hence, models for peritectic solidification usually consider an instantaneous peritectic reaction, neglecting the reaction step [29]. Moreover, to date, no dedicated analytical model exists for peritectic growth kinetics. Thus, we use here a simple adaptation of a dendrite tip growth kinetics theory [26], in which the liquid solute supersaturation, i.e. the non dimensional composition gap in front of the solid/liquid interface, accounts for the several liquids facing the growing peritectic solid [13] – e.g. l(0) and l(1) facing s(2)

2 , in Figure 2. The analytical expressions for the microstructure growth kinetics are given hereafter.

The dendrite tip growth kinetics is estimated with the classical law from Kurz et. al. [26]:

vsi = - D l ml/si (wl/si - wsi/l)

π 2 Γ si/l ⎣⎢⎡

⎦⎥⎤|

|Iv-1(Ω si/l)

2 , (18)

Ω si/l = ( wl/si - wsi

l ∞ ) / ( wl/si - wsi/l ) . (19)

Where the liquid composition away from the tip, wsi=1 l ∞, is that of the bulk liquid, <wl(0)>l(0).

The peritectic growth is adapted from the dendrite tip kinetics, assuming a composition of the

liquid away from the growing solid as an average of the several liquid facing the solid [13]:

wsi

l ∞= Σ0≤ j<i( )|

|gl(j)<wl(j)>l(j) / Σ

0≤ j<i( )||g

l(j) . (20)

For instance, for the solid s2 in Figure 2, wsi=2

l ∞=(gl(0)<wl(0)>l(0)+gl(1)<wl(1)>l(1))/(gl(0)+gl(1)).

The growth kinetics of a eutectic (α+β) is computed from Jackson and Hunt model [27]:

v(α+β)/l = ΔTE 2 / ( )|

| 4 Kr Kc , (21)

with the alloy properties, Kc and Kr, the equivalent liquidus slope, m , and the α-phase eutectic fraction, fα

eut:

Kc = ( )|| m (wβ/α

-wα/β) P' / ( )||fα

eut ( 1 - fα eut ) D l , (22)

Kr = 2 m ⎝⎜⎜⎛

⎠⎟⎟⎞Γ α/l sin(θ α )

fα eut ml/α

+ Γ β/l sin(θ β )

( 1 - fα eut) ml/β

, (23)

m =⎝⎛

⎠⎞ ml/α ml/β / ⎝

⎛⎠⎞ ml/α + ml/β , (24)

fα eut = ( wl/α

-wα/β ) / ( wβ/α-wα/β ) . (25)

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Numerical solution

For each interface α/β between two phases, knowing the equilibrium compositions, wα/β and wβ/α, the interface velocity vα/β is computed from the interface mass and solute balances (5) and (6). For each boundary α/β between two zones, knowing the velocity vα/β from microstructure growth kinetics, the composition wα/β = wβ/α is computed from the equations (5) and (6). Then, the system of partial derivative equations describing the evolution of the system involves the averaged mass balance in phase α, equation (2), the solute mass balance in phase α, equation (5) and the global heat balance of the system, equation (16). Its unknowns are the volume fraction of phase α, gα, the average composition of phase α, <wα>α, and the temperature, T. The system consists of first order ordinary derivative equations and may be steep for high nucleation undercoolings. It is solved using an implicit multiple step backward Euler method (also known as Gear method [30]). From the chosen initial value of the time step, Δt0, optimization is further achieved by bounding the current time step, Δt, to a critical value Δtc regarding the growth velocities of the zones:

Δt = Min {Δt0, Δtc} with Δtc = R / ( Maxi

{|v si|} Ncriterion ) . (26)

where Ncriterion is a chosen integer value. Phenomenologically, this means that the fastest growing zone needs at least Ncriterion time steps to go across the entire domain of radius R. The model is implemented in language C++ within Microsoft Visual Studio 2005. Thermodynamic calculations are coupled in language Fortran (with an Intel Visual Fortran Compiler) with the interface TQ 7.0 of ThermoCalc version S [16].

5. EML sample

The sample solidified under electromagnetic levitation is an Al - 25 at.% Ni alloy droplet, also noted Al75Ni25. Figure 4 shows the results of the SEM and image analyses on the droplet. Individual images are captured and combined into a total image of size of 9600*8052 pixels, with a resolution of 1.055 µm/pixel. In Figure 4a is presented the combined secondary electron image of the whole droplet, with a zoom on one individual micrograph. The assumed axis of symmetry of the droplet (z) – i.e. the axis of the levitating coil, vertical ascending during the experiment – is sketched. Four different gray levels appear: (i) a white structure, with a dendritic shape, i.e. the primary dendritic solid s1=Al3Ni2; (ii) a light gray layer surrounding this primary solid, i.e. the peritectic solid s2=Al3Ni; (iii) a dark gray zone filling most of the remaining interdendritic space, i.e. the eutectic structure made of two solids s3=Al3Ni+Al; and (iv) black zones which correspond to the porosity due to solidification shrinkage. The microscope resolution at this magnitude is not high enough to distinguish a

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two-phase lamellar eutectic, thus appearing as homogeneous dark gray. The fractions of dendritic, peritectic and eutectic structures are thus measured, not the fractions of phases.

Figure 4b is the gray level distribution of the full image from 0 to 255, showing the three distinct minima used as threshold values for image processing: 25 (porosity/eutectic), 135 (eutectic/peritectic) and 233 (peritectic/dendritic). The uncertainty ranges, used for error estimation, are marked in dashed lines. The measured volume fraction of porosity is gPor. = 0.0088 ± 0.0007. Since the solidification shrinkage is not taken into account in the modeling, the fractions of microstructures are normalized with respect to the total solid fraction. Then, the volume fractions are: gs1 = 0.295 ± 0.009, gs2 = 0.538 ± 0.028 and gs3 = 0.167 ± 0.020. Figure 4c shows the colored image after the threshold procedure. From the eutectic/peritectic threshold, drawing 100 evenly spaced horizontal lines through the full image, 16014 elementary spacings are counted, and the average DAS is estimated as 35.78 µm.

The solidification of the Al75Ni25 droplet displayed in Figure 4 is simulated with the parameters summarized in Table 1. In order to evaluate the heat transfer coefficient hext, taken constant during the whole solidification sequence of an electromagnetic levitation experiment, the cooling curve recorded for the liquid prior to any solidification event (i.e. above the liquidus temperature) is fitted to the solution of a purely convective heat exchange for a sphere of radius R, density ρ and heat capacity cp:

T(t) = Text + ⎝⎛

⎠⎞|

|T0 - Text exp ⎩⎪⎨⎪⎧

⎭⎪⎬⎪⎫

- 3 hext R ρ cp

t . (27)

The external temperature is set to Text = 293 K. The heat capacity of the melt, cp = 944 J kg-1 K-1, is computed at the liquidus temperature from a CalPhaD calculation at the alloy composition. The Al75Ni25 alloy density is estimated from the densities of pure aluminum and nickel, with the formula given in Table 1, leading to ρ = 3818.5 kg m-3. The mass of the droplet is 890.7 milligrams, which corresponds to an equivalent sphere of radius 3.8187 mm. The fitted heat transfer coefficient is thus hext = 213.8 W m-2 K-1.

The simulation predictions (time evolution of temperature and fraction of microstructures) are plotted and compared with experimental data in Figure 5. Experimental measurements are represented with open symbols, while simulation predictions appear in filled symbols. Figure 5a shows the temperature history. As explained above, the heat transfer coefficient is extracted from the experimental cooling curve by fitting equation (27) in the range 0 ≤ t ≤ 5 s. The apparent undercoolings prior to recalescences, ΔT si, are also measured. Since peritectic and eutectic recalescences are very steep, the measured values are assumed to be nucleation undercoolings with ΔT s2

n = 150 K and ΔT s3n = 15 K. The primary recalescence,

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rather small and smooth, is assumed to be a growth related undercooling and its nucleation undercooling ΔT s1

n is taken equal to 0 K. Figure 5b and 5c show the evolution of the phase fractions and of the normalized radii of the zones, respectively. The latter correspond to the positions of the growing dendritic, peritectic and eutectic envelopes of the microstructures. The first step of the dendritic solidification is the growth of the mushy zone. Then, a slope change in the evolution of the Al3Ni2 fraction occurs when the mushy zone is well developed. Upon nucleation of the peritectic Al3Ni phase at around 25 s, a steep recalescence on the cooling curve is observed. It corresponds to a strong decrease of the primary Al3Ni2 solid fraction, due on the one hand to the remelting of the primary dendritic phase in zone (1) during the peritectic growth in the liquid (very fast, 25 ≤ t ≤ 25.1 s) and on the other hand to the peritectic transformation Al3Ni2 → Al3Ni in zone (2) that continues up to the end of solidification. The SEM measured fractions are spotted as open symbols with their error ranges at the end of the graph. As can be seen, predictions come very close to measurements for the fraction of phases as well as for the temperature evolution.

A detailed analysis by varying simulation parameters shows that the reasons listed hereafter are involved in obtaining this good result.

- Measured cooling rate prior to solidification for adjustment of the heat transfer coefficient, as well as nucleation temperature. These crucial parameters that define the effective heat extraction rate of the system and its ability to nucleate new phases could be adjusted thanks to the measured temperature evolution of the system available in an EML experiment.

- Measured dendrite arm spacing entering the diffusion lengths, as this is crucial for the diffusion kinetics in the solid phase.

- Access to materials properties at equilibrium through the thermodynamic database.

- Dependence of the enthalpy of liquid with composition, ∂<Hl>l/∂<wl>l, for all liquid phases. Explanation involves the large variation of composition of the liquid phases during solidification, while the dependence for the solid phases is less. Similarly, it is found that the role of the variation of enthalpy with temperature, or heat capacity, do not play a significant role here.

- Adjustment of the diffusion coefficient for the solid and liquid phases. In the absence of measured values, these properties were only broadly adjusted (constant typical values for all solid phases and the liquid phase).

- Adjustment of the Gibbs-Thomson coefficient for the Al3Ni2/l interface, with similar consideration as for the diffusion coefficient.

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Figure 4. Scanning Electron Microscopy (SEM) and image analyses of a solidified Al75Ni25 droplet. (a) Combined SEM secondary electron images (full image size: 9600*8052 pixels),

with droplet symmetry axis (z) and detail of one individual micrograph (size: 1024*768 pixels); (b) gray level histogram, showing thresholds and uncertainty ranges (dashed lines);

(c) image after multiple thresholding, showing microstructures and porosity [14].

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Table 1. Simulation parameters Parameter Symbol Unit Source Material properties ThermoCalc database PBIN - Solid structure diffusion coefficient Ds 5×10-11 m2 s-1 [14]Liquid phase diffusion coefficient Dl 10-9 m2 s-1 [14] Gibbs-Thomson coefficient of Γ Al3Ni2/l 2 10-8 m K [14] solid/liquid interface Γ Al3Ni/l 3.5×10-8 m K [31] Γ Al/l 1.86×10-7 m K [31] Eutectic structure contact angle θ Al3Ni/l 19.22 ° [31] θ Al/l 11.29 ° [31] Jackson-Hunt model series P' 0.335(fα

eut (1-fα eut))1.65 [27, 32]

Density of the particle ρ ( w0/ρAl + (1-w0)/ρNi )-1 kg m-3 Density of Nickel ρNi 8909 kg m-3 [33] Density of Aluminum ρAl 2700 kg m-3 [33]

Heat transfer data EML, Fig.5 IA, Fig.7 External temperature Text 293 293 K Heat transfer coefficient hext 213.8 - W m-2 K-1 Droplet initial velocity v0 - 0 m s-1 Density of Helium ρf - 37.303 T -0.9559 kg m-3 [34] Viscosity of Helium µf - (4.3679 T 0.6702)10-7 Pa s [34] Thermal conductivity of Helium κf - 0.002778 T 0.7025 W m-1 K-1 [34] Prandtl number Pr - 0.713 - [34] Coefficients in Whitaker’s relation B - 2.778×10-3 - [34]

m - 0.7025 - [34]

Alloy and process data EML, Fig.5 IA, Fig.7 Particle radius R 3818.7 (a) 275 µm (b) 82.5 Nominal compositions x0 75 80 at.% Al w0 57.97 64.78 wt% Al

Measured or adjusted data EML, Fig.5 IA, Fig.7 Secondary dendrite arm spacing λ2 35.78 (a) 10 µm (b) 1 µm Nucleation undercooling ΔT Al3Ni2

n 0 0 K ΔT Al3Ni

n 150 0 K ΔT Al

n 15 0 K

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Figure 5. Simulation results for the Al75Ni25 droplet solidified under EML (Figure 4) compared with experimental data showing (a) temperature of the droplet versus pyrometer

measurements calibrated at different temperatures and (b) volume fraction of microstructures versus measured SEM fractions with error range and (c) the normalized radii of the zones, i.e. the position of the growing dendritic, peritectic and eutectic envelopes of the microstructures.

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6. IA particles

Intensive characterization has been conducted on IA Al-Ni particles. The particle size, the atomization gas and the alloy compositions were systematically varied. The microstructures of the produced particles were studied using Transmission Electron Microscope (TEM), Scanning Electron Microscope (SEM), Electron Backscatter Diffraction (EBSD), Neutron Diffraction (ND) and X-ray Diffraction (XRD). Rietveld analysis was also used to find the phase fractions in Al-Ni alloys. While this will not be presented into details in the present contribution, it is useful to notice a systematic trend for the Al80Ni20 composition illustrated in Figure 6. The formation of primary Al3Ni2 was suppressed in small size particles and Al3Ni formed directly from the melt. For a given alloy, thermodynamic equilibrium predicts a unique fraction of phases at room temperature, whatever the size of the system. This is obviously not the case in the experimental results presented in Figure 6. Not only the distribution of phases varies with the diameter of the particles, but other experimental data show that it is a function of the atomization gas [11]. Explanations involve competition between diffusion in phases and growth of dendritic, peritectic and eutectic microstructures, combined at various heat extraction rates. The model was used for the understanding of the several physical effects involved that permit explanations of the measured fractions of phases.

Figure 6. IA Al80Ni20 in helium with particle radius (a) 275 μm and (b) 82.5 μm.

In order to account for the atomization process, a time dependent heat transfer coefficient is computed with a model dedicated to the atomization process [13, 34, 35]. The Nusselt number of the heat exchange is Nu=hext D/κf, with hext the heat transfer coefficient, D the particle diameter and κf the thermal conductivity of the gas. From the modified Whitaker correlation [34], the Nusselt number is expressed as a function of the Reynolds number of the flow, Re=D ρf v/μf, the Prandtl number, Pr=μf /(ρf αf ), and the viscosity of the gas μf, where

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ρf and αf are the density and viscosity of the atomizing gas, respectively. The velocity of the falling droplet, v, appearing in Re, is integrated with respect to time from the balance between gravity and drag forces in the atomization fluid. The convective heat transfer coefficient is thus directly correlated to the velocity of the droplet [13]. All data for the computation of the heat transfer coefficient are provided in Table 1.

Figure 7 presents the result of simulations performed for atomization of an Al80Ni20 alloy corresponding to the particle sizes given in Figure 6. Each column stands for one simulation, namely (a) R = 275 μm and (b) 82.5 μm. The three graphs on each column share the same x-axis and show the time evolution of (a) the temperature of the droplet, with the equilibrium temperatures for the formation of a new solid phase (horizontal lines with filled symbols) and the predicted nucleation temperatures (horizontal lines with empty symbols); (b) the average volume fraction of the phases and (c) the radius of each zone, i.e. dendritic Al3Ni2, peritectic Al3Ni and eutectic (Al+Al3Ni), normalized by the total radius of the particle, R. The normalized radius of a zone illustrates the extent of the expansion of a microstructure within the droplet. For instance, when the radius RAl3Ni2/R reaches unity, the mushy zone – or grain envelope of the primary dendritic structure – is fully developed and no bulk liquid remains.

We notice first the difference in cooling rate and the resulting solidification time being inversely proportional to the particle size. Secondly, the predicted nucleation temperature for the peritectic phase is systematically higher than the equilibrium temperature defined by the phase diagram, while the nucleation of the eutectic occurs close to its equilibrium temperature. We detail hereafter the solidification scenarios for the two simulations, in order of complexity.

Regime of distinct successive growth The most common solidification behavior is presented for the larger particle in

Figure 7. It also corresponds to the scenario of the EML droplet presented before and in earlier analyses [7, 24]. The cooling curve exhibits three recalescences, corresponding to the successive growths of the primary dendritic Al3Ni2, the peritectic Al3Ni and the eutectic (Al+ Al3Ni) microstructures. Direct solidification of the primary dendritic Al3Ni2 occurs between t = 20 and 50 milliseconds. The graph (c) with the radii of zones shows that the peritectic zone starts to grow at around t = 40 ms, but its expansion is quickly stopped by the increase in temperature due to the recalescence associated with the primary dendritic Al3Ni2 phase shown in graph (a). After the primary recalescence, the temperature decreases again and, at around t = 80 ms, the peritectic zone grows significantly and leads to a peritectic recalescence and a clear increase of the average volume fraction of the Al3Ni phase as shown in (b). The decrease in primary Al3Ni2 fraction is first due to the remelting of the primary solid induced by the peritectic recalescence and then to the peritectic transformation Al3Ni2→ Al3Ni, which

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lasts up to complete solidification. After 100 ms, the eutectic structure grows and fills in the remaining liquid. Finally, the solidification is complete when the solid eutectic zone is fully developed, i.e. when RAl/R reaches unity.

Regime of shortcut of the primary growth On the other hand, the small particle follows a different solidification path as illustrated

in Figure 7. The composition gradients involved are of the same order of magnitude, and then the diffusion and growth kinetics are comparable, while the cooling rate is almost one order of magnitude higher. Hence, the growth-induced primary undercooling is so high that the temperature is already below the peritectic nucleation temperature before any primary solid has significantly started to grow. On the radii graph (c), the peritectic zone starts expanding very shortly after the primary mushy zone, but with a higher growth velocity. Hence, the peritectic zone quickly overtakes the primary zone (RAl3Ni ≥ RAl3Ni2), the growth of the latter then being stopped as the primary dendrites are completely surrounded by the peritectic phase. Additionally, the peritectic recalescence takes place up to temperatures higher than the peritectic equilibrium temperature TP

Al3Ni. After the peritectic recalescence, cooling continues and the solidification ends with eutectic growth. The final microstructure only contains peritectic Al3Ni and eutectic (Al+ Al3Ni) as shown in (b).

As can be concluded from a first comparison between the simulated solidification sequence presented in Figure 7 and discussed above, the model predicts well the qualitative phase fractions observed in Figure 6. Several comments can be done with respect to this results:

- Modified Whitaker’s correlation. This correlation seems well suited for computation of the heat transfer coefficient for atomized particles. It is simple to implement in modeling of heat balance and its ability to describe the dependence of the nature of the gas was demonstrated in the past [34, 35].

- Dendrite arm spacing was only broadly estimated from Figure 6.

- Nucleation undercooling. Because these parameters could not be measured upon atomization, there were systematically set to zero.

It is remarkable that this explanation for the presence of the peritectic phase can be reached while no other parameters were adjusted. In particular, all unknown values adjusted when applying the model to the EML sample were not changed for application to IA. The only drastic change is the nucleation undercooling, revealing in this system a limited effect on the overall solidification sequence and the final microsegregation state.

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Figure 7. Simulation of the solidification of Al80Ni20 impulse-atomized in helium with particle radius (left) R=275 μm and (right) R=82.5 μm, with (a) the temperature; (b) the

average volume fraction of phases and (c) the normalized radii of the zones, i.e. the position of the growing dendritic, peritectic and eutectic envelopes of the microstructures.

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7. Conclusion

A microsegregation solidification model for multiple phase transformations in binary alloy coupled with thermodynamic equilibrium calculations (CalPhaD) [13, 14]. The model predicts the evolution of temperature, volume fraction and average composition of the different phases, in the occurrence of several solidification reactions (primary dendritic, peritectic and eutectic). Given a thermodynamic database is available, the equilibrium calculations make it possible to: (i) apply the model to any alloy without prior knowledge of the solidification sequence; (ii) compute non-linear phase diagrams and the stable and metastable compositions of the interfaces between phases; (iii) estimate the stable and metastable enthalpies of phases, and their partial derivatives (e.g. heat capacities).

The temperature-time profiles measured during Electro-Magnetic Levitation solidification experiments and the fraction of the solidified microstructures could be compared with simulation results for a Al75Ni25 sample. The simulated temperature-time-profile exhibiting three recalescences is in good agreement with measurements and the predicted amount of microstructures is better than the standard lever rule and Gulliver-Scheil approximations. The model thus correlates quantitatively the final microstructures with the processing conditions. Application to impulse atomization has only started. The results demonstrate the potency of the model and permits interpretation of the measured fraction of phases as a function of the particle size.

Major improvements would include the addition of convection and porosity, and the extension to multicomponent alloys [36] for which the equilibrium calculations are essential too. Moreover, the introduction of such a segregation model in a macroscopic simulation would mark a step forward in the solidification simulation of peritectic alloys at larger scales.

Acknowledgements

This work has been partly funded by the European Union – Framework Program 6 integrated project IMPRESS (Contract No. NMP3-CT-2004-500635), by the European Space Agency project NEQUISOL (15236/02/NL/SH), by the Canadian Space Agency (9F007-08-0154 and SSEP Grant 2008), by the Grant for Young European Materials Science & Engineering Scientists from the German Federation of Materials Science & Engineering (Bundesvereinigung Materialwissenschaft und Werkstofftechnik e.V., BV MatWerk), the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) and the Federal Institute for Materials Research and Testing (Bundesanstalt für Materialforschung und -prüfung, BAM).

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