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Solute trapping in non-equilibrium solidification: A comparative model study
Materialia 6 (2019) 100256
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Solute trapping in non-equilibrium solidification: A comparative model study Klemens Reuther a,∗, Stephan Hubig b, Ingo Steinbach b, Markus Rettenmayr a a b
Otto Schott Institute of Materials Research, Friedrich Schiller Universität Jena, Löbdergraben 32, Jena 07743, Germany Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universität Bochum, Universitätsstr. 150, Bochum 44801, Germany
a r t i c l e
i n f o
a b s t r a c t
Keywords: Rapid solidification Solid/liquid interface Solute trapping Modeling Phase field models,
A sharp interface model and a diffuse interface model describing rapid solidification are compared. Both models are based on the assumption of two independent processes at the solid/liquid interface that together consume the available driving force. These two processes are (1) the interface motion and (2) the redistribution of atoms between the two phases. The sharp interface model is based on an interface thermodynamics model (Hillert and Rettenmayr, [3,4]) and derives its driving force for the phase transformation directly from Gibbs free energy formulations of the liquid and solid phases. In contrast, the Finite Interface Disspiation model (Steinbach et al., [10]), a diffuse interface model, is based on a grand potential formulation. The models are benchmarked against experimental data on solute trapping in the Al–Sn system (Smith and Aziz, [23]). The sharp interface model reproduces the experimental data with a single set of kinetic parameters, the diffuse interface model requires a velocity dependent interface permeability. It is concluded that due to the investigated high Péclet number regime the permeability in the diffuse interface model does not only describe the transport processes through the interface, but also the diffusion processes in the layer directly in front of it.
1. Introduction
Kaplan [5]. Both models define the solidification process of a binary alloy assuming two thermodynamic processes: the actual growth of the solid phase by the motion of the interface, and the redistribution of atoms, describing the change of composition caused by the change in chemical potentials in the newly formed solid. Both of these processes together are responsible for the Gibbs free energy decrease at the moving interface. Starting from these two processes, Cahn and Aziz developed analytical expressions for the liquidus slope and partition coefficient based on simplified chemical potentials assuming dilute solutions. In contrast, Hillert and Rettenmayr considered Gibbs free energies without restricting assumptions and incorporated diffusive fluxes in both phases. This resulted in a wider applicability as compared to the CGM, as the model is able to describe the thermodynamics of transient phase transformation processes and non-dilute alloys or alloys of strongly interacting species. Buchmann and Rettenmayr [6] incorporated the thermodynamic considerations in a kinetic model and describe both rapid solidification and melting [6–9]. The second model is a diffuse interface (Phase Field) model, which describes the interface as a transition region of finite thickness. The Phase Field model investigated in this work is the Finite Interface Dissipation model presented in Ref. [10].
The solidification of alloy melts is a prominent field of research in materials science, and due to the large variety of technical processes ranging from standard casting to quenching techniques and additive manufacturing [1,2] it gave rise to numerous challenging scientific problems. Whereas for a large range of processing parameters the assumption of thermodynamic equilibrium at the solid/liquid interface was shown to be an appropriate approximation for both analytical and numerical models, it loses its validity once the solidification front velocity reaches or exceeds the velocity of the involved diffusion processes. In this situation, partitioning according to the equilibrium phase diagram cannot occur anymore, and solute trapping occurs. A quantitative and predictive description of this phenomenon is one of the scientific problems mentioned above. In this paper we investigate rapid solidification of an Al-0.02at.%Sn system with two fundamentally different models. The first model is a sharp interface model, which considers the interface to be a mathematically sharp division between the solid and liquid phase. The interface thermodynamics of the model used in this work was developed by Hillert and Rettenmayr [3,4]. Although developed independently, it is closely related to the classical “continuous growth model” (CGM) by Aziz and ∗
Corresponding author. E-mail addresses: [email protected] (K. Reuther), [email protected] (S. Hubig), [email protected] (I. Steinbach), [email protected] (M. Rettenmayr). https://doi.org/10.1016/j.mtla.2019.100256 Received 21 September 2018; Accepted 16 February 2019 Available online 21 February 2019 2589-1529/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
K. Reuther, S. Hubig and I. Steinbach et al.
Materialia 6 (2019) 100256
The development of diffuse interface models for alloy solidification was a challenging task by itself [11,12], and initially the assumption of local equilibrium [13,14] was generally employed. More recent diffuse interface models consider solute trapping by either incorporating second order time derivatives, or by including exchange terms for solute redistribution [15]. The Finite Interface Dissipation model belongs to the latter category. It was designed for the description of phase transformations far from equilibrium on a mesoscopic scale. Its first derivation was presented in the context of two-phase binary systems [10] and later extended to the general multiphase multicomponent case [16]. The model is not based on the usual assumption of equal chemical potentials at the interface, but instead links the solute concentration in the phases by a kinetic equation. The centrepiece of both the sharp and the diffuse interface models is their incorporation of a general thermodynamic description of the phases provided in the form of Gibbs free energy functions [17]. The similar thermodynamic description of both models provides the opportunity for deeper insight into the thermodynamic processes. Establishing a common ground of understanding for both modeling approaches is beneficial in several ways, since both models supplement each other. The diffuse interface model is easily extended to three dimensions and can directly describe e.g., dendritic morphology [18], whereas the sharp interface model is crucial for providing a benchmark that diffuse interface models should match [19]. 2. Model description 2.1. The sharp interface model The two processes defined by the CGM and Ref. [4] are described by two fluxes, 𝐽 m and 𝐽 r , related to the interface motion and to the redistribution of solute through the interface, respectively. They are defined by: 𝑣 𝐽m = (1) 𝑉𝑚 𝐽r =
𝑣 (𝑐 − 𝑐s ), 𝑉𝑚 l
(2)
where v is the velocity of the moving solid/liquid interface, 𝑐l and 𝑐s are the solute concentrations on the liquid and solid side of the interface, respectively, and Vm is the molar volume. Note that in this formulation diffusion in the solid state is neglected, an assumption which is generally reasonable in metallic alloy solidification and which will be upheld throughout the present work. The thermodynamic forces 𝐷m and 𝐷r driving the two fluxes are given by ) 1 ( (1 − 𝑐l )Δ𝜇 A + 𝑐l Δ𝜇 B 𝐷 =− 𝑉𝑚 m
𝐷r = −
(3)
1 (Δ𝜇 A − Δ𝜇 B ). 𝑉𝑚
(4)
is the change of chemical potential 𝜇 of a mole of respectively A or B atoms changing from liquid to solid phase: Δ𝜇 A,B
Δ𝜇 A,B = 𝜇sA,B − 𝜇lA,B The sum of the two driving forces yields the total driving force ( ) 𝐷tot = 𝐷m + 𝐷r = (1 − 𝑐s )Δ𝜇 A + 𝑐s Δ𝜇 B
(5) 𝐷tot : (6)
This driving force is equal to the total Gibbs free energy decrease of the system caused by the moving interface [4]. The two driving forces are proportional to the fluxes, connected by the mobilities L and M: 𝐽 r = 𝐿𝐷r
(7)
𝐽 m = 𝑀𝐷m
(8)
Following Buchmann and Rettenmayr [6] these mobilities can be related to physical quantities: the mobility M for interface motion is connected to the maximum crystallization velocity v0 by 𝑣 𝑀= 0 , (9) 𝑅𝑇 where R is the universal gas constant and T the temperature. For L, the mobility for the redistribution of atoms, a slightly differing definition than that from Ref. [6] is used in the present work: 𝑣 𝐿 = D (𝑐l − 𝑐s ) (10) 𝑅𝑇 𝑣D is the velocity of the atoms during the redistribution across the interface. This velocity can be described as diffusion through an interface layer of thickness a0 with an interfacial diffusion coefficient 𝐷i 𝑣D =
𝐷i 𝑎0
(11)
Following Buchmann and Rettenmayr, the diffusivity in the interface layer is assumed to be equal to that in the liquid phase, 𝐷i = 𝐷l . The sharp interface model then states that in any diffusion controlled solidication process, Eqs. (7) and (8) have to be consistently fulfilled at all times. If the thermodynamic potentials are known, these two equations have four free variables: the temperature T, the velocity v and the interface concentrations 𝑐l and 𝑐s . Given a temperature and interface velocity, the interface liquidus and solidus concentrations can be calculated. If the equations have no solution, no phase transformation can occur at that temperature and velocity. The sharp interface model contains only two adjustable parameters, one in each kinetic coefficient: the maximum crystallization velocity and the effective thickness of the interface layer. While the maximum crystallization velocity is assumed to be of the order of magnitude of the velocity of sound in metals [6], the interface thickness is not well defined. However, to retain the sharp interface nature of the model its value should be within one order of magnitude of the interatomic distance. 2.2. The diffuse interface model The equations of motion of the Finite Interface Dissipation model are [10]: ( ) 𝜕𝑓l 𝜕𝑓s 𝜙s 𝑐̇ s = ∇ ⋅ (𝜙s 𝐷s ∇𝑐s ) + 𝑃 𝜙s 𝜙l − + 𝜙s 𝜙̇ s (𝑐l − 𝑐s ), (12) 𝜕𝑐l 𝜕𝑐s (
𝜕𝑓s 𝜕𝑓l − 𝜕𝑐s 𝜕𝑐l
)
+ 𝜙l 𝜙̇ l (𝑐s − 𝑐l ),
(13)
( [ ) ( )] 𝜋2 1 𝜋√ 𝜙̇ s = 𝐾 𝜎 Δ𝜙s + 𝜙s (1 − 𝜙s )ΔΦphi , − 𝜙s − 2 𝜂 𝜂2
(14)
𝜙l 𝑐̇ l = ∇ ⋅ (𝜙l 𝐷l ∇𝑐l ) + 𝑃 𝜙s 𝜙l
where 𝜙s is the Phase Field variable of the solid and 𝜙l of the liquid phase. Note that in our two-phase case 𝜙s = 1 − 𝜙l holds. 𝐷s and 𝐷l are the diffusivities in the solid and liquid phase, respectively, and 𝑓s and 𝑓l are their free energy densities that depend on their individual solute concentrations 𝑐s and 𝑐l . The interface permeability P is a measure for how easily solute can be exchanged between the solid and the liquid phase. The parameter 𝜂 is the width of the diffuse interface and can be chosen for numerical convenience. 𝜎 is the interfacial energy, K the corrected interfacial mobility, which is given as 𝐾=
8𝑃 𝜂𝑀sl 8𝑃 𝜂 + 𝑀sl 𝜋 2 (𝑐s − 𝑐l )2
,
(15)
where 𝑀sl is the physical mobility of the solid/liquid interface. Note that the parameter K does not originate from a thin interface analysis [20], but is a mere consequence of the model assumptions [10].
K. Reuther, S. Hubig and I. Steinbach et al.
Fig. 1. Driving forces as formulated for the sharp interface model (𝐷tot = 𝐷m + 𝐷r ) and the diffuse interface model (ΔΦtot = ΔΦphi + ΔΦredis ). The quantity 𝑐̃ = 𝜙l 𝑐s + 𝜙s 𝑐l is a measure of interface composition.
Just as in the sharp interface model, the motion of the interface is described as a process that is independent of the solute redistribution through the interface. The driving force for interface motion is in principle independent ( ) 𝜕𝑓 𝜕𝑓 ΔΦphi = 𝑓l − 𝑓s + 𝜙s s + 𝜙l l (𝑐s − 𝑐l ). (16) 𝜕𝑐s 𝜕𝑐l 𝜕𝑓s 𝜕𝑐s
and
𝜕𝑓l 𝜕𝑐l
are the diffusion potentials of the solid and liquid phase,
respectively. The third term proportional to 𝜙̇ s,l in Eqs. (12) and (13) ensures solute conservation. The second term in Eqs. (12) and (13), proportional to P, describes the solute exchange between locally coexisting phases, i.e., inside the diffuse interface. This contribution locally drives the free energy towards the minimum and corresponds to the solute exchange described by Jr and L between the adjacent phases in the sharp interface model. Standard solute diffusion is described in the first term in Eqs. (12) and (13) via the divergence of the diffusion fluxes for each phase ∇ ⋅ (𝜙s,l 𝐷s,l ∇𝑐s,l ). By including both the permeability related process and the diffusion process, the model equations drive the system towards global thermodynamic equilibrium. Note that the total driving force ΔΦtot is given as the grand potential difference [10] ( ) 𝜕𝑓 𝜕𝑓 ΔΦtot = 𝑓s − 𝑐s s − 𝑓l − 𝑐l l . (17) 𝜕𝑐s 𝜕𝑐l This total driving force is consumed by both processes of interface migration and solute redistribution ΔΦtot = ΔΦphi + ΔΦredis . Therefore, the driving force for solute redistribution is ( ) 𝜕𝑓s 𝜕𝑓l ΔΦredis = −(𝜙s 𝑐l + 𝜙l 𝑐s ) − . (18) 𝜕𝑐s 𝜕𝑐l 3. Thermodynamic comparison Although both models draw on the same assumption of two distinct processes that together are responsible for consuming the available free energy, the formulation of the driving forces is different, as illustrated in Fig. 1. The diffuse interface model uses the grand potential difference ΔΦtot , i.e., the chemical potential difference of the A atoms between liquid and solid phase, as the total driving force. In contrast, the sharp interface model assumes the total driving force Dtot to be described by the Gibbs free energy released by the growth of solid with a solute concentration of 𝑐s . The driving force for the interface motion, ΔΦphi , changes across the interface in connection with the mixture composition 𝑐̃ = 𝜙l 𝑐s + 𝜙s 𝑐l .
Materialia 6 (2019) 100256
ΔΦphi exhibits a strong relation with the sharp interface model: on the liquid side it is equal to the total driving force of the sharp interface model Dtot , whereas on the solid side it is equal to the partial driving force for interface motion of the sharp interface model 𝐷m . The driving force for interface motion is therefore larger towards the liquid side of the diffuse interface and smaller towards the solid side. Such spurious behaviour is inherent to Phase Field models and is usually corrected by mobility correction schemes [20]. However, since the Finite Interface Dissipation model operates under strong non-equilibrium conditions, no such correction scheme has been developed yet. The partial driving force for redistribution, ΔΦredis , fills the balance to the total driving force as given by the grand potential ΔΦtot . Thus the driving force for redistribution is strongest at the solid side and weakest at the liquid side of the diffuse interface. The diffuse interface model therefore always tends towards the prediction of a larger driving force for interface motion than the sharp interface model. The same holds for the total driving force. Depending on the respective chemical potentials, the driving force for redistribution can be larger or smaller in either model. The interface mobility can therefore be expected to be significantly lower in the diffuse interface model than the coefficient for interface motion in the sharp interface model, while the interface permeabilites should be of a comparable order of magnitude. 4. Modeling rapid solidification of Al-0.2at.%Sn In the present work, steady state rapid solidification of Al-0.2at.%Sn, a model alloy for which experimental data are available [23], is investigated by both models. The thermodynamic data for the solid (Al) phase and the liquid mixture phase were taken from the COST database [22], the Gibbs free energies of the pure elements were taken from Ref. [21]. The equilibrium solidus and liquidus lines for the fcc-Al phase calculated by these data are shown in Fig. 2. The fcc phase exhibits a retrograde solidus below temperatures of ∼ 865 K, with a maximum Sn solubility of ∼ 0.21at.%. The calculated liquidus temperature Tl is 932.18K for the chosen composition of 0.2at.%Sn. Two equilibrium solidus temperatures Ts exist at 878.47 K and 848.34 K, corresponding to two solidification intervals Δ𝑇0 = 𝑇𝑙 − 𝑇𝑠 with magnitudes of 53.72 K and 83.84 K, respectively. For the investigation of steady state solidification of an alloy with a fixed initial composition, the sharp interface model can be reduced to a single equation. The velocity enters both fluxes (Eqs. (7) and (8)) linearly and can be eliminated, yielding 𝐿𝐷r = (𝑐l − 𝑐s ) 𝑀𝐷m .
(19)
Since in the steady state 𝑐s is fixed to 𝑐0 = 0.2at.%, only two parameters, temperature and liquidus concentration, remain. The equation was solved numerically for a fixed range of temperatures using Maple 2016 and the DirectSearch package. The corresponding solidification velocities were then calculated by Eq. (7). Phase Field simulations with the diffuse interface model were carried out by initializing both concentration fields to the nominal alloy concentration of 0.2at.%Sn. The phase field variable was initialized with its known sinus-shaped traveling wave profile [20]. The temperature T is an input parameter of the Phase Field model and uniform in space and time. A reasonable range for choosing the temperature T is obtained from the equilibrium phase diagram. Each chosen temperature results in a different steady state velocity v and partition coefficient k (see below). The interfacial mobility 𝑀sl was chosen such that the range of temperatures maps to the range of velocities reported in the experimental results. For numerical integration, a finite difference scheme with second order spatial discretization and first order forward time discretization was used. Explicit timesteps were performed until the system converged to a steady state. A moving frame was used to save computational resources and evaluate the velocity v. The moving frame moves in growth direction, i.e.
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Materialia 6 (2019) 100256
Fig. 2. Equilibrium solidus and liquidus lines for fcc-Al in the Al-Sn system, as calculated from the data in Refs. [21,22]. The initial concentration c0 and the respective solidus and liquidus temperature(s) are marked. Note that the equilibrium solidus line has a retrograde trend below 865K and that the liquidus line ends in an eutectic point at a temperature of 502.4 K and a concentration of 98.0at.%Sn.
Fig. 3. Top: 𝑘 − 𝑣 curves calculated by the sharp interface model (circles) and by the diffuse interface model, the latter using two different values for the interface permeability (squares and diamonds, respectively), and experimental results of Smith and Aziz [23] (stars). Bottom: 𝑣 − 𝑇 -curves calculated with both models. The two equilibrium solidification intervals are indicated by dashed lines.
shifts the entire solutions of the phase and concentration fields against the growth direction as soon as the solid phase field variable 𝜙s increases to above 1/2 at a designated gridpoint. The inverse time intervals between shifts, together with the grid spacing, provide a direct measurement of the interface velocity. The steady state was determined by an observation of the concentration and phase field profiles. If no further change was visible, steady state was assumed to be reached, and the simulation stopped. From the steady state profile the solute partition coefficient was evaluated via [10] 𝑘(𝑣) =
𝑐s far-field solid concentration = . max[𝑐l ] maximum liquid concentration
(20)
It was found that the maximum liquid concentration is typically located at the solid side of the diffuse interface, where 𝜙s ≈ 1. Note that in this work the Finite Interface Dissipation model is operated in a regime of high interface Péclet-numbers Pe = 𝜂∕𝛿 = 𝜂𝑣∕𝐷l ≈ 105 . The smallest resolved length scale of the interface (determined by 𝜂) is therefore much larger than the characteristic length of solute diffusion 𝛿. In that case, all diffusive effects are described by the permeability term, which can be interpreted as a coarse-grained representation of the diffusion occuring in front of the interface [15]. 5. Results and discussion In Fig. 3a the velocity-dependent partition coefficients for the rapid solidification of Al-0.2at.%Si as calculated by both models are shown together with the experimental results from Ref [23]. The parameters of both models are listed in Table 1. The sharp interface model reproduces the experimental data in a satisfactory manner with the chosen
Table 1 Parameters used for the Al–Sn system. Sharp interface model Diffusion coefficient in the melt Effective interface thickness Maximum crystallization velocity
Dl a0 v0
m2 /s m m/s
1.3 × 10−7 exp(23.8kJ∕RT) [24] 1.4 × 10−9 1000 [6]
Dl 𝐷s Δt Δx P 𝜂 𝑀sl 𝜎
m2 /s m2 /s s m m3 /Js m m4 /Js J/m2
1.3 × 10−7 exp(23.8kJ/RT) [24] 0 10−11 10−6 varying, see text 10−5 5 × 10−8 104
Diffuse interface model Diffusion coefficient in the melt Diffusion coefficient in the solid Timestep Grid spacing Interface permeability Interface width Interface mobility Interfacial energy
parameters. Complete solute trapping (i.e., 𝑘 = 1) is predicted at velocities exceeding ∼ 13 m/s. Additionally, the model predicts a second branch of lower partition coefficients at lower temperatures. The diffuse interface model yields a differentiated result: For solidification velocities of ∼ 1.5 m/s, the experimental results are reproduced by an interface permeability of 3 × 10−4 m3 ∕Js. For higher velocities of ∼ 4 m/s a significantly lower interface permeability of 1.5 × 10−5 m∕Js is required to attain an agreement with the experimentally measured partition coefficients. By performing simulations with different interface widths 𝜂 (down to the order of magnitude of the sharp interface effective interface thickness) it was confirmed that the results for k(v) are invariable with respect
K. Reuther, S. Hubig and I. Steinbach et al.
to a change in 𝜂 as long as the permeability is changed simultaneously according to P∝1/𝜂 and the interface Péclet numbers are sufficiently high, i.e., Pe > 102 . This means that in the regime of high interface Péclet numbers the k(v)-curve is effectively a function of the product P · 𝜂 only. A second result is the drastically different description of the temperature dependence of rapid solidification by both models (Fig. 3b). The diffuse interface model predicts a monotonous relationship between undercooling and growth velocity, which is only weakly influenced by the different interface permeability parameters. The two undercoolings connected to equilibrium solidification do not appear to be special points in this curve, the equilibrium case is not represented as a limiting case. In fact it is not possible for the diffuse interface model to reach the equilibrium case at solidus temperature in the current setting. To reproduce the equilibrium case with the diffuse interface model, an infinitely small mobility 𝑀sl ≈ 0 and an infinitely high permeability P ≈ ∞ would be necessary. The sharp interface model does reproduce equilibrium solidification as a limiting case for vanishing interface velocities. The two observed branches are consequently connected to undercoolings above the upper and below the lower equilibrium solidus temperature. Between the two solidus temperatures, no solutions to Eq. (19) were found. The upper branch, corresponding to the experimentally observed values, exhibits a decreasing undercoooling with higher velocities, from equilibrium solidification (𝑣 = 0) at the equilibrium solidification interval of ∼ 54 K down to an undercooling of ∼ 34 K at velocities up to ∼ 7 m/s. Only at that point does the relation invert, and higher undercoolings are again required for higher velocities. It is pertinent to note that for the parameter set chosen for the diffuse interface model, the parameter K is essentially constant and equal to 𝑀sl . By taking into account the actual values of the different contributions in Eq. (15), it can be seen that the limit 𝐾 ≈ 𝑀sl is indeed a good approximation. Additional simulation runs were conducted with the K-parameter replaced by the constant 𝑀sl with no observable changes in the results. For different alloy systems which do not fulfill this limit, however, the diffuse interface model may produce qualitatively different results. 6. Conclusions Although the two models investigated in this work are similar in their concept of two separate processes at the solid/liquid interface of a rapidly solidifying alloy, they are based on distinctly different descriptions of the involved thermodynamics. They consequently provide different descriptions of the steady state solute trapping behaviour in the Al–Sn system. With the chosen kinetic parameters, the sharp interface model reproduces the experimentally measured velocity-dependent partition coefficients quite well. Due to its sharp interface, the model is not influenced by the change in width of the diffusion zone in front of the interface with varying interface velocity. The diffuse-interface model in contrast is affected by the change in diffusion zone width, as the diffusion zone is completely situated inside the diffuse interface. The permeability parameter describing solute redistribution both in the bulk liquid near the interface and between the phases therefore has to vary with velocity to describe the experimental values. In the present study only the sharp interface model converges towards thermodynamic equilibrium at the interface for v → 0 at the solidus temperatures.
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Declaration of interest None. Acknowledgment This work was funded by the Deutsche Forschungsgemeinschaft (DFG) under grants nos. Re1261/8-2 and Ste1116/10-2. References [1] T.R. Anantharaman, C. Suryanarayana, Review: a decade of quenching from the melt, J. Mater. Sci. 6 (8) (1971) 1111–1135, doi:10.1007/BF00980610. [2] M. Ramsperger, R.F. Singer, C. Körner, Microstructure of the nickel-base superalloy CMSX-4 fabricated by selective electron beam melting, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 47 (3) (2016) 1469–1480, doi:10.1007/s11661-015-3300-y. [3] M. Hillert, Solute drag, solute trapping and diffusional dissipation of Gibbs energy, Acta Mater. 47 (18) (1999) 4481–4505. [4] M. Hillert, M. Rettenmayr, Deviation from local equilibrium at migrating phase interfaces, Acta Mater. 51 (10) (2003) 2803–2809. [5] M.J. Aziz, T. Kaplan, Continuous growth-model for interface motion during alloy solidification, Acta Metall. 36 (8) (1988) 2335–2347. [6] M. Buchmann, M. Rettenmayr, Rapid solidification theory revisited - a consistent model based on a sharp interface, Scripta Mater. 57 (2) (2007) 169–172. [7] M. Rettenmayr, M. Buchmann, Modeling rapid liquid/solid and solid/liquid phase transformations in Al alloys, Int. J. Mater. Res. 99 (6) (2008) 613–617. [8] M. Buchmann, M. Rettenmayr, Non-equilibrium transients during solidification–a numerical study, Scripta Mater. 58 (2) (2008) 106–109. [9] M. Buchmann, M. Rettenmayr, Numerical study of steady state melting of a binary alloy, J. Cryst. Growth 310 (21) (2008) 4623–4627. [10] I. Steinbach, L. Zhang, M. Plapp, Phase-field model with finite interface dissipation, Acta Mater. 60 (6–7) (2012) 2689–2701, doi:10.1016/j.actamat.2012.01.035. [11] A.A. Wheeler, W.J. Boettinger, G.B. McFadden, Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A 45 (10) (1992) 7424–7439, doi:10.1103/PhysRevA.45.7424. [12] R.F. Almgren, Second-order phase field asymptotics for unequal conductivities, SIAM J. Appl. Math. 59 (6) (1999) 2086–2107, doi:10.1137/S0036139997330027. [13] S.G. Kim, W.T. Kim, T. Suzuki, Phase-field model for binary alloys., Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60 (6 Pt B) (1999) 7186–7197, doi:10.1103/PhysRevE.60.7186. [14] B. Echebarria, R. Folch, A. Karma, M. Plapp, Quantitative phase-field model of alloy solidification, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 70 (6) (2004) 22, doi:10.1103/PhysRevE.70.061604. [15] L. Zhang, E.V. Danilova, I. Steinbach, D. Medvedev, P.K. Galenko, Diffuse-interface modeling of solute trapping in rapid solidification: predictions of the hyperbolic phase-field model and parabolic model with finite interface dissipation, Acta Mater. 61 (11) (2013) 4155–4168, doi:10.1016/j.actamat.2013.03.042. [16] L. Zhang, I. Steinbach, Phase-field model with finite interface dissipation: extension to multi-component multi-phase alloys, Acta Mater. 60 (2012) 2702–2710, doi:10.1016/j.actamat.2012.02.032. [17] M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, in: Materials Science, Cambridge University Press, 2007, p. 524, doi:10.1017/ CBO9780511812781. [18] R. Kobayashi, A numerical approach to three-dimensional dendritic solidification, Exp. Math. 3 (1) (1994) 59–81, doi:10.1038/066030b0. [19] A. Karma, W.-J. Rappel, Quantitative phase field modelling of dendritic growth in two and three dimensions, Phys. Rev. E 57 (4) (1998) 4323. [20] I. Steinbach, Phase-field models in materials science, Modell. Simul. Mater. Sci. Eng. 17 (7) (2009) 073001, doi:10.1088/0965-0393/17/7/073001. [21] A.T. Dinsdale, SGTE data for pure elements, Calphad-Comput. Coupling Phase Diagr. Thermochem. 15 (4) (1991) 317–425. [22] I. Ansara, A. Dinsdale, M. Rand (Eds.), COST 507 Thermochemical database for light metal alloys, Vol. 2, Office for Official Publications of the European Communities, Luxembourg, 1998. EUR 18499. [23] P.M. Smith, M.J. Aziz, Solute trapping in aluminum-alloys, Acta Metallurgica Et Materialia 42 (10) (1994) 3515–3525. [24] A.D. LeClaire, G. Neumann, 3.2.13 Aluminum group metals, Diffusion in Solid Metals and Alloys. Landolt-Börnstein - Group III Condensed Matter (Numerical Data and Functional Relationships in Science and Technology), 26, 1990, doi:10.1007/10390457_41.
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Full Length Article
Solute trapping in non-equilibrium solidification: A comparative model study Klemens Reuther a,∗, Stephan Hubig b, Ingo Steinbach b, Markus Rettenmayr a a b
Otto Schott Institute of Materials Research, Friedrich Schiller Universität Jena, Löbdergraben 32, Jena 07743, Germany Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universität Bochum, Universitätsstr. 150, Bochum 44801, Germany
a r t i c l e
i n f o
a b s t r a c t
Keywords: Rapid solidification Solid/liquid interface Solute trapping Modeling Phase field models,
A sharp interface model and a diffuse interface model describing rapid solidification are compared. Both models are based on the assumption of two independent processes at the solid/liquid interface that together consume the available driving force. These two processes are (1) the interface motion and (2) the redistribution of atoms between the two phases. The sharp interface model is based on an interface thermodynamics model (Hillert and Rettenmayr, [3,4]) and derives its driving force for the phase transformation directly from Gibbs free energy formulations of the liquid and solid phases. In contrast, the Finite Interface Disspiation model (Steinbach et al., [10]), a diffuse interface model, is based on a grand potential formulation. The models are benchmarked against experimental data on solute trapping in the Al–Sn system (Smith and Aziz, [23]). The sharp interface model reproduces the experimental data with a single set of kinetic parameters, the diffuse interface model requires a velocity dependent interface permeability. It is concluded that due to the investigated high Péclet number regime the permeability in the diffuse interface model does not only describe the transport processes through the interface, but also the diffusion processes in the layer directly in front of it.
1. Introduction
Kaplan [5]. Both models define the solidification process of a binary alloy assuming two thermodynamic processes: the actual growth of the solid phase by the motion of the interface, and the redistribution of atoms, describing the change of composition caused by the change in chemical potentials in the newly formed solid. Both of these processes together are responsible for the Gibbs free energy decrease at the moving interface. Starting from these two processes, Cahn and Aziz developed analytical expressions for the liquidus slope and partition coefficient based on simplified chemical potentials assuming dilute solutions. In contrast, Hillert and Rettenmayr considered Gibbs free energies without restricting assumptions and incorporated diffusive fluxes in both phases. This resulted in a wider applicability as compared to the CGM, as the model is able to describe the thermodynamics of transient phase transformation processes and non-dilute alloys or alloys of strongly interacting species. Buchmann and Rettenmayr [6] incorporated the thermodynamic considerations in a kinetic model and describe both rapid solidification and melting [6–9]. The second model is a diffuse interface (Phase Field) model, which describes the interface as a transition region of finite thickness. The Phase Field model investigated in this work is the Finite Interface Dissipation model presented in Ref. [10].
The solidification of alloy melts is a prominent field of research in materials science, and due to the large variety of technical processes ranging from standard casting to quenching techniques and additive manufacturing [1,2] it gave rise to numerous challenging scientific problems. Whereas for a large range of processing parameters the assumption of thermodynamic equilibrium at the solid/liquid interface was shown to be an appropriate approximation for both analytical and numerical models, it loses its validity once the solidification front velocity reaches or exceeds the velocity of the involved diffusion processes. In this situation, partitioning according to the equilibrium phase diagram cannot occur anymore, and solute trapping occurs. A quantitative and predictive description of this phenomenon is one of the scientific problems mentioned above. In this paper we investigate rapid solidification of an Al-0.02at.%Sn system with two fundamentally different models. The first model is a sharp interface model, which considers the interface to be a mathematically sharp division between the solid and liquid phase. The interface thermodynamics of the model used in this work was developed by Hillert and Rettenmayr [3,4]. Although developed independently, it is closely related to the classical “continuous growth model” (CGM) by Aziz and ∗
Corresponding author. E-mail addresses: [email protected] (K. Reuther), [email protected] (S. Hubig), [email protected] (I. Steinbach), [email protected] (M. Rettenmayr). https://doi.org/10.1016/j.mtla.2019.100256 Received 21 September 2018; Accepted 16 February 2019 Available online 21 February 2019 2589-1529/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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Materialia 6 (2019) 100256
The development of diffuse interface models for alloy solidification was a challenging task by itself [11,12], and initially the assumption of local equilibrium [13,14] was generally employed. More recent diffuse interface models consider solute trapping by either incorporating second order time derivatives, or by including exchange terms for solute redistribution [15]. The Finite Interface Dissipation model belongs to the latter category. It was designed for the description of phase transformations far from equilibrium on a mesoscopic scale. Its first derivation was presented in the context of two-phase binary systems [10] and later extended to the general multiphase multicomponent case [16]. The model is not based on the usual assumption of equal chemical potentials at the interface, but instead links the solute concentration in the phases by a kinetic equation. The centrepiece of both the sharp and the diffuse interface models is their incorporation of a general thermodynamic description of the phases provided in the form of Gibbs free energy functions [17]. The similar thermodynamic description of both models provides the opportunity for deeper insight into the thermodynamic processes. Establishing a common ground of understanding for both modeling approaches is beneficial in several ways, since both models supplement each other. The diffuse interface model is easily extended to three dimensions and can directly describe e.g., dendritic morphology [18], whereas the sharp interface model is crucial for providing a benchmark that diffuse interface models should match [19]. 2. Model description 2.1. The sharp interface model The two processes defined by the CGM and Ref. [4] are described by two fluxes, 𝐽 m and 𝐽 r , related to the interface motion and to the redistribution of solute through the interface, respectively. They are defined by: 𝑣 𝐽m = (1) 𝑉𝑚 𝐽r =
𝑣 (𝑐 − 𝑐s ), 𝑉𝑚 l
(2)
where v is the velocity of the moving solid/liquid interface, 𝑐l and 𝑐s are the solute concentrations on the liquid and solid side of the interface, respectively, and Vm is the molar volume. Note that in this formulation diffusion in the solid state is neglected, an assumption which is generally reasonable in metallic alloy solidification and which will be upheld throughout the present work. The thermodynamic forces 𝐷m and 𝐷r driving the two fluxes are given by ) 1 ( (1 − 𝑐l )Δ𝜇 A + 𝑐l Δ𝜇 B 𝐷 =− 𝑉𝑚 m
𝐷r = −
(3)
1 (Δ𝜇 A − Δ𝜇 B ). 𝑉𝑚
(4)
is the change of chemical potential 𝜇 of a mole of respectively A or B atoms changing from liquid to solid phase: Δ𝜇 A,B
Δ𝜇 A,B = 𝜇sA,B − 𝜇lA,B The sum of the two driving forces yields the total driving force ( ) 𝐷tot = 𝐷m + 𝐷r = (1 − 𝑐s )Δ𝜇 A + 𝑐s Δ𝜇 B
(5) 𝐷tot : (6)
This driving force is equal to the total Gibbs free energy decrease of the system caused by the moving interface [4]. The two driving forces are proportional to the fluxes, connected by the mobilities L and M: 𝐽 r = 𝐿𝐷r
(7)
𝐽 m = 𝑀𝐷m
(8)
Following Buchmann and Rettenmayr [6] these mobilities can be related to physical quantities: the mobility M for interface motion is connected to the maximum crystallization velocity v0 by 𝑣 𝑀= 0 , (9) 𝑅𝑇 where R is the universal gas constant and T the temperature. For L, the mobility for the redistribution of atoms, a slightly differing definition than that from Ref. [6] is used in the present work: 𝑣 𝐿 = D (𝑐l − 𝑐s ) (10) 𝑅𝑇 𝑣D is the velocity of the atoms during the redistribution across the interface. This velocity can be described as diffusion through an interface layer of thickness a0 with an interfacial diffusion coefficient 𝐷i 𝑣D =
𝐷i 𝑎0
(11)
Following Buchmann and Rettenmayr, the diffusivity in the interface layer is assumed to be equal to that in the liquid phase, 𝐷i = 𝐷l . The sharp interface model then states that in any diffusion controlled solidication process, Eqs. (7) and (8) have to be consistently fulfilled at all times. If the thermodynamic potentials are known, these two equations have four free variables: the temperature T, the velocity v and the interface concentrations 𝑐l and 𝑐s . Given a temperature and interface velocity, the interface liquidus and solidus concentrations can be calculated. If the equations have no solution, no phase transformation can occur at that temperature and velocity. The sharp interface model contains only two adjustable parameters, one in each kinetic coefficient: the maximum crystallization velocity and the effective thickness of the interface layer. While the maximum crystallization velocity is assumed to be of the order of magnitude of the velocity of sound in metals [6], the interface thickness is not well defined. However, to retain the sharp interface nature of the model its value should be within one order of magnitude of the interatomic distance. 2.2. The diffuse interface model The equations of motion of the Finite Interface Dissipation model are [10]: ( ) 𝜕𝑓l 𝜕𝑓s 𝜙s 𝑐̇ s = ∇ ⋅ (𝜙s 𝐷s ∇𝑐s ) + 𝑃 𝜙s 𝜙l − + 𝜙s 𝜙̇ s (𝑐l − 𝑐s ), (12) 𝜕𝑐l 𝜕𝑐s (
𝜕𝑓s 𝜕𝑓l − 𝜕𝑐s 𝜕𝑐l
)
+ 𝜙l 𝜙̇ l (𝑐s − 𝑐l ),
(13)
( [ ) ( )] 𝜋2 1 𝜋√ 𝜙̇ s = 𝐾 𝜎 Δ𝜙s + 𝜙s (1 − 𝜙s )ΔΦphi , − 𝜙s − 2 𝜂 𝜂2
(14)
𝜙l 𝑐̇ l = ∇ ⋅ (𝜙l 𝐷l ∇𝑐l ) + 𝑃 𝜙s 𝜙l
where 𝜙s is the Phase Field variable of the solid and 𝜙l of the liquid phase. Note that in our two-phase case 𝜙s = 1 − 𝜙l holds. 𝐷s and 𝐷l are the diffusivities in the solid and liquid phase, respectively, and 𝑓s and 𝑓l are their free energy densities that depend on their individual solute concentrations 𝑐s and 𝑐l . The interface permeability P is a measure for how easily solute can be exchanged between the solid and the liquid phase. The parameter 𝜂 is the width of the diffuse interface and can be chosen for numerical convenience. 𝜎 is the interfacial energy, K the corrected interfacial mobility, which is given as 𝐾=
8𝑃 𝜂𝑀sl 8𝑃 𝜂 + 𝑀sl 𝜋 2 (𝑐s − 𝑐l )2
,
(15)
where 𝑀sl is the physical mobility of the solid/liquid interface. Note that the parameter K does not originate from a thin interface analysis [20], but is a mere consequence of the model assumptions [10].
K. Reuther, S. Hubig and I. Steinbach et al.
Fig. 1. Driving forces as formulated for the sharp interface model (𝐷tot = 𝐷m + 𝐷r ) and the diffuse interface model (ΔΦtot = ΔΦphi + ΔΦredis ). The quantity 𝑐̃ = 𝜙l 𝑐s + 𝜙s 𝑐l is a measure of interface composition.
Just as in the sharp interface model, the motion of the interface is described as a process that is independent of the solute redistribution through the interface. The driving force for interface motion is in principle independent ( ) 𝜕𝑓 𝜕𝑓 ΔΦphi = 𝑓l − 𝑓s + 𝜙s s + 𝜙l l (𝑐s − 𝑐l ). (16) 𝜕𝑐s 𝜕𝑐l 𝜕𝑓s 𝜕𝑐s
and
𝜕𝑓l 𝜕𝑐l
are the diffusion potentials of the solid and liquid phase,
respectively. The third term proportional to 𝜙̇ s,l in Eqs. (12) and (13) ensures solute conservation. The second term in Eqs. (12) and (13), proportional to P, describes the solute exchange between locally coexisting phases, i.e., inside the diffuse interface. This contribution locally drives the free energy towards the minimum and corresponds to the solute exchange described by Jr and L between the adjacent phases in the sharp interface model. Standard solute diffusion is described in the first term in Eqs. (12) and (13) via the divergence of the diffusion fluxes for each phase ∇ ⋅ (𝜙s,l 𝐷s,l ∇𝑐s,l ). By including both the permeability related process and the diffusion process, the model equations drive the system towards global thermodynamic equilibrium. Note that the total driving force ΔΦtot is given as the grand potential difference [10] ( ) 𝜕𝑓 𝜕𝑓 ΔΦtot = 𝑓s − 𝑐s s − 𝑓l − 𝑐l l . (17) 𝜕𝑐s 𝜕𝑐l This total driving force is consumed by both processes of interface migration and solute redistribution ΔΦtot = ΔΦphi + ΔΦredis . Therefore, the driving force for solute redistribution is ( ) 𝜕𝑓s 𝜕𝑓l ΔΦredis = −(𝜙s 𝑐l + 𝜙l 𝑐s ) − . (18) 𝜕𝑐s 𝜕𝑐l 3. Thermodynamic comparison Although both models draw on the same assumption of two distinct processes that together are responsible for consuming the available free energy, the formulation of the driving forces is different, as illustrated in Fig. 1. The diffuse interface model uses the grand potential difference ΔΦtot , i.e., the chemical potential difference of the A atoms between liquid and solid phase, as the total driving force. In contrast, the sharp interface model assumes the total driving force Dtot to be described by the Gibbs free energy released by the growth of solid with a solute concentration of 𝑐s . The driving force for the interface motion, ΔΦphi , changes across the interface in connection with the mixture composition 𝑐̃ = 𝜙l 𝑐s + 𝜙s 𝑐l .
Materialia 6 (2019) 100256
ΔΦphi exhibits a strong relation with the sharp interface model: on the liquid side it is equal to the total driving force of the sharp interface model Dtot , whereas on the solid side it is equal to the partial driving force for interface motion of the sharp interface model 𝐷m . The driving force for interface motion is therefore larger towards the liquid side of the diffuse interface and smaller towards the solid side. Such spurious behaviour is inherent to Phase Field models and is usually corrected by mobility correction schemes [20]. However, since the Finite Interface Dissipation model operates under strong non-equilibrium conditions, no such correction scheme has been developed yet. The partial driving force for redistribution, ΔΦredis , fills the balance to the total driving force as given by the grand potential ΔΦtot . Thus the driving force for redistribution is strongest at the solid side and weakest at the liquid side of the diffuse interface. The diffuse interface model therefore always tends towards the prediction of a larger driving force for interface motion than the sharp interface model. The same holds for the total driving force. Depending on the respective chemical potentials, the driving force for redistribution can be larger or smaller in either model. The interface mobility can therefore be expected to be significantly lower in the diffuse interface model than the coefficient for interface motion in the sharp interface model, while the interface permeabilites should be of a comparable order of magnitude. 4. Modeling rapid solidification of Al-0.2at.%Sn In the present work, steady state rapid solidification of Al-0.2at.%Sn, a model alloy for which experimental data are available [23], is investigated by both models. The thermodynamic data for the solid (Al) phase and the liquid mixture phase were taken from the COST database [22], the Gibbs free energies of the pure elements were taken from Ref. [21]. The equilibrium solidus and liquidus lines for the fcc-Al phase calculated by these data are shown in Fig. 2. The fcc phase exhibits a retrograde solidus below temperatures of ∼ 865 K, with a maximum Sn solubility of ∼ 0.21at.%. The calculated liquidus temperature Tl is 932.18K for the chosen composition of 0.2at.%Sn. Two equilibrium solidus temperatures Ts exist at 878.47 K and 848.34 K, corresponding to two solidification intervals Δ𝑇0 = 𝑇𝑙 − 𝑇𝑠 with magnitudes of 53.72 K and 83.84 K, respectively. For the investigation of steady state solidification of an alloy with a fixed initial composition, the sharp interface model can be reduced to a single equation. The velocity enters both fluxes (Eqs. (7) and (8)) linearly and can be eliminated, yielding 𝐿𝐷r = (𝑐l − 𝑐s ) 𝑀𝐷m .
(19)
Since in the steady state 𝑐s is fixed to 𝑐0 = 0.2at.%, only two parameters, temperature and liquidus concentration, remain. The equation was solved numerically for a fixed range of temperatures using Maple 2016 and the DirectSearch package. The corresponding solidification velocities were then calculated by Eq. (7). Phase Field simulations with the diffuse interface model were carried out by initializing both concentration fields to the nominal alloy concentration of 0.2at.%Sn. The phase field variable was initialized with its known sinus-shaped traveling wave profile [20]. The temperature T is an input parameter of the Phase Field model and uniform in space and time. A reasonable range for choosing the temperature T is obtained from the equilibrium phase diagram. Each chosen temperature results in a different steady state velocity v and partition coefficient k (see below). The interfacial mobility 𝑀sl was chosen such that the range of temperatures maps to the range of velocities reported in the experimental results. For numerical integration, a finite difference scheme with second order spatial discretization and first order forward time discretization was used. Explicit timesteps were performed until the system converged to a steady state. A moving frame was used to save computational resources and evaluate the velocity v. The moving frame moves in growth direction, i.e.
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Fig. 2. Equilibrium solidus and liquidus lines for fcc-Al in the Al-Sn system, as calculated from the data in Refs. [21,22]. The initial concentration c0 and the respective solidus and liquidus temperature(s) are marked. Note that the equilibrium solidus line has a retrograde trend below 865K and that the liquidus line ends in an eutectic point at a temperature of 502.4 K and a concentration of 98.0at.%Sn.
Fig. 3. Top: 𝑘 − 𝑣 curves calculated by the sharp interface model (circles) and by the diffuse interface model, the latter using two different values for the interface permeability (squares and diamonds, respectively), and experimental results of Smith and Aziz [23] (stars). Bottom: 𝑣 − 𝑇 -curves calculated with both models. The two equilibrium solidification intervals are indicated by dashed lines.
shifts the entire solutions of the phase and concentration fields against the growth direction as soon as the solid phase field variable 𝜙s increases to above 1/2 at a designated gridpoint. The inverse time intervals between shifts, together with the grid spacing, provide a direct measurement of the interface velocity. The steady state was determined by an observation of the concentration and phase field profiles. If no further change was visible, steady state was assumed to be reached, and the simulation stopped. From the steady state profile the solute partition coefficient was evaluated via [10] 𝑘(𝑣) =
𝑐s far-field solid concentration = . max[𝑐l ] maximum liquid concentration
(20)
It was found that the maximum liquid concentration is typically located at the solid side of the diffuse interface, where 𝜙s ≈ 1. Note that in this work the Finite Interface Dissipation model is operated in a regime of high interface Péclet-numbers Pe = 𝜂∕𝛿 = 𝜂𝑣∕𝐷l ≈ 105 . The smallest resolved length scale of the interface (determined by 𝜂) is therefore much larger than the characteristic length of solute diffusion 𝛿. In that case, all diffusive effects are described by the permeability term, which can be interpreted as a coarse-grained representation of the diffusion occuring in front of the interface [15]. 5. Results and discussion In Fig. 3a the velocity-dependent partition coefficients for the rapid solidification of Al-0.2at.%Si as calculated by both models are shown together with the experimental results from Ref [23]. The parameters of both models are listed in Table 1. The sharp interface model reproduces the experimental data in a satisfactory manner with the chosen
Table 1 Parameters used for the Al–Sn system. Sharp interface model Diffusion coefficient in the melt Effective interface thickness Maximum crystallization velocity
Dl a0 v0
m2 /s m m/s
1.3 × 10−7 exp(23.8kJ∕RT) [24] 1.4 × 10−9 1000 [6]
Dl 𝐷s Δt Δx P 𝜂 𝑀sl 𝜎
m2 /s m2 /s s m m3 /Js m m4 /Js J/m2
1.3 × 10−7 exp(23.8kJ/RT) [24] 0 10−11 10−6 varying, see text 10−5 5 × 10−8 104
Diffuse interface model Diffusion coefficient in the melt Diffusion coefficient in the solid Timestep Grid spacing Interface permeability Interface width Interface mobility Interfacial energy
parameters. Complete solute trapping (i.e., 𝑘 = 1) is predicted at velocities exceeding ∼ 13 m/s. Additionally, the model predicts a second branch of lower partition coefficients at lower temperatures. The diffuse interface model yields a differentiated result: For solidification velocities of ∼ 1.5 m/s, the experimental results are reproduced by an interface permeability of 3 × 10−4 m3 ∕Js. For higher velocities of ∼ 4 m/s a significantly lower interface permeability of 1.5 × 10−5 m∕Js is required to attain an agreement with the experimentally measured partition coefficients. By performing simulations with different interface widths 𝜂 (down to the order of magnitude of the sharp interface effective interface thickness) it was confirmed that the results for k(v) are invariable with respect
K. Reuther, S. Hubig and I. Steinbach et al.
to a change in 𝜂 as long as the permeability is changed simultaneously according to P∝1/𝜂 and the interface Péclet numbers are sufficiently high, i.e., Pe > 102 . This means that in the regime of high interface Péclet numbers the k(v)-curve is effectively a function of the product P · 𝜂 only. A second result is the drastically different description of the temperature dependence of rapid solidification by both models (Fig. 3b). The diffuse interface model predicts a monotonous relationship between undercooling and growth velocity, which is only weakly influenced by the different interface permeability parameters. The two undercoolings connected to equilibrium solidification do not appear to be special points in this curve, the equilibrium case is not represented as a limiting case. In fact it is not possible for the diffuse interface model to reach the equilibrium case at solidus temperature in the current setting. To reproduce the equilibrium case with the diffuse interface model, an infinitely small mobility 𝑀sl ≈ 0 and an infinitely high permeability P ≈ ∞ would be necessary. The sharp interface model does reproduce equilibrium solidification as a limiting case for vanishing interface velocities. The two observed branches are consequently connected to undercoolings above the upper and below the lower equilibrium solidus temperature. Between the two solidus temperatures, no solutions to Eq. (19) were found. The upper branch, corresponding to the experimentally observed values, exhibits a decreasing undercoooling with higher velocities, from equilibrium solidification (𝑣 = 0) at the equilibrium solidification interval of ∼ 54 K down to an undercooling of ∼ 34 K at velocities up to ∼ 7 m/s. Only at that point does the relation invert, and higher undercoolings are again required for higher velocities. It is pertinent to note that for the parameter set chosen for the diffuse interface model, the parameter K is essentially constant and equal to 𝑀sl . By taking into account the actual values of the different contributions in Eq. (15), it can be seen that the limit 𝐾 ≈ 𝑀sl is indeed a good approximation. Additional simulation runs were conducted with the K-parameter replaced by the constant 𝑀sl with no observable changes in the results. For different alloy systems which do not fulfill this limit, however, the diffuse interface model may produce qualitatively different results. 6. Conclusions Although the two models investigated in this work are similar in their concept of two separate processes at the solid/liquid interface of a rapidly solidifying alloy, they are based on distinctly different descriptions of the involved thermodynamics. They consequently provide different descriptions of the steady state solute trapping behaviour in the Al–Sn system. With the chosen kinetic parameters, the sharp interface model reproduces the experimentally measured velocity-dependent partition coefficients quite well. Due to its sharp interface, the model is not influenced by the change in width of the diffusion zone in front of the interface with varying interface velocity. The diffuse-interface model in contrast is affected by the change in diffusion zone width, as the diffusion zone is completely situated inside the diffuse interface. The permeability parameter describing solute redistribution both in the bulk liquid near the interface and between the phases therefore has to vary with velocity to describe the experimental values. In the present study only the sharp interface model converges towards thermodynamic equilibrium at the interface for v → 0 at the solidus temperatures.
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