Modeling of non-equilibrium solute diffusion upon rapid solidification: Effective mobility versus kinetic energy

Abstract

The effective mobility approach is compared with the kinetic energy approach in terms of sharp interface modeling and phase-field modelling of non-equilibrium solute diffusion upon rapid solidification of binary alloys. The two approaches are equivalent for modelling of long range solute diffusion in bulk phases, but only the effective mobility approach can introduce the non-equilibrium solute diffusion effect to short range solute diffusion at a sharp interface or within a diffuse interface. Addition of the kinetic energy terms results in an unreasonable non-bilinear expression of the flux and thermodynamic driving force in the free energy production of interface migration or phase field propagation, whereas the effective mobility approach allows the thermodynamic extremal principle workable.

Keywords

Solute diffusion Thermodynamics Kinetics Rapid solidification Theory

This paper is part of a special issue on Applications of Irreversible Thermodynamics in Metallurgy and Materials Science

Introduction

Rapid solidification is extremely interesting for both scientific and technological reasons. It is not only a good technique for preparing materials of potential novel properties but also a good subject for studying non-equilibrium phenomena.¹ Upon rapid solidification, several irreversible dissipative processes occur concurrently, e.g. solute diffusion in bulk phases, trans-interface diffusion (or solute redistribution) and interface migration at the growing sharp interface.^2,3 The first and second dissipative processes belong respectively to long and short range solute diffusion.^4,5

The solute diffusion velocity in bulk liquid (of order 0.1–10 m s^− 1)⁶ determines how fast a solute atom can travel in liquid.⁷ The interfacial solute diffusion velocity is comparable with but is smaller than .⁸ [The diffusion velocity is a characteristic velocity above which solute diffusion cannot occur any more. It can be defined as V_D = (D/τ)^1/2 with D the diffusion coefficient and τ the relaxation time of solute diffusion to its equilibrium state.] The maximal interface velocity V_o is equal to the speed of sound if interface migration is collision limited.¹ Practically, the measured interface velocity V, which approaches 10–100 m s^− 1,¹ can be comparable with or even larger than . In this case, solute diffusion is not only determined by its instantaneous concentration gradient but also relevant to its local evolution history, i.e. non-equilibrium solute diffusion (NESD).⁹ Modelling of NESD upon rapid solidification, therefore, becomes quite important for the understanding of non-equilibrium phenomena.^3,6–24 Two different recipes have been proposed, i.e. the kinetic energy approach¹³ and the effective mobility approach.²³

Kinetic energy approach

In order to describe NESD by phase field modelling, the ‘slow’ variable in the classical irreversible thermodynamics {c} and the ‘fast’ variable in the extended irreversible thermodynamics,⁹ can be integrated to form a general space of variables .¹³ This is equivalent to add a so called kinetic energy term to the classical free energy density. Here, c is the overall solute molar fraction, is the overall solute flux and is the kinetic coefficient that is independent of .¹³ As a result, a second derivation of c with respect to time is added to the classical Cahn–Hilliard equation.²⁵ To show more about the kinetic energy approach¹³ 1 the Maxwell–Cattaneo equation⁹ for NESD in bulk phases is reformulated as 2 One can see clearly that there is an extra contribution from the kinetic energy term to the thermodynamic driving force of solute diffusion if its mobility is taken as the independent variable on the NESD effect. Here, ‘S’ and ‘L’ denote respectively the value in solid and liquid, D_i is the solute diffusion coefficient, v_m the atomic volume is assumed to be the same for solvent A and solute B, c_i is the solute molar fraction, is the relaxation time of , is the solute diffusion potential and is a kinetic parameter independent on . The kinetic energy¹³ suggests that the NESD effect changes the thermodynamic state in the case of a prescribed kinetic mobility. This approach was widely used to propose the sharp interface model,^3,16 phase field model^13,17,18,21 and phase field crystal models.^7,19,20

Effective mobility approach

Noting that the Maxwell–Cattaneo equation⁹ equation (1) can also be reformatted as 3 an effective mobility approach²³ was proposed very recently for modelling of NESD. In contrast to the kinetic energy,¹³ the thermodynamic driving force was assumed to be the independent variable on the NESD effect. In this case, the effective mobility of solute diffusion changes from to an effective one with as a dimensionless effective mobility term (see equation (3). The effective mobility approach²³ in the case of a prescribed thermodynamic state changes the kinetic mobility.

If the driving force for solute diffusion is taken as the gradient of solute molar fraction , equation (3) reduces to 4 An effective solute diffusion coefficient can then be defined by comparing equation (4) with the classical Fick's law . In the case of one-dimensional steady state growth, a standard coordinate transformation with V as a constant interface velocity reduces to 5 which is the effective diffusion coefficient of Sobolev.¹⁰ Substituting D_i by , the classical theory can be extended directly to the case of NESD. For example, if is defined approximately as , with η the interface thickness, the solute trapping model of Aziz^26,27 can be modified as^8,10 6 where k_e and k are the equilibrium and non-equilibrium solute partition coefficients.Compared with Aziz's model^26,27 (i.e. setting in equation (6) in which as , complete solute trapping happens abruptly at a finite interface velocity . This solute trapping model with the NESD effect has been verified by experiments^8,10–12 and atomistic simulations.²⁴ As compared with the effective diffusion coefficient of Sobolev,¹⁰ both the kinetic energy approach¹³ and the effective mobility approach²³ are applicable to the case of non-steady state growth.

This work aims to carry out a comparative study between the kinetic energy approach¹³ and the effective mobility approach²³ in terms of examples for sharp interface modelling and phase field modelling of NESD upon rapid solidification of binary alloys. It will be shown that the two approaches are equivalent for modelling of long range solute diffusion in bulk phases, but only the effective mobility approach²³ can introduce the NESD effect to short range solute diffusion (or solute redistribution) at a sharp interface or within a diffuse interface. The effective mobility approach²³ therefore should be dominant in modelling of NESD upon rapid solidification.

Sharp interface modelling of rapid solidification of binary alloys

Total free energy and its change rate

In the case of a sharp interface, a generalised total free energy in the entire volume (see Fig. 1) is given by 7 where with (i = S, L; j = A, B) the chemical potential, and the associated Lagrange multiplier λ is included to ensure the global solute conservation in a closed system (i.e. ). The introduction of the generalised total free energy by combining of the total free energy F with that is not a real contribution to total free energy is helpful to describe the additional constraints thermodynamically consistently as will be shown in what follows.

Schematic diagram for isothermal rapid solidification of binary alloys; solute jump happens at sharp interface

Differentiating equation (7) with respect to time by the help of the transport theorem²⁸ and using the Gibbs–Duhem relation (i = S, L) and the solute conservation law , the change rate of total free energy is obtained as 8 where and (i = S, L) are respectively the solute diffusion potentials in bulk phases and at the interface. The change rate of total free energy equation (8) is a bilinear expression in the fluxes and thermodynamic driving forces. The first and second terms on the right hand side of equation (8) correspond to long range solute diffusion in solid and liquid whose fluxes and conjugative driving forces are and ( ). The third term is for the dissipative processes at the interface, i.e. short range solute diffusion in solid (liquid) whose flux and driving force are and ( ) and interface migration whose crystallisation flux and driving force are and . The three dissipative processes are constrained by the solute balance, which can be derived from the terms with the Lagrange multiplier λ in equation (8) 9 If equation (9) is substituted into the third term on the right hand side of equation (8), the total molar driving free energy at the interface is obtained as 10 where is the solute molar fraction transferred across the interface,^29,30 and Δ denotes the difference between the variable in solid and liquid. The dissipations at the interface are summarised in Fig. 2.

Molar Gibbs free energy diagram for isothermal solidification of binary alloys: if solute diffusion in solid is significant at sharp interface, there are three dependent dissipative processes (i.e. solute diffusion in solid and liquid at interface and interface migration with fluxes respectively as , and ) and one additional constraint from solute balance at sharp interface (i.e. equation (9))

If the kinetic energy terms, i.e. ( ), are added to the total free energy of the system, equation (7) is rewritten as 11 Assuming a time independent ( )¹³ and following a similar derivation procedure as equation (8), we have 12 For the conjugative driving force of long range solute diffusion in bulk phases, there is an extra contribution ( ) from the kinetic energy; please see the first and second terms on the right hand side of equation (12). For the dissipative processes at the interface, the NESD effect is not introduced to short range solute diffusion in solid and liquid; please see the first and second terms in the brace of equation (12). The free energy production of interface migration, i.e. the third term in the brace of equation (12), however, becomes a non-bilinear expression in the flux and thermodynamic driving force.

Evolution equations

Owing to its particular advantages in modelling of non-equilibrium dissipative systems with additional constraints, the thermodynamic extremal principle (TEP)^31–37 is adopted here to derive the evolution equations. For a general understanding and its extensive application to various fields of materials science, the readers are referred to a recent review of Fischer et al.³⁷ The evolution of the system according to the TEP of Svoboda et al.^34–37 for the linear irreversible thermodynamics is given by 13 The total free energy dissipation in terms of the effective mobility approach is expressed as 14 where the mobilities are 15 16 17 Here, is the interfacial solute diffusion velocity, R is the gas constant and T is the temperature.

Substituting equations (8) and (14) into the extremal condition equation (13) and then eliminating the Lagrange multiplier λ by equation (9), one obtains the evolution equations for the bulk phases 18 and for the interface 19 where the kinetic coefficients are 20 21 22 23 24 25 A substitution of equation (15) into equation (18) leads to the Maxwell–Cattaneo equation equation (1)) for long range solute diffusion in solid and liquid, and the three governing equations for short range solute diffusion in solid and liquid at the interface and interface migration equation (19)) follow the Onsager's reciprocal relation³¹

Adopting the effective mobilities ((equations (19)–(25)), by setting and using equations (9) and (16), are rewritten as 26 27 In this case, the three dependent dissipative processes at the interface reduce to two independent processes, i.e. interface migration (equation (26)) and trans-interface diffusion (or solute redistribution) (equation (27)). Trans-interface diffusion becomes relevant to its evolution history by the dimensionless effective mobility term . The two dissipative processes at the interface are summarised in Fig. 3. In the case of one-dimensional steady state growth, a standard coordinate transformation reduces equation (27) to 28 which is similar to the solute trapping models^3,16 in the case of linear irreversible thermodynamics. At , and , i.e. complete solute trapping happens.

Molar Gibbs free energy diagram for isothermal solidification of binary alloys: if solute diffusion in solid is negligible at sharp interface, there are two independent dissipative processes, i.e. trans-interface diffusion and interface migration with fluxes respectively as and ;^2,3 please see equations (26) and (27)

If the kinetic energy approach¹³ is adopted, the total free energy dissipation Q is defined as 29 The extremal condition equation (13), however, cannot be used due to the non-bilinear expression for free energy production of interface migration in equation (12). If the driving force for interface migration is assumed arbitrarily to be independent on and , substituting equations (12) and (29) into equation (13) and then eliminating the Lagrange multiplier λ by equation (9) lead to the Maxwell–Cattaneo equation (equation (1)) for long range solute diffusion in bulk phases and 30 for the dissipative processes at the interface. The kinetic coefficients in equation (30) are the same as (equations (20)–(25)) except that ( ) thereof is substituted by ( ). Setting and using equation (9), equation (30) reduces to 31 32 Noting that equation (32) is similar to Aziz's solute trapping model²⁷ in the case of linear irreversible thermodynamics, the NESD effect is not enclosed into trans-interface diffusion. This is the reason why the kinetic energy approach¹³ needs to be combined with the effective diffusion coefficient of Sobolev¹⁰ to derive the solute trapping models with the NESD effect for concentrated alloys in recent works.^3,16 It must be pointed out that equation (31) for interface migration is the same as the previous work³ in the case of linear irreversible thermodynamics.

If the interface dissipation by solute diffusion in solid (i.e. the first term in the brace of (equations (19) and (30)) reduce to the previous work,³ which rederives the sharp interface model of Hillert et al.^29,30 by TEP.^31–37 The consideration of interface dissipation by solute diffusion in solid in the current work makes the sharp interface models equations (19) and (30) completely comparable with the phase field models^21,23 as described in what follows.

Phase field modelling of rapid solidification of binary alloys

Phase field modelling of NESD upon rapid solidification of binary alloys was carried out recently.^17,21,23 It was found that the kinetic energy approach¹³ introduces the NESD to long range solute diffusion but not short range solute redistribution, thus resulting in an inhomogeneous solute profile within the interface even when diffusion less solidification has occured.^17,21 The effective mobility approach²³ is able to introduce the NESD to both long range solute diffusion and short range solute redistribution and thus predicts an abruptly concurrent occurrence of complete solute trapping and absence of solute drag. To show clearly the difference between the kinetic energy approach¹³ and the effective mobility approach²³ as well as the relation between phase field model and sharp interface model, the previous work^21,23 is described concisely in what follows.

Total free energy and its change rate

In the case of a diffuse interface, the generalised total free energy in the entire volume Ω (see Fig. 4) is expressed as^38,39 33 where the contributions from the interface, bulk phases and additional constraint are 34 35 36 Here, σ_SL is the interface energy, h_i ( , ) is the interpolation function, is defined as the double obstacle potential⁴⁰ such that the interfacial width η remains finite uponsolidification, f_bulk is assumed to be a mixture of g_S and g_L, and f_ad with the associated Lagrange multiplier λ is included such that the solute conservation is ensured to describe the additional constraint thermodynamically consistently. The change rate of total free energy is²³ 37 where thevariational derivatives are given by 38 39 Equation (37) is also a bilinear expression in the fluxes and thermodynamic driving forces. There are three dissipative processes, i.e. phase field propagation, solute diffusion in solid and liquid. The bulk contribution to the conjugative driving force of phase field propagation (e.g. in equation (38)) is similar to that of interface migration at a sharp interface(e.g. in equation (8)). The conjugative driving force for solute diffusion can be rewritten as . The first term reduces to for solid where and for liquid where (i.e. the driving forces for long-rage solute diffusion), whereas the second term corresponds to the driving forces for short range solute diffusion by noting that is similar to for short range solute diffusion at the sharp interface (see equation (8)). In other words, long and short range solute diffusions are integrated uniformly by phase field modelling but not described separately.⁴¹

Schematic diagram for isothermal rapid solidification of binary alloys: in such diffuse interface separated by phase field , the overall molar fraction c and solute diffusion flux change continuously from boundary between interface and solid where and to the boundary between interface and liquid where and

The three dissipative processes are constrained by the solute balance that is given by the terms with the Lagrange multipliers λ in equation (37) 40 Equation (40) within the diffusion interface is similar to equation (9) at the sharp interface. Substituting equation (40) into equation (37), the free energy production from the bulk contribution is obtained as²³ 41 The local total molar driving free energy for phase field propagation, short range solute diffusion in solid and liquid is with that is similar to in equation (10). The second term on the right hand side of equation (41) is the free energy production for long range solute diffusion in solid and liquid. The molar free driving energies from the bulk contribution are summarised in Fig. 5.

Molar Gibbs free energy diagram for isothermal solidification of binary alloys: if solute diffusion in solid is significant, there are three dependent dissipative processes (i.e. phase field propagation, solute diffusion in solid and liquid) and one additional constraint from solute balance within diffuse interface (i.e. equation (40)), which is similar to sharp interface as shown in Fig. 2

Adding the kinetic energy terms, i.e. ( ), to equation (33), we have 42 where 43 Then, assuming similarly a time independent ( ),¹³ the change rate of total free energy density is²¹ 44 In this case, an extra contribution ( ) from the kinetic energy is introduced to long range solute diffusion (i.e. the second term on the right hand side of equation (44)), whereas for phase field propagation, its free energy production is a non-bilinear expression in the flux and thermodynamic driving force (i.e. the first term on the right hand side of equation (44)). These outcomes are also comparable with the sharp interface equation (12).

Evolution equations

The evolution equations for the phase field propagation, solute diffusion in solid and liquid^21,23 follow also the Onsager's reciprocal relation³¹ and are similar to equations (19) and (30) for sharp interface. In bulk phases, the governing equation for solute diffusion is the same as the Maxwell–Cattaneo evolution equation (equation (1)). In the case of negligible solute diffusion in solid, i.e. , the phase field model derived from the effective mobility approach²³ reduces to 45 46 The first and second terms on the right hand side of equation (46) are the equilibrium and non-equilibrium long range solute diffusion fluxes. The third term is the short range solute redistribution flux whose driving force and mobility are and . In the case of one-dimensional steady state growth, equation (46) in terms of the dimensionless coordinate reduces to 47 One can see that both short and long range solute diffusions disappear abruptly at a finite interface velocity . Furthermore, at , and , i.e. complete solute trapping happens. Therefore, the effective mobility approach is able to predict a sharp concurrence of diffusionless solidification and absence of solute drag.²³ In this case, the local total molar driving free energy for phase field propagation and solute redistribution is from the bulk contribution. The parts corresponding to the former and latter are respectively and ; please see Fig. 6.

Molar Gibbs free energy diagram for isothermal solidification of binary alloys: if solute diffusion in solid is negligible, there are two independent dissipative processes, i.e. phase field propagation and solute diffusion in liquid; please see equations (45) and (46); two independent dissipative processes are totally comparable with interface migration and trans-interface diffusion at sharp interface as shown in Fig. 3

The phase field model derived from the kinetic energy approach²¹ in the case of negligible solute diffusion in solid reduces to 48 49 The local molar driving free energy for phase field propagation from the bulk and kinetic energy contributions is similar to the case at the sharp interface (equation (31)).The NESD effect is introduced to long range solute diffusion (i.e. the second term on the right hand side of equation (49)) but not short range solute redistribution (i.e. the third term). This is the reason why solute diffusion still happens within the diffuse interface even at . For example, in the case of one-dimensional steady state growth, equation (49) in terms of the dimensionless coordinate reduces to 50 One can see clearly that the solute profile within the diffuse interface is not uniform even at . However, holds at , i.e. complete solute trapping occurs.

Conclusions

The effective mobility approach²³ is compared with the kinetic energy approach¹³ in terms of sharp interface modelling and phase field modelling of NESD upon rapid solidification of binary alloys. The current work is helpful not only for modelling of NESD but also for understanding of non-equilibrium phenomena (e.g. in undercooled metals).⁴² Our main conclusions are as follows. 1

The effective mobility approach²³ introduces the NESD effect to both long range solute diffusion in bulk phases and short range solute diffusion at the sharp interface or within the diffuse interface. It allows complete solute trapping in the case when a sharp interface happens abruptly at a finite interface velocity that is equal to the solute diffusion velocity in liquid, and in the case when a diffuse interface occurs together with the absence of solute drag.

The kinetic energy approach¹³ includes the NESD effect into long range solute diffusion but not short range solute diffusion either at the sharp interface or within the diffuse interface. For the sharp interface modelling, it needs to be combined with the effective diffusion coefficient of Sobolev,¹⁰ which is a special case of effective mobility to predict complete solute trapping. For phase field modelling, it allows complete solute trapping to happen at a finite interface velocity but with unreasonable solute diffusion within the interface.

As a useful principle for modelling of non-equilibrium dissipative systems with additional constraints, the TEP of Svoboda et al.^34–37 is adopted to derive the evolution equations. Compared with the kinetic energy approach,¹³ which results in a non-bilinear expression in the flux and thermodynamic driving force in the free energy production of interface migration (equation (12)) or of phase field propagation (equation (44)), the effective mobility approach²³ allows the TEP of Svoboda et al.^34–37 workable for modelling of NESD without any arbitrary approximation.

Acknowledgements

Haifeng Wang thanks the support of Alexander von Humboldt Foundation for a research fellowship and Professor P. K. Galenko for useful discussions. Authors are grateful to the National Basic Research Program of China (973 Program, grant no. 2011CB610403), National Science Funds for Distinguished Young Scientists (grant no. 51125002), Natural Science Foundation of China (grant no. 51371149), the Fundamental Research Funds for the Central Universities (No. 3102015AX003) and Free Research Fund of State Key Lab. of Solidification Processing (grant no. 92-QZ-2014).

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