Driving force for binary alloy solidification under far from local equilibrium conditions

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ScienceDirect Acta Materialia 93 (2015) 256–263 www.elsevier.com/locate/actamat

Driving force for binary alloy solidification under far from local equilibrium conditions S.L. Sobolev Institute of Problems of Chemical Physics, Academy of Sciences of Russia, Chernogolovka, Moscow Region 142432, Russia Received 23 March 2015; revised 17 April 2015; accepted 17 April 2015

Abstract—In this study we have incorporated the local nonequilibrium term into the Gibbs equation to obtain the local nonequilibrium correction to the driving force for solidification of binary alloys. The local nonequilibrium correction has been used to calculate the effective liquidus slope and the interface temperature for the model with and without solute drag effects. It has been demonstrated that both the local nonequilibrium correction and the solute drag play the most important role in the intermediate rage of the interface velocity V. When V increases up to V D , where V D is the characteristic diffusive velocity, a sharp transition to completely diffusionless and partitionless solidification occurs, which implies that the local nonequilibrium correction and the solute drag effects can be ignored. The transition is accompanied by a sharp change in the effective liquidus slope and the interface temperature as functions of interface velocity. The model was applied to describe initial transient and steady-state solidification of Si–As alloy and a good agreement between the model predictions and the available experimental data was obtained. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Rapid solidification; Binary alloys; Gibbs energy; Liquidus; Extended thermodynamics

1. Introduction In the past few decades, modeling of rapid solidification and solute trapping has gained attention for its role in many modern solidification processes such as thermal spraying, spin coating, and laser melting [1–3]. At deep undercooling the solidification process involves extremely fast heat and mass transfer at very small time and length scales, which implies that the process occurs under far from equilibrium conditions [3–30]. In such a case the classical thermodynamics cannot be applied due to violation of the local-equilibrium hypothesis and the local-nonequilibrium theory should be used. The most popular version of the local-nonequilibrium thermodynamics is the so-called extended irreversible thermodynamics (EIT) [31], which includes dissipative fluxes in the set of independent variables. Review of some other local nonequilibrium approaches can be found in [6,7,20,32–35]. It is remarkable that all the extended approaches lead to a hyperbolic diffusion equation as the simplest generalization of the classical diffusion equation of parabolic type to the local nonequilibrium case. The local-nonequilibrium transfer equation of hyperbolic type has been used to describe traveling waves of phase transformations under far from local equilibrium conditions [6]. This approach has been successfully applied to rapid solidification of binary alloys [8–12]. It has been demonstrated that the local nonequilibrium diffusion effects E-mail: [email protected]

are responsible for a sharp transition from diffusion controlled to diffusionless solidification with complete solute trapping at a finite interface velocity [8–12]. In recent years the local nonequilibrium diffusion approach has been extended to study different types of solute trapping models [13–16], rapid colloidal solidification [17], non-equilibrium dendrite growth [18,29,30], multicomponent alloy solidification [19], space nonlocal effects due to diffusion-stress coupling in liquid phase [20,21], solute drag effects [23– 26], morphological stability [27,28], nonlinear liquidus and solidus [23,27,29,30]. The interface velocity V is related with the kinetic driving force for the interface motion DG through the reaction rate theory as V ¼ V 0 ½1  expðDG=RT Þ

ð1Þ

where DG is the free energy change per mole between the solid and liquid phases, i.e. the driving force for solidification, R is the gas constant, T is the interface temperature, V 0 is the interface velocity at an infinite driving force. Under local nonequilibrium condition DG can be represented as [10,36] DG ¼ DG0 þ DGneq 0

ð2Þ

where DG is the local equilibrium part of the Gibbs free energy change, which can be calculated using classical irreversible thermodynamics, and DGneq is the local nonequilibrium correction to the Gibbs free energy change, which arises due to local nonequilibrium effects and should be calculated on the basis of EIT [31]. The additional term, DGneq ,

http://dx.doi.org/10.1016/j.actamat.2015.04.028 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

S.L. Sobolev / Acta Materialia 93 (2015) 256–263

was taken into account for calculation of the effective liquidus slope [26], which has been widely used in theoretical modeling of rapid alloy solidification but some uncertainties exist [23,27–30]. In this paper we calculate the local nonequilibrium correction to the change in free energy of transformation during rapid alloy solidification using EIT approach. We also calculate the effective liquidus slope and the interface temperature with allowance for the local nonequilibrium and solute drag effects and compare the model predication with available experimental data. 2. Local nonequilibrium thermodynamic functions 2.1. Local nonequilibrium entropy and free energy for binary system The local nonequilibrium Gibbs equation for binary system is [31]: Tds ¼ du þ pdq1  ldC  saq2 D1 JdJ

ð3Þ

where T is temperature, s is a local specific entropy, C is concentration of component 1, a ¼ @l=@C, l ¼ l1  l2 , li is chemical potential of component, s is relaxation time to local equilibrium of the diffusion flux J, D is diffusion coefficient, q is mass density, p is pressure, u is specific internal energy. The local nonequilibrium correction to the classical Gibbs equation is represented by the last term in Eq. (3), which arises due to the diffusion flux J and depends on its relaxation time to local equilibrium s. If the characteristic time of the process t0  s, the nonequilibrium correction is small and can be ignored. In this case Eq. (3) reduces to the classical (local equilibrium) Gibbs equation. Far from local equilibrium t0  s and the nonequilibrium correction should be taken into account. Integration of Eq. (3) allows one to obtain the local nonequilibrium Gibbs free energy G in the form [31] G ¼ G0 þ Gneq where G0 is the local-equilibrium Gibbs free energy, and Gneq is the local non-equilibrium correction, which is given as   s @l neq 2 ð4Þ G ¼J 2Dq2 @C The local nonequilibrium entropy can be written in the analogous form: S ¼ S 0 þ S neq S neq ¼ J 2

s 2Dq2 T



@l @C



2.2. Local nonequilibrium correction to the driving force for solidification The local nonequilibrium correction to the driving force for solidification in Eq. (2), DGneq , is the difference between the local nonequilibrium correction to the Gibbs free energy in the liquid (L) Gneq and in the solid (S) Gneq L S . Using Eq. (4) one can obtain     s @l sS @lS DGneq ¼ J 2L  J 2S 2D @C 2DS @C S

257

Taking into account that the solute diffusion flux in the solid phase is very small in comparison with that in the liquid phase, the nonequilibrium corrections for the driving force for solidification at the interface can be written as   s @l ð5Þ DGneq ¼ J 2L 2D @C Note that all the parameters in these expressions refer to the liquid phase. Thus the local nonequilibrium driving force for solidification and the entropy change at the interface of binary mixture take the form   s @l ð6Þ DG ¼ GL  GS ¼ DG0 þ J 2L 2D @C For ideal solution Eq. (5) reduces to DGneq ¼ s RTJ 2L =2DC L

ð7Þ

The mass balance law leads to the following interface condition V ðC L  C S Þ ¼ J L  J S

ð8Þ

where V is interface velocity. Taking into account that J S is much smaller than J L , Eq. (8) gives J L ¼ VC L ð1  KÞ, where K ¼ C S =C L is solute partition coefficient. Substituting this expression for JL in Eq. (7), we obtain the nonequilibrium corrections to the Gibbs free energy change: DGneq ¼ C L RT ð1  KÞ2 V 2 =2V 2D

ð9Þ

where V D ¼ ðD=sÞ1=2 is the characteristic diffusion velocity [8–10]. Thus the total Gibbs free energy change at the interface during rapid alloy solidification with allowance for the local nonequilibrium effects is given as DG=RT ¼ DG0 =RT þ C L ð1  KÞ2 V 2 =2V 2D

ð10Þ

2.3. Steady-state regime The exact mathematical treatment of the rapid alloy solidification problem may be found by solving the time-dependent governing equations, subject to appropriate initial and boundary conditions. But to avoid mathematical difficulties the steady-state approximation is usually used. The steady-state assumption assumes that after a short initial transient period the solid–liquid interface begins to move with constant velocity and the solute concentration is time-independent in the reference frame attached to the interface. Steady-state assumption significantly simplifies the mathematical formulation of the problem. Strictly speaking, the mathematical analysis shows that steady-state solidification takes an ‘infinite’ amount of time and distance to be achieved [1]. The local nonequilibrium approach gives the estimate of the characteristic time tst to achieve steady-state as follows [22]: tLNDM ¼ cDð1  V 2 =V 2D Þ=KV 2 ; st

V
ð11Þ

here c is a constant such as at t ¼ tst the interface solute concentration in the liquid is about 70% and 90% of the steady state concentration for c ¼ 1 and c ¼ 2, respectively [1]. When V P V D , alloy solidification occurs in the diffu¼ 0. Eq. (11) sionless regime, which implies that tLNDM st implies that the local nonequilibrium effects shrink the initial transient period in binary alloy solidification and for many practical situations, especially at high interface

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S.L. Sobolev / Acta Materialia 93 (2015) 256–263

velocity, the departure from the steady-state may be ignored but at low interface velocity and small partition coefficient the initial transient period can be significant. For steady-state solidification with planar interface the solute concentration in the solid C S is equal to the initial solute concentration in the liquid C o , which gives the solute concentration in the liquid at the interface as C L ¼ C o =K. This allows us to rewrite Eq. (9) as follows DGneq =RT ¼ C 0 ð1  KÞ2 V 2 =2KV 2D

ð12Þ

neq

As the value of DG in Eq. (12) depends on K, it becomes necessary to obtain K as a function of the interface velocity V with allowance for the deviation from local nonequilibrium conditions. The local nonequilibrium diffusion model (LNDM) [8–14], which takes into account the deviation of solute diffusion in the liquid phase from local equilibrium using hyperbolic diffusion equation [5–7,20,31–34], gives partition coefficient for dilute solution in the form ( K ð1V 2 =V 2 ÞþV =V i E D ; V V D eq where K E ¼ C eq S =C L is the equilibrium partition coefficient, Vi is the so-called interface diffusive velocity, i.e. the characteristic solute trapping velocity above which K deviates strongly from KE. Eq. (13) can be generalized for non-dilute solution as ( ½k þð1k ÞC ð1V 2 =V 2 ÞþV =V o i E E D ; V V D

where kE is the equilibrium partition coefficient for the solute divided by the equilibrium partition coefficient for the solvent [13,14]. The more recent local nonequilibrium approach [15] describes K(V) as a power law given by ( 1=½1þAV =ðV D V Þ KE ; V V D where A is a parameter analogous to the reciprocal of Vi. Note that Vi as well as A is a kinetic rate parameter characterizing solute trapping at the solid–liquid interface, whereas VD is a purely diffusive parameter, which is equal to the propagation velocity of concentration disturbances in the bulk liquid [8–14]. Eqs. 13–15 predict a sharp transition to complete solute trapping K ¼ 1 at a finite interface velocity V = VD, which is a consequence of the local nonequilibrium diffusion effects [8–16]. As the interface velocity V increases, these effects depress solute diffusion in the liquid phase leading to completely diffusionless and partitionless solidification at V > VD [8–16]. Note that the local equilibrium models predict K ! 1 only asymptotically at V ! 1 (see Refs. [13,14] and references wherein). This result, i.e. the sharp transition to partitionless and diffusionless solidification at a finite interface velocity, is confirmed by molecular dynamic simulations [37–39], phase-field [40], and phase-field-crystal models [41,42]. In further calculation we use Eq. (15) keeping in mind that Eqs. (13) and (14) demonstrate analogous asymptotic behavior and predict K-values that differ insignificantly for the system under consideration. Substituting Eq. (15) in Eq. (12) we have

DGneq ¼

RTC 0 V 2D

 ( 2=½1þAV ðV D V Þ 1=½1þAV =ðV D V Þ =2K E 1  KE ; 0;

V < VD

ð16Þ

V >VD

Eq. (16) demonstrates that DGneq tends to zero both at low interface velocity V ! 0 and at relatively high interface V ! V D . At low interface velocity the solidification occurs under local equilibrium conditions and the local nonequilibrium effects can be ignored at all. At V ! V D the local nonequilibrium effects depress solute diffusion flux J, which, in turn, decreases the local nonequilibrium correction. When the interface velocity exceeds the critical point V ¼ V D , there is no solute diffusion and solute flux in the liquid phase at all, which implies that DGneq ¼ 0. Thus the local nonequilibrium correction plays an important role at intermediate values of the interface velocity 0 < V < V D. 3. Interface temperature and effective liquidus slope under local nonequilibrium conditions 3.1. Interface temperature The thermodynamic constraints on the compositions at the interface give the magnitude of the free energy change for the formation of one mole of solid of composition C S from a liquid of composition C L in the following form [43,44] DG0 ¼ ð1  C S ÞDlA ðC L Þ þ C S DlB ðC L Þ lSA

lLA

where DlA ¼  and DlB ¼ tion Eq. (17) reduces to [43,44]

lSB



ð17Þ lLB .

For ideal solu-

DG0 =RT ¼ ð1  K E ÞðC L  C eq L Þ þ C L ðK E  KÞ þ C L K ln K=K E

ð18Þ

Substituting Eq. (18) in Eq. (10) and taking into account eq eq that the interface temperature T ¼ T A þ meq L C L , where mL is the equilibrium liquidus slope and T A is the melting temperature of major component, we obtain DG ð1  K E Þðmeq L CL  T þ T A Þ ¼ RT meq L     2 K 2 V þ ð1  KÞ þ C L K E  K 1  ln KE 2V 2D

ð19Þ

Combining Eq. (19) with kinetic law in the form of Eq. (1) we obtain the interface temperature as follows     meq C L K V2 þ ð1  KÞ2 2 T ¼ TA þ L 1  K 1  ln KE 1  KE 2V D   meq V L ln 1   ð20Þ V0 1  KE In the diffusionless regime at V > V D the complete solute trapping K = 1 is reached and Eq. (20) gives 1

T ¼ T 0  meq L ð1  K E Þ

lnð1  V =V 0 Þ

meq L C 0 ð1

1

ð21Þ

where T 0 ¼ T A   K E Þ ln K E is the temperature of equal free energy of two phases [43]. Eq. (21) implies that in the completely diffusionless and partitionless regime (V > VD) the alloy behaves as a “pure metal” with melting temperature T0.

S.L. Sobolev / Acta Materialia 93 (2015) 256–263

3.2. Effective liquidus and solidus slope

1,2

ð22Þ which rewrites Eq. (21) for the interface temperature in the following form

meq L 1  KE

meq L m3 ¼ 1  KE

 (

bð1  KÞ ln K=K E ; 0; V > V D ð1  KÞ2 V 2 =2V 2D ; 0; V > V D

V
V
0,2 2

ð24Þ

where b is a solute drag parameter defined as zero for the model without solute drag effects and unity for the model with solute drag effects. Eq. (24) for the effective liquidus slope mneq allows for both the local nonequilibrium and L solute drag effects. The effective liquidus slope, Eq. (24), contains three parts: m1 ¼ meq L ð1  K þ K ln K=K E Þ= ð1  K E Þ is the main term, m2 ¼ meq L ½bð1  KÞ ln K=K E = 2 ð1  K E Þ is the solute drag term, and m3 ¼ meq L ð1  KÞ 2 2 V =2V D ð1  K E Þ is the part of the effective liquidus slope due to the local nonequilibrium correction to the Gibbs free energy. Taking into account that K = 1 at V > VD, we obtain  1  K þ K ln K=K E ; V < V D meq L ð25Þ m1 ¼ 1  K E  ln K E ; V > V D

m2 ¼

0,8

ð23Þ

For relatively low interface velocity V  V D , when the local nonequilibrium correction is negligible, Eqs. (22) and (23) reduce to the well-known expressions for the interface temperature T ¼ T A þ mL C L  V =l and velocity-dependant liquidus slope mL ¼ meq L ð1  Kþ K ln K=K E Þ=ð1  K E Þ [43,44], where l ¼ V o ðK E  1Þ=meq L is the interface kinetic coefficient. Taking into account the solute drag effects (see, for example, Ref. [45]), we can rewrite Eq. (22) as eq mneq L ¼ mL ½1  K þ K ln K=K E þ bð1  KÞ ln K=K E i þ ð1  KÞ2 V 2 =2V 2D =ð1  K E Þ

1 1,0

Liquidus slope

Eq. (21) allows us to introduce the effective liquidus slope mneq L as h i 2 2 neq 2 mL ¼ meq L 1  K þ K ln K=K E þ ð1  KÞ V =2V D =ð1  K E Þ

1 eq T ¼ T A þ mneq lnð1  V =V 0 Þ L C L  mL ð1  K E Þ

259

ð26Þ

ð27Þ

Note that the local nonequilibrium correction to the effective liquidus slope, Eq. (27), is proportional to ðV =V D Þ2 , whereas the result given by Galenko [26] states that the correction is proportional to V =V D , which seems to be ether a mathematical mistake or misprint. The difference does not quantitatively affect the effective liquidus slope very much, but it may be important in the context of other problems, such as the interface stability or phase transformation near the critical points. In Fig. 1 we plot m1 (solid curve 1), Eq. (25), m2 with complete solute drag term b = 1 (solid curve 2) and with partial solute drag b = 0.3 (dashed curve 2a), Eq. (26), and m3 (solid curve 3) scaled with meq L =ð1  K E Þ as functions of the nondimensional interface velocity V/VD calculated using the effective

2a 3 0,0 0,0

0,2

0,4 0,6 0,8 Nondimensional interface velocity

1,0

1,2

Fig. 1. Components of the effective liquidus slope mneq scaled with L meq L =ðK E  1Þ as functions of the nondimensional interface velocity V/ VD: curves 1 – the main term m1 for the local nonequilibrium case (solid line) and local equilibrium case (dashed line); curves 2 - the solute drag term m2 ; curve 3 – the nonequilibrium correction term m3 . Curves 2 – complete (b = 1) solute drag with local nonequilibrium effects (solid line) and without local nonequilibrium effects (dashed curve); dashed curve 2a – partial solute drag (b = 0.3) with local nonequilibrium effects.

partition coefficient Eq. (15). For comparison we also plot m1 and m2 calculated without local nonequilibrium effects (dashed curves 1 and 2 in Fig. 1, respectively). Fig. 1 demonstrates that as the interface velocity V increases, the main part of the liquidus slop m1 increases monotonically and reaches the slope of the T 0 line at V ¼ V D , whereas the solute drag term m2 and the term due to the local nonequilibrium correction m3 go through a maximum and equal zero at V P V D . As the interface velocity V increases, m2 and m3 first increase because the deviation from local equilibrium is proportional to the value of the interface velocity V. But further increase of the deviation from local equilibrium leads to the depressed solute diffusion due to more pronounced solute trapping effects, which decreases m2 and m3 at V ! V D . It implies that close to the critical point V ¼ V D , the contribution of m2 and m3 to the total effective liquidus slope mneq L is relatively weak in comparison with the contribution of m1 , which value is affected by the local nonequilibrium effects through the effective solute partition coefficient K (see Eq. (25)) represented by Eqs. 13–15. Note that the local equilibrium liquidus slope m1 reaches the T 0 line only asymptotically at V ! 1 (see dashed curve 1 in Fig. 1). The solute drag term m2 also comes to zero only asymptotically at V ! 1 (see dashed curve 2 in Fig. 1). The importance of solute drag in rapid alloy solidification is still under discussion [23–30,45–48] but in any case at high interface velocity V  VD near the transition to diffusionless solidification the effects of solute drag are relatively small.

4. Application to Si–As alloy solidification In recent years rapid solidification of Si–As alloys is one of the most important subjects of great practical and scientific interest [22–26,40,46–48]. The Si–As systems have been studied extensively, but large discrepancies still exist in the

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experimental results and theoretical modeling. Rapid solidification of polycrystalline Si-4.5 at.% As and Si-9 at.% As alloys at velocities ranging from 0.25 to 2 m/s was studied by Kittl et al. [46–48] using the pulsed laser melting technique. The congruent melting temperature T0 [46], the solute partition coefficient [47,48], and the interface temperature [48] were measured as functions of the interface velocity. Theoretical description of Kittl et al. experimental data is under an extensive discussion [22–26,40,46–48] but there are some difficulties in describing the experimental data at low interface velocity due to breakdown of the steady state assumption [22–25]. Moreover the role of solute drag effects during rapid alloy solidification is still unclear and there are some controversial points in theoretical modeling of the local nonequilibrium phase diagram [22–25,46–48].

the slope of the T0 line, which implies that the alloy solidifies completely diffusionless and partitionless. Both the local nonequilibrium correction and the solute drag increase the effective liquidus and solidus slopes at a given interface velocity (see Fig. 2). When V ! V D , the local nonequilibrium effects depress solute diffusion near the interface, which, in turn, decreases the local nonequilibrium correction and the solute drag effects. Note that despite the local nonequilibrium correction to the driving force being negligible at V ! V D , the local nonequilibrium solidus and liquidus slopes differ substantially from their local equilibrium counterparts through the effective partition coefficient, Eqs. 13–15.

4.1. Effective liquidus and solidus slopes

To describe the Kittl et al. experiment [46–48] it is more convenient to express the interface temperature, Eq. (23), in terms of the T0 temperature. Taking into account solute drag effects (see Eqs. (24) and (26)), Eq. (23) gives

To find the interface temperature T as a function the interface velocity V for the Kittl et al. experiments [46– 48] we first calculate corresponding effective liquidus and solidus slopes using Eqs. 25–27 with allowance for the effective partition coefficient K, Eq. (17). The measured K(V) relations with the commonly accepted value of K E ¼ 0:3 [48] has been fitted to the theoretical prediction to obtain the diffusion velocity in the bulk liquid VD, which falls in the range from 2.45 m/s to 2.8 m/s [9,10,13–15]. For the current calculations VD = 2.5 m/s has been used. The preneq dicted effective liquidus mneq solidus slopes scaled L and mS eq with mL =ð1  K E Þ are shown in Fig. 2. Solid curves represent mneq and mneq without solute drag effects (b = 0), L S dash-dotted curves – with partial solute drag effects (b = 0.3), and dotted curves – with solute drag effects (b = 1). The dashed curves are the effective liquidus and solidus slope without solute drag effects and local nonequilibrium correction. As the interface velocity V increases, the effective liquidus slope mneq monotonically increases L whereas the effective solidus slope mneq monotonically S decreases. At V P V D ¼ 2:5 m/s they coincide reaching

2,4

Liquidus and solidus slopes

2,1

1,8

1,5

1,2

0,9

0,6 0,0

0,5

1,0

1,5

2,0

2,5

Interface velocity

Fig. 2. Effective liquidus mneq and solidus mneq slopes scaled with L S meq =ð1  K Þ as functions of the interface velocity V m/s calculated for E L neq Si–As experiment [48]. Solid curves - mneq L and mS without solute drag effects (b = 0); dash-dotted curves – with partial solute drag effects (b = 0.3); dotted curves – with solute drag effects (b = 1); dashed curves – without solute drag and nonequilibrium correction to the driving force; VD = 2.5 m/s.

4.2. Interface temperature

T ¼ T A þðT A T 0 Þ 1K þK lnK=K E þbð1KÞlnK=K E þð1KÞ2 V 2 =2V 2D  ðC 0 =C L ÞlnK E 

V l

!

ð28Þ

The effective partition coefficient K as a function of V in Eq. (28) is given by Eq. (15) whereas CL is unknown function of V. To find CL as a function of V let us first compare the characteristic time of the experiment tex with the characteristic time tst , needed for the interface to reach the steady-state conditions. The characteristic time of the experiment is the ratio of the interface depth, d = 30 nm, at which the interface temperature was measured [48], and the interface velocity V, i.e. tex ¼ d=V . The characteristic time needed for the interface to approach steady-state condition, tst , is given by Eq. (11) using the parameters, which correspond to the Kittl et al. experiment, i.e. D ¼ ð1; 5  0; 5Þ  108 m2/s and K E ¼ 0:3 [48]. Note that there is some uncertainty in K E , which value falls within the range from K E ¼ 7  104 to K E ¼ 0:3 [48]. For further calculations we use here the commonly accepted value of K E ¼ 0:3 [48], keeping in mind that K E ¼ 7  104 would give higher value for tst . Comparing the characteristic times tst and tex , we obtain that at V < V 1 1:1 m/s for c ¼ 1 and at V < V 2 2:2 m/s for c ¼ 2 the characteristic time of the experiment tex is less than the characteristic time tst to achieve steady-state conditions. This implies that the interface depth d, at which the interface temperature was measured in Kittl et al. experiments [48], was not kept high enough to ensure the steady-state conditions at low interface velocity. Thus the solidification process in the Kittl et al. experiments [46–48] occurs in the steady state regime at relatively high interface velocity and in the initial transient, i.e. non-steady-state regime, at relatively low interface velocity. To take it into account in our theoretical modeling, we calculate two T vs V functions: T1, which corresponds to steady-state tex > tst , and T2, which corresponds to the non-steady-state conditions tex < tst . In the steady-state the solute concentration in the solid C S equals to the initial solute concentration in the liquid C o , and, hence, the ratio C o =C L in Eq. (28) gives the solute partition coefficient K. Thus, substituting the effective partition

S.L. Sobolev / Acta Materialia 93 (2015) 256–263

coefficient K, Eq. (15), into Eq. (28) and taking into account that C o =C L ¼ K, one obtains the interface temperature T1 as a function of the interface velocity V for steady-state. The resulting T(V) functions are shown in Fig. 3 in comparison with the experimental measurements of Kittl et al. [47,48]. Solid curve 1 represents the interface temperature without solute drag, dash-dotted curve 1 – with partial solute drag b = 0.3, and dashed curve 1 – without solute drag and local nonequilibrium correction to the driving force. The value for interface kinetic coefficient l was taken from the experiment of Kittl et al. [48], which adopted l1 ¼ 15 K s/m. The solid curve 1 in Fig. 3 demonstrates a good agreement between the model prediction and the experimental data at relatively high interface velocities with fitting parameter T 0 ¼ 1423 K, which corresponds to the experimental result T 0 ¼ 1425  25 K determined by Kittl et al. experiments [46]. At relatively low interface velocity the steady-state is not reached yet, as it has been discussed above, which leads to disagreement between the steady-state model prediction (solid curve 1) and the experimental results. To calculate T2(V) function at relatively low interface velocity one should take into account that in the beginning of the solidification process before the interface starts to move, the solute concentration at the interface equals to the initial solute concentration C o . During initial transient the solute concentration at the interface increases from C L ¼ C o at the beginning of the initial transient to its steady-state value C L ¼ C o =K at the end of the initial transient. This implies that during initial transient the solute concentration at the interface CL in Eq. (28) can be approximated by the average solute concentration C L , which lies in the range C 0 < C < C 0 =K, i.e. C L ¼ eC o , where e > 1. Fig. 3 shows T2 vs V function for initial transient with partial solute drag effects b = 0.3 (dashed curve 2), without solute drag (solid curve 2), and without solute drag and local nonequilibrium correction (dash-dotted curve 2) calculated with the same T 0 temperature (1423 K) and fitting parameter e ¼ 1:25. The main feature observed when comparing the model predictions with the experimental data in the low range of the measured 1500

Interface temperature, K

1450 1

1400

2

1350

1300 0,0

0,5

1,0

1,5

2,0

2,5

Interface velocity, V m/s

Fig. 3. Interface temperature T as a function of the interface velocity V m/s. Curves 1 – steady-state; curves 2 – initial transient; thick gray line – transition between initial transient and steady-state; spheres – experimental points [48]. Parameters: K E ¼ 0:3; T 0 ¼ 1423 K; VD = 2.5 m/s; e ¼ 1:25. Solid curves – without solute drag (b = 0), dash-dotted curves – with partial solute drag (b = 0.3), and dashed curves – without solute drag and local nonequilibrium correction.

261

interface velocity is the increase of the calculated initial transient temperature T2 with decreasing interface velocity V, which corresponds to the experimental data (see curves 2 in Fig. 3). In contrast, the steady-state interface temperature T1 decreases with decreasing interface velocity V in the low-velocity regime, which contradicts the experimental data (see curves 1 in Fig. 3). Thus, the non-steady-state approach provides better qualitative and quantitative agreement with the experimental data in the low-velocity regime than the steady-state one. This confirms our prediction made by comparison between the characteristic times tst and tex that at low velocities the steady-state in the Kittl et al. experiment is not reached. Thus the experimental data of Kittl et al. [46–48] for rapid solidification of Si-As alloy can be interpreted as follows: (i) in the low-velocity regime V < V 1 , the characteristic time of the experiment tex is less than the characteristic time needed to reaches the steady-state tst , which can be treated as initial transient (ii) at high interface velocity V > V 2 the inverse inequality holds (tex > tst ), i.e. the solidification occurs in steady-state; (iii) in the range V 1 < V < V 2 , the characteristic times are of the same order (tex tst ) and transition from initial transient to steady-state regime occurs. The gray line in Fig. 3 approximates the interface temperature in the intermediate regime between the initial transient (solid curve 2) and the steady-state (solid curev1), which occurs between V 1 0:8 m/s and V 2 1:7 m/s. This corresponds to the condition tex tst , which holds within the range between V 1 and V 2 . 4.3. Comparison with other models Li et al. [23] proposed thermodynamic approach for silicon–arsenic alloy with non-straight liquidus and solidus curves. The model is consistent with the available experimental data if significant solute drag effects are introduced. The differences between the model and other typical models for Si–As alloy solidification were investigated by using the site fraction f as a fitting parameter to obtain a better description of the experiment data for each model. In another paper the same authors considered a generalized solute drag model for binary alloy solidification with a planar phase interface proposed as an extension of Hillert– Sundman model [24]. Using a new thermodynamic parameter set of the Si–As system, the model can, with the introduction of three new adjustable parameters, fit the available experimental data. Note that the difference between Li et al. model and other discussed models [23,24] is of the order of the error of the experiment (see, for example, Figs. 2 and 3 in Ref. [23] and Fig. 2 in Ref. [24]), that makes it difficult to judge which model provides more realistic description of the experiment. The model calculation of the interface thickness based on steady-state assumption gave unphysically large values at low interface velocity, which led to conclusion that the steady-state condition in the Kittl et al. experiments may not be achieved at these velocities [24]. To correct their steady-state model Li et al. approximated the solute concentration in the solid phase at the interface by velocity dependant function for V < 0.8 m/s [23,24]. This corresponds to our prediction that solidification in the Kittl et al. experiments occurs in non-steady-state regime at V 6 0:8 m/s. Note that the effective partition coefficients K(V) predicted by the generalized solute drag model of Li et al. [24] and our early LNDM without solute drag [8–13] are nearly indistinguishable.

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Wang et al. [25] described the steady-state planar solidification of Si–As alloy using a maximal entropy production principle. They analyzed the phase diagram of the Si–As system and concluded that the steady-state planar solidification with C S ¼ C o is not possible for Si–9 at.% As alloy if V is not sufficiently high. They also noticed that their prediction of the interface temperature T based on the steady state approach is good for V > 0.8 m/s but not that good for V < 0.8 m/s for any sets of parameters (see, for example, Figs. 7b and 9b in Ref. [25]). These conclusions correspond to the present model, which predicts the non-steady-state regime at low interface velocity V < V 1 0:8 m/s (see Fig. 3). Prediction of the effective partition coefficient K(V) based on the maximal entropy production principle [25] corresponds to the LNDM results [8–13]. Thus, the extended models [23–25] with allowance for non-straight liquidus-solidus lines and maximal entropy production principle, may give more realistic solution at high interface velocity, but still have difficulties in describing low-velocity regimes of Si-As alloy solidification due to the steady-state assumption. The non-steady-state effects qualitatively change the behavior of the interface temperature vs. interface velocity function in the low-velocity regime. Combination of the non-steady state approach with the extended Si-As solidification models [23–25] is an interesting topic for future work and will be presented elsewhere. More experimental work and direct measurements of interface response functions and phase diagrams for Si-As alloys are required to confirm the results of theoretical modeling and distinguish between different physical effects. 5. Conclusion At high interface velocity the Gibbs free energy and entropy of the phases involved in alloy solidification depend on the dissipative (diffusion) fluxes. This yields the local nonequilibrium correction to the driving force for solidification, which is proportional to ðV =V D Þ2 at V < VD. The local nonequilibrium correction, as well as the solute drag effects, plays the most important role at an intermediate value of the interface velocity 0 < V < VD, whereas at V  VD and at V P V D it can be ignored. When V ! VD, the local nonequilibrium effects decrease solute diffusion flux near the interface (see Ref. [8–14]) and increase the effective partition coefficient K, Eqs. 13– 15, which, in turn, decreases the local nonequilibrium correction and the solute drag. At the same time, the depressed solute diffusion leads to a sharp transition to diffusionless and partitionless (K = 1) solidification at V = VD, which is the main manifestation of the local nonequilibrium diffusion effects. In the intermediate range of the interface velocity the local nonequilibrium correction to the thermodynamic functions may also be important in the phase-field and phase-field-crystal modeling, which describe the interface region in rapid alloy solidification by minimization of some specified free energy functional [40–43]. Acknowledgments The reported study was partially supported by RFBR, research Project No. HR 14-48-03535n14.

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