Rapid phase transformation under local non-equilibrium diffusion conditions

Abstract

Phase transformation with a moving interface occurs under far from local equilibrium conditions when the interface moves with sufficiently high velocity. This deviation cannot be adequately described by the classical irreversible thermodynamics with diffusion equation of parabolic type because it assumes local equilibrium hypothesis. The local non-equilibrium diffusion model has been developed to take into account the deviation from local equilibrium during binary alloy solidification using the hyperbolic diffusion equation. The model introduces a finite propagation velocity of concentration disturbances in the bulk liquid V_D as the characteristic diffusion parameter and predicts a sharp transition from diffusion controlled to diffusionless and partitionless solidification at V = V_D due to the deviation from local equilibrium. This review does not intend to be extensive, but rather to illustrate the main advances of the local non-equilibrium diffusion approach in rapid alloy solidification. Applications of the local non-equilibrium model to other phase transformation processes such as solid–solid transformation, colloidal and multicomponent alloy solidification have been briefly discussed. The parallels and possible combinations of the model with molecular dynamics, phase field, phase field crystal and cellular automata methods have been considered.

Keywords

Phase transformation Rapid alloy solidification Solid state transformation Colloidal solidification Local non-equilibrium Hyperbolic diffusion Diffusionless solidification Partition coefficient Solute trapping

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

Introduction

The modelling of phase transformation with moving phase interface such as binary and multicomponent alloy solidification, colloidal solidification and solid state transformation typically considers the transport of mass and heat through the bulk of the one or both of the phases as well as the phase transformation kinetics at the interface.^1–6 In the near equilibrium limit, transformation occurs slowly enough that the growing phase has the equilibrium composition, given by the equilibrium phase diagram. But when the interface velocity V increases, the transformation process begins to deviate from equilibrium.^1–97 The non-equilibrium nature of phase transformation has made it possible to produce supersaturated solid solutions, metastable compounds, highly disordered intermetallic compounds, ceramic materials, glasses and other advanced engineering materials with unique characteristics.^1–97 The most obvious manifestation of the deviation from local equilibrium during rapid alloy solidification is solute trapping, which reduces the solute segregation at the solid/liquid interface.^1–37 The solute trapping phenomenon is often accompanied by grain refinement^{7–9,11,14–17} and is analogous to disorder trapping^14–17 and density trapping.³⁸ Therefore, the development of capability in predicting solute trapping phenomena, especially at far from local equilibrium conditions, is an important task in designing new materials and new processes. The degree of solute trapping is usually quantified by the partition coefficient K, defined as the ratio of the concentration of solute in the solid C_S to that in the liquid C_L at the interface. Several models of the solute trapping during solidification of binary alloys have been proposed.^18–37 Continuous growth model (CGM) by Brice³³ and Aziz et al.^34,35 (see also discussion in Ref. 36) leads to the following K versus V dependence 1 where K_E is the equilibrium partition coefficient, V_st is the characteristic solute trapping velocity. The Brice model³³ assumes that the rate at which solute atoms can join the solid is proportional to the so called diffusion velocity , where D is the solute diffusion coefficient in the bulk liquid, and λ is the interatomic spacing. In this case, the characteristic solute trapping velocity is given as , where f is a fraction of solute atoms that reach the solid surface and join the solid phase, i.e. a fraction of active surface growth sites. The model of Aziz et al. treats V_st as the interface diffusive velocity V_i, which was given by the ratio D_i/λ, where D_i is the coefficient for interdiffusion across the interface, and λ is the interatomic spacing³⁴ or the interface width.³⁵ In spite of some arbitrariness about how V_i is treated, it can be considered as a solute partitioning parameter, which is the growth rate at which K(V) is in midtransition between K_E and unity.³⁵ Both D_i and V_i are not subject to direct experimental determination: the values for V_i have been inferred by fitting the observed dependence of K on V.³⁵ Assuming that V_st is identical for both models, we obtain , which corresponds to experimentally observed inverse correlation between V_i and K_E.³⁵

According to the most recent solute trapping theory of Jackson et al.,³⁷ K(V) is described as a power law given as 2 where A is a parameter analogous to the reciprocal of the characteristic solute trapping velocity. Both CGM and the model of Jackson et al. as well as most of other theoretical treatments of rapid alloy solidification^24,25 are based on classical irreversible thermodynamics (CIT), which assumes the local equilibrium assumption as a fundamental assumption and leads to the classical parabolic type diffusion equation (PDE)^1,2,39 where C is the solute concentration. PDE is one of the most popular equations in mathematical physics, and it provides a good description of heat mass transfer processes in a great variety of practically important situations as long as the local equilibrium assumption holds. However, some experimental results^7–17 and theoretical treatments^18–32,40 on rapid alloy solidification indicate that, at high solid/liquid interface velocity, the local equilibrium assumption breaks down, and the solute diffusion in the bulk liquid occurs under far from local equilibrium conditions. In this case, CIT and PDE cannot adequately describe solute diffusion in the bulk liquid, and a local non-equilibrium approach should be used.

Local non-equilibrium diffusion approach to rapid alloy solidification

Local non-equilibrium diffusion model (LNDM): Simplest version

There exists an extensive body of literature on the heat mass transfer theories that go beyond the local equilibrium assumption (see, for example, Refs. 39–44 and references therein). The extended irreversible thermodynamics³⁹ is one of the most elaborated versions of the thermodynamic theories. It is remarkable that all these approaches lead to the hyperbolic diffusion equation (HDE) as the first step generalisation of PDE for the local non-equilibrium case 3 where τ is the relaxation time of solute diffusion flux to local equilibrium.^39–44 Hyperbolic diffusion equation takes into account both dissipative and wave properties of the transport processes under local non-equilibrium conditions, and it transmits waves of concentration (waves of concentration disturbances) with a finite speed .^{18–32,39–44} When , HDE reduces to PDE, which implies an infinite velocity of concentration disturbances . The local equilibrium assumption holds for relatively slow processes when the characteristic time of the process t_chand breaks down for fast processes when . In terms of the solid/liquid interface velocity V, the local equilibrium assumption holds at and breaks down at V∼V_D. In the last case, the diffusion process occurs under far from local equilibrium conditions, and it should be described by HDE (equation (3)).^{18–22,39–44} For metallic alloys, V_D is of the order 10–20 m s^− 1,^7–9,11 whereas for intermetallic compounds, it is in the range from 0.2 to 3.8 m s^− 1.^12,14–17 Note that the diffusion velocity V_D defined by HDE as the speed of concentration wave (concentration disturbances) in the bulk liquid is a purely diffusion parameter. In contrast, the interface diffusive velocity V_i in CGM characterises the solute segregation at the interface and is a solute trapping parameter. To stress the difference between V_i and V_D, we sometimes call the later as the bulk liquid diffusion velocity.^18–22

In the context of the moving phase transformation zone, HDE has been first used to describe temperature distribution around the ultrafast moving energy source (see Ref. 40 and references wherein). Later, this approach has been modified to develop the LNDM^18–22 for rapid solidification of binary alloys, which takes into account the local non-equilibrium solute diffusion in the bulk liquid at high interface velocity V∼V_D.

The exact mathematical treatment of the solute diffusion problem may be found by solving the time dependent HDE (equation (3)), subject to appropriate initial and boundary conditions in both the solid and the liquid phases. But to avoid mathematical difficulties, the steady state approximation is usually used. It assumes that, after a short initial transient period, the solid/liquid interface begins to move with constant velocity V, and the solute concentration is time independent in the reference frame attached to the interface. In the steady state approximation, equation (3) gives^18–27,40 4 The solution to equation (4) for semi-infinite liquid phase 0 < X < ∞ has the form^18–22,40 5 where C₀ and C_i are the solute concentration far from and at the interface (X = 0) respectively. Equation (5) predicts that the local non-equilibrium diffusion effects qualitatively change the solute diffusion head of the fast moving interface: the solute diffusion field remains undisturbed at V ≥ V_D. This result has a clear physical meaning: a source of concentration disturbances, i.e. the interface, moving with a velocity higher than the maximum velocity of the disturbances, i.e. the diffusive velocity V_D, cannot affect the media ahead of itself.⁴⁰ This phenomenon is analogous to supersonic effect. Thus, when the interface velocity V exceeds the critical point V = V_D, the local non-equilibrium diffusion effects lead to an abrupt transition from diffusion controlled to diffusionless solidification.^18–22 Figure 1 plots the steady state local non-equilibrium concentration profiles around the interface region (equation (5)), for different ratios V/V_D (solid curves). The corresponding local equilibrium solute concentration distributions obtained from PDE (see, for example, Refs. 1 and 2) are placed for comparison. When V < V_D (Fig. 1a and b), there is a concentration spike C(X)>C₀ ahead of the interface due to the solute segregation at the interface and solute diffusion in the bulk liquid. In such a case, the solidification mechanism is essentially controlled by solute diffusion. As the interface velocity increases, the local non-equilibrium diffusion effects became more significant, and the concentration spike is getting smaller and narrower (compare solid and dashed lines in Fig. 1a and b). When the interface velocity V exceeds the diffusion velocity V_D, the solute diffusion ahead of the moving interface is totally depressed by the local non-equilibrium effects and the solute concentration in the liquid phase does not change, i.e. and C(X) ≡ C₀ (see solid line in Fig. 1c). In contrast, the local equilibrium models^1,2 predict that the diffusion layer and the concentration spike at the interface exist even at V>V_D (see dashed line in Fig. 1c). The value of the local non-equilibrium characteristic diffusing length (boundary layer) δ^LNDM is defined by equation (5) as^18–22 (see curve 3 in Fig. 2) 6 where is the characteristic diffusion length for the local non-equilibrium case^1,2 (curve 4 in Fig. 2). As interface velocity V increases, the effective diffusion length δ^LNDM (equation (6)), shrinks more rapidly than δ₀ and equals zero at V>V_D, whereas δ₀>0 at any value of V (see curves 3 and 4 in Fig. 2). A comparison of the local non-equilibrium solute distribution ahead of the moving interface (equation (5)) with the local equilibrium one allows us to introduces the effective (velocity dependent) diffusion coefficient in the form^18–22 7 The effective diffusion coefficient (equation (7)) decreases with increasing interface velocity (see curve 2 in Fig. 2) due to increasing deviation from local non-equilibrium. When the interface velocity exceeds the critical value V = V_D, the effective diffusion coefficient D^LNDM equals zero manifesting the sharp transition to diffusionless solidification.^18–22 The effective diffusion coefficient (equation (7)) can be used to convert the results of the local equilibrium models to the local non-equilibrium case by substitution D for D^LNDM. For example, LNDM extends the solute partition coefficient of CGM (equation (1)) for the local non-equilibrium case as follows^19,20,22 8 Equation (8) describes a sharp transition to complete solute trapping , when the interface velocity V passes through the critical point V = V_D (curve 5 in Fig. 2).^18–29 The result is in agreement with experimental data^19,20,22 and has been confirmed by molecular dynamics (MD),^58–60 phase field (PF) model⁶¹ and phase field crystal (PFC) model.^62–64 Note that the local equilibrium models (equations (1) and (2)) predict complete solute trapping K → 1 only asymptotically at V → ∞ (curve 1 in Fig. 2). Substituting D for D^LNDM in the non-dilute CGM expression for partition coefficient,³⁶ one can easily obtain the non-dilute LNDM partition coefficient in the form^24,25 9 where k_E is the equilibrium partition coefficient for the solute divided by that for the solvent.³⁶ The effective diffusion coefficient (equation (7)) has also been used to demonstrate that the local non-equilibrium effects stabilise the plane solid/liquid interface.^19,20,22 For high interface velocities, when the local non-equilibrium effects play the most decisive role, the classical Mullins and Sekerka result for the absolute stability limit takes the form (see, for example, Refs. 1 and 2) where C_a is the critical bulk concentration in the liquid, V_a is the velocity of absolute stability limit, T_m is the melting point of pure solvent and Γ is the capillarity constant. Figure 3 shows the critical concentration C_a versus V_a taking into account that the liquidus slope m is a relatively weak function of V in comparison with K(V) and D^LNDM(V). Dashed curve represents the Mullins and Sekerka equation for the local equilibrium case, i.e. with D = const and K(V) expressed by equation (1). Solid curve represents its extension to the local non-equilibrium case with D^LNDM(V) and K^LNDM(V) (Refs. 19, 20 and 22). Figure 3 demonstrates that at a given C_a, which implies that the local non-equilibrium diffusion effects stabilise the interface due to depressed solute diffusion. Moreover, the local non-equilibrium diffusion effects set the limit .^19,20,22

a V/V_D = 0·5: long range diffusion growth; b V/V_D = 0·9: short range diffusion growth; c V>V_D: diffusionless growthConcentration profiles near interface region versus position obtained at different values of non-dimensional interface velocity V/V_D [solid lines: prediction of LNDM with HDE (eq. (5); dashed lines: prediction of PDE (V_D → ∞)]

Local equilibrium partition coefficient K, equations (1) and (2) (curve 1); non-dimensional effective diffusion coefficient D^LNDM/D, equation (7) (curve 2); non-dimensional local non-equilibrium boundary layer δ^LNDM, equation (6) (curve 3); non-dimensional local equilibrium boundary layer (curve 4); and local non-equilibrium partition coefficient K^LNDM, equation (8) (curve 5) as functions of non-dimensional interface velocity V/V_D

Critical concentration C_a as function of non-dimensional velocity V_a/V_D (solid curve: prediction of LNDM; dashed curve: local equilibrium Mullins and Sekerka theory)

The slope of the local non-equilibrium liquidus line m^LNDM can be written as^19–22 10 where m_L is the equilibrium liquidus line. In this case, the effective (local non-equilibrium) liquidus and solidus temperatures are and respectively, where T_A is the melting temperature of the major component. As the interface velocity V increases, the effective solidus temperature increases, and the effective liquidus temperature decreases due to the solute trapping effects. At V = V_D (when K^LNDM = 1), they coincide reaching the T₀ temperature, i.e. the temperature of equal free energy of two phases, and at V>V_D, they do not change any more. The interface temperature for steady state planar growth with allowance for local non-equilibrium effects is given as 11 where β is interface kinetic coefficient. The interface temperature T_i as a function of the interface velocity V is shown in Fig. 4 for the steady state solute concentration at the interface in the liquid (curve 1) and for the fixed solute concentration at the interface in the liquid (curve 2). The sharp change in the effective partition coefficient K^LNDM (equation (8) or (9)), during the transition to diffusionless solidification at leads to the corresponding sharp change in the effective liquidus slope m^LNDM (equation (10)), which, in turn, leads to the sharp break of the interface temperature versus interface velocity function (equation (11)). At , the growth is mainly solute diffusion controlled (second term on the right hand side of equation (11)). When the interface velocity V exceeds the critical point , equation (11) can be rewritten in terms of T₀ temperature as , which implies that, at these velocities, the alloy solidifies as a ‘pure metal’ with melting temperature T₀. The transition to diffusionless and partitioned solidification occurs at , i.e. rather close to the T₀ line.^19–22,27 The prediction of LNDM that the local non-equilibrium diffusion effects lead to the sharp transition from diffusion controlled solidification to diffusionless and partitionless growth at , which is accompanied by the break in the function, is consistent with experimental data.^{7–17,19–27} For example, measurements of interface velocity as a function of undercooling during rapid solidification of Cu–Ni and Cu–Ni–B,^7,8 Ni–B,^9,11,14 Ag–Cu,¹⁰ Co–Si,^11,12,14 Ti–Ni,¹³ Fe–Ge (Ref. 16) and β Ni3Ge (Ref. 17) reveal a sharp break in the velocity–undercooling relationship at relatively high interface velocity, which is accompanied by almost partitionless solidification.^9,11 The experiments show that it is not the critical undercooling that initiates the break, but rather the critical solidification velocity, which approximately equals the diffusive speed V_D. Note that the classical solidification approach based on PDE can describe the transition to diffusionless solidification with complete solute trapping only asymptotically at V → ∞.

Non-dimensional interface temperature T/T_A versus non-dimensional interface velocity V/V_D: curve 1, steady state, i.e. at interface; curve 2, at interface

The transition from diffusion controlled to thermally controlled growth with the sharp break in the (or interface velocity versus interface undercooling) function is often accompanied by disorder trapping and grain refinement.^{7–9,11,14–17} Disorder trapping is a phenomenon quite analogous to solute trapping: at high interface velocity, the atoms can no longer sort themselves by diffusion to build up the superlattice of the intermetallic phase, which leads to a disordered structure. This implies that the local non-equilibrium diffusion effects set the limit V_crit < V_D on the critical velocity V_crit at which an ordered structure can still form. Indeed, at V>V_D, there is no diffusion ahead of the interface, and an ordered structure cannot form at such velocity. Moreover, estimates of a solute boundary layer under local non-equilibrium conditions demonstrate that, if one or two diffusive jumps of solute atoms are needed to build up the superlattice, the critical velocities are V_crit = 0.6V_D or V_crit = 0.4V_D respectively.²⁷ Thus, as the interface velocity increases, the deviation from equilibrium also increases, which affects the solute partitioning at the interface and solute diffusion in the bulk liquid. The hierarchy of deviations from equilibrium with increasing interface velocity V is summarised in Table 1.

Table 1

Hierarchy of deviations from equilibrium during steady state solidification of binary alloys

V = 0	V V _D	V < V _D	V ≤ V _D	V>V _D
Full equilibrium	Quasi-equilibrium regime	No equilibrium at the interface but local diffusion equilibrium in the bulk liquid	No equilibrium at the interface and local non-equilibrium in the bulk liquid	Diffusionless and partitionless solidification
K = K _E	K = K _E	K = K(V) − CGM	K = K(V)^LNDM − LNDM	K = K^LNDM = 1
C _L = C ₀	C _L = C ₀ /K _E	C _L = C ₀ /K(V)	C_L= C₀/K(V)^LNDM	C _L = C _S = C ₀
C _S = C ₀ K _E	C _S = C ₀	C_S= C₀; D = const	C_S= C₀; D(V)^LNDM	D^LNDM = 0
No diffusion	Solute diffusion field is described by PDE	Solute diffusion field is described by PDE	Solute diffusion field is described by HDE	No diffusion ahead of the interface
	, the characteristic time to reach the steady state is relatively long for small V and K		Local non-equilibrium effects shrink the characteristic time to reach steady state³²

Extensions of LNDM

Local non-equilibrium diffusion model,^18–22 briefly referred to in the previous section, is the simplest model to describe the local non-equilibrium diffusion effects during rapid solidification of binary alloys. It assumes that the plane interface is sharp and the liquidus and solidus are linear lines. Many recent papers^{3–5,23–32,45–56} extend LNDM to more general cases. A number of solute trapping models beside CGM have been modified for the local non-equilibrium case using the effective diffusion coefficient (equation (7)).^24,25 All the modified models predict complete solute trapping K = 1 at V>V_D due to depressed solute diffusion (see equation (7)). The dimensionless growth parameter found in the Monte Carlo simulations as the important parameter for solute trapping³⁷ has been modified for the local non-equilibrium diffusion case taking into account that the probability that an atom will be trapped into the crystal under local non-equilibrium conditions depends on V_D.²⁶ In such a case, the Jackson et al. model (equation (2))³⁷ yields²⁶ 12 This equation also predicts the transition to complete solute trapping K = 1 at V = V_D. Several recent papers^{45–48,50,51,56,57} study the effect of non-linear liquidus and solidus on rapid alloy solidification, which provide more accurate description of rapid alloy solidification with real phase diagram. Application of the maximal entropy production principle to rapid solidification with a sharp interface⁵² confirms the LNDM result that the transition to complete solute trapping occurs at a finite interface velocity and demonstrates the significance of solute drag. Extended interface stability problem has been also investigated.^47–49 It has been demonstrated that the local non-equilibrium diffusion effects significantly influence the interface stability. Applications of local non-equilibrium approach to dendritic growth usually consider the local non-equilibrium condition at the dendrite tip,^{20,22,45,46,49,50,56,57} which also leads to a sharp transition to diffusionless solidification at V = V_D. But strictly speaking, the normal component of the interface velocity varies along the dendrite side, which leads to the correspond change of the partition coefficient.²⁸ This implies that the transition to diffusionless solidification during dendrite growth is not sharp as in the case of the plane interface^18–22 but occurs in a range of interface velocities.²⁸ The precise free boundary problem for dendritic growth under local non-equilibrium diffusion conditions must be determined from the solution of HDE with allowance for the changing normal component of the interface velocity along the dendrite side. It is expected that the local non-equilibrium effects may change the dendrite shape significantly.

Applications to other systems

Multicomponent alloy solidification

All the above analysis focuses on binary alloys, while most industrial alloys are multicomponent alloys.^{29,66–71,94,96} Local non-equilibrium diffusion model version for multicomponent alloys²⁹ leads to the solute diffusion equations in the form 13 where C_i and τ_i are the concentration and the relaxation time of the component i respectively, and D_ij are diffusion coefficients, which constitute an diffusion matrix. In general, the diffusion coefficients are not symmetric . The diagonal terms D_ii are so called the ‘main terms’ of the diffusion matrix because they are commonly larger than the off-diagonal terms D_ij and similar in magnitude to the binary values.

For a steady state growth of a ternary system with plane interface, equation (13) reduces to²⁹ where . If the diffusional interaction between the species is ignored, the solute distributions are independent from each other, and a two-step transition to diffusionless solidification occurs: at , the growth is partly diffusionless, i.e. it is diffusionless with regards to the solute 1, and at , the growth is completely diffusionless. At a relatively weak diffusional interaction, the solute concentrations ahead of the interface change monotonically with the distance, but a strong interaction between the components may lead to a maximum or a minimum of solute concentrations in the liquid phase.^29,68 Diffusion velocity values in the multicomponent systems change due to the interaction between the components,²⁹ which, in turn, changes condition for solute trapping at the interface. Wang et al.^70,71 modified the local non-equilibrium approach to the multicomponent concentrated alloys. For the time being, however, use of analytical solutions is limited by the lack of measured diffusion coefficients and methods to determine the off-diagonal terms.

Solid state transformations

Solid state transformation is receiving increasing interest in the material science community both from experimental and theoretical point of view.^{3,36,72,73,93} Solid state transformations provide an interesting setting to test theoretical models in new regimes that are not usually accessible in a solidification context, such as the transitions to absolute morphological stability and to diffusionless solidification with complete solute trapping at high interface velocity. As the critical velocities required for transition to absolute stability V_a and to diffusionless (and partitionless) solidification V_D are proportional to the solute diffusion coefficient, these typical velocities are very high in alloy solidification and very difficult to produce. The solid state diffusion coefficients, and hence the critical velocities in solid state transformations, are orders of magnitude smaller and can be easily reached. For example, in directional transformation of a hypoperitectic Fe–17.5 at-Co alloy, flat cells were observed at 7 μm s^− 1, while the plane front γ was clearly seen at 10 μm s^− 1.⁷² The authors concluded that the experimentally determined V_a value is between these two growth rates, which agrees well with the calculated value of V_a = 9.2 μm s^− 1 obtained from the local equilibrium theory with . At relatively low interface velocity V = 1 μm s^− 1, the concentration spike was observed at δ/γ interface, but at higher rate V = 10 μm s^− 1, no concentration spike was detected.⁷² However, it was concluded that a spike existed according to classical approach, but it was not detected with the electron probe micro-analyser (EPMA) technique because it was very thin. It should be stressed that there is an appreciable uncertainty in the experimental values of the diffusion coefficient in Fe–Co system. For example, assuming , we obtain V_D = 8 μm s^− 1, which is less than the higher rate V = 10 μm s^− 1 obtained in the experiment.⁷² This implies that at V = 10 μm s^− 1, the diffusionless transformation regime with complete solute trapping was reached in the experiment. The absence of a concentration spike at δ/γ interface (Fig. 6b in Ref. 72) corresponds to this conclusion. The authors⁷² also proposed that the interface velocity must be >10 cm s^− 1 to reach the regime of complete solute trapping, whereas LNDM predicts V_D = 0.825 cm s^− 1 at the same value of the diffusion coefficient, which is at least one order of magnitude lower. Using the Gibbs energy dissipation approach, Hillert et al.^36,73 estimated that the partitionless growth of α (bcc) phase from γ (fcc) phase in the FeNi system at 900 K and an alloy composition of 6 mol.-Ni can be reached at a velocity ∼10^− 12 m s^− 1. This approach also predicts that diffusionless and partitionless transformation can occur rather close to the T₀ line, which corresponds to our prediction .^19–22,27 Thus, owing to the small diffusion coefficients in solids, and hence low corresponding diffusion velocity, the solid state transformation appears to be a promising field to experimental study of local non-equilibrium diffusion effects.

Colloidal solidification

In materials science, the rapid solidification of colloidal or nanosuspensions is receiving increasing interest owing to the novel microaligned structures that can be produced for a wide range of applications such as organic electronics, microfluidics, molecular filtration, nanowires, cryopreservation and tissue engineering.^23,74–78 Using the freezing of colloids to template the porosity in materials is known as ice templating or freeze casting.⁷⁴ The phenomenon is very similar to unidirectional solidification of binary alloys, with ice playing role of a fugitive second phase. Typical velocities used when processing materials through solidification of colloidal suspensions are of the order of 10–100 μm s^− 1, which implies that the system in these conditions is likely to be out of equilibrium.^23,74–78 The investigation of the structure and dynamics of the crystal/fluid interface of colloidal hard spheres in real space by confocal microscopy demonstrates that the deviation from equilibrium leads also to the interface broadening from 8 to 9 particle diameters close to equilibrium to 15 particle diameters far from equilibrium.⁷⁸ The diffusive speed V_D in colloidal solidification can be treated as the maximum speed with which an individual particle moves in the bulk liquid, i.e. the ‘molecular’ velocity of the particles, which depends on the particle size.^23,75 For alumina particle size of 2–20 μm, the estimated diffusion velocity falls in the range of 10–125 μm s^− 1,²³ which can be likely of the order of the experimentally observed interface velocity in colloidal solidification.⁷⁶ Partitioning of polystyrene impurity particles during colloidal solidification has been studied by Nozava et al.⁷⁷ It was observed that, for each sample with different impurities, K approached unity as the growth rate increased. The diffusion coefficient of each particle was determined using the Stokes–Einstein equation, and the results were found to be similar for the different particles and on the order of 10^− 12 m² s^− 1. Assuming that the particle jump distance in the liquid phase is of the order of the host particles size R = 500 nm, the diffusion velocity can be estimated as , which gives V_D = 7.2 mm h^− 1. This value corresponds to the critical velocity for complete solute trapping observed in the experiment.⁷⁷ Thus, the low values of diffusion velocity and the big size of colloidal particles make colloidal suspensions increasingly important as model systems to study local non-equilibrium phase transformations.

Parallels and combination with other approaches

Molecular dynamics simulations

In recent years, a fundamental understanding of alloy solidification has been gained through MD.^{1,2,58–60,79,80} Cook and Clancy^58,59 have used non-equilibrium MD simulation method to study solute trapping in an A_xB_1 − x Lennard–Jones alloy. The complete solute trapping was found when the interface velocity attained its steady state value of 4 m s^− 1. It has been stressed that neither the CGM nor periodic stepwise growth theories of solute segregation, which are based on CIT, predicted the complete trapping observed in the MD simulations. The most recent work Yang et al.⁶⁰ calculated solute concentration profile around the interface region from an “LJ simulation with a relatively low velocity and the highest velocity considered. For the slow V, the concentration profile demonstrates a peak on the liquid side of the interface, which corresponds to LNDM predictions at V < V_D. For the high V case, the MD simulation results in the nearly flat concentration profile, indicating clearly solute trapping behaviour (Fig. 1 in Ref. 60), which is consistent with diffusionless solidification predicted by LNDM at V>V_D. Yang et al.⁶⁰ also calculated solute partition coefficient K versus V and compared the result with theoretical models. Continuous growth model and the model of Jackson et al. fit the data only for the lower velocities. At higher interface velocity, the MD date of Yang et al.⁶⁰ predicts transition to partitionless solidification at a finite value of the interface velocity (see the upper panel of Fig. 2 in Ref. 60), which is in agreement with LNDM predictions^18–27 (see also equations (8) and (12)) and cannot be explained in the frameworks of CGM and the model of Jackson et al.

In recent years, particular interest has centred on the hybrid continuum atomistic models of laser–materials interactions, which combine the advantages of MD method with a continuum description.^79,80 Molecular dynamics is used in the very phase transformation region, where active processes of rapid non-equilibrium phase transformations, damage and ablation take place, whereas the continuum equations are solved in a much wider region affected by the heat mass transfer in the system. In particular, the local non-equilibrium phase transformations in metals have been studied by MD simulations combined with the continuum two-temperature (2T) model.^79,80 The continuum 2T model has been widely used to describe energy transfer in metals under ultrashort laser irradiation.^{41,44,79,80,90} It assumes that the lattice and electrons can be described by separate temperatures, which space–time evolution is described by two coupled heat conduction equations. The parabolic 2T model, which assumes that the electron gas is at local equilibrium, uses the classical transfer equation of parabolic type for the electron gas. For far from local equilibrium processes, when the relaxation time of the electrons is of the order of the characteristic time of the process (e.g. the laser pulse duration), the hyperbolic 2T model^41,44,90 should be used. A combination of the hyperbolic 2T model with MD methods provides better description of ultrafast laser beam interaction with metals.⁷⁹

Phase field models

The PF models consider the diffuse character of the solid/liquid interface with a finite thickness and describe the dynamical phenomena in both the bulk phases and the interface region by the minimisation of some specified free energy functional.^{1,2,61,81–85} The jump in the concentration at the interface, which is typical for a sharp interface formulation disappears and is replaced by a continuous profile. The detailed description of the interface region seems to be more realistic, but it is achieved at the expense of rather complicated mathematical formulation. Phase field studies with the parabolic form of the dynamic equations^1,2,81–84 have been modified to their hyperbolic form using HDE (equation (3)), for solute concentration and PF variables.^61,85 The hyperbolic PF model^61,85 confirms the result contained earlier by LNDM^18–22 that the sharp transition to diffusionless and partitionless solidification occurs at V = V_D; yet, the mathematical formulation of the PF model becomes rather complicated. Indeed, the transition is caused by the local non-equilibrium diffusion effects in the bulk liquid, which depress solute diffusion at high interface velocity V → V_D.^18–27 It implies that the transition to diffusionless solidification has a purely diffusion nature, and it does not depend on details of the phase transformation kinetics, whereas solute segregation at the interface depends on transformation kinetics at V < V_D.

Phase field crystal models

The PFC method,^{38,62–65,86} a continuum method operating on atomistic length scales and diffusive time scales, has helped bridge the multiple scale gap between MD and phase field. The PFC model captures the atomic scale structure analogous to MD, yet its diffusive time scale is of the same order as regular PF models. Furthermore, the PFC is motivated from classical density functional theory and contains only a minimal number of parameters, which, in principle, can be derived from fundamental liquid state properties through the direct correlation functions controlling the excess energy. Humadi et al.^62–64 have incorporated two time scales into PFC model of a binary alloy to explore different solute trapping properties as a function of crystal/melt interface velocity. They considered inertial dynamics in both the concentration and density field using HDE (equation (3)), which converts the parabolic type PFC model into the hyperbolic one. Humadi et al.^62–64 calculated the solute concentration profiles for both parabolic and hyperbolic cases. With only diffusive (parabolic) dynamics, the concentration ahead of the interface is higher than the solid concentration even in extremely fast velocities (see Fig. 3b in Ref. 63), which implies that the segregation coefficient K approaches unity in the limit of infinite velocity. When the second time derivative is activated in the concentration and density fields, there is a substantial amount of trapping even in the low velocity regime (see Fig. 3c in Ref. 63). In the high velocity regime, the liquid concentration ahead of the interface equals the averaged solid concentration through the interface, i.e. there is no solute concentration peak ahead of the interface (compare Fig. 3d in Ref. 63 with Fig. 1c in the present paper). The complete solute trapping K = 1 is achieved at a finite interface velocity (compare Figs. 4 and 5 in Ref. 63 with Fig. 2 curve 5 in the present paper).

The next step to extend the hyperbolic PF and PFC models is to include the third order mixed derivative in the dynamical equations, which is responsible for space non-local effects.^{30,31,41–44} In such a case, the governing equation for the dimensionless local number density n (and concentration) can be rewritten as 14 where β and l are characteristic time and length scales respectively. For zero correlation length (l = 0), equation (14) reduces to the hyperbolic PFC model. The effect of non-zero correlation lengths is to smooth the discontinuity, which corresponds to purely wave-like (hyperbolic) dynamics, leading to diffusion-like dynamics in the high frequency limit.^30,31

Note that the conventional modelling of rapid solidification phenomena as well as PF and PFC approaches have used thermodynamic potentials to calculate the rate of non-equilibrium phase transformation. Although a non-zero free energy change means that there is a driving force for solidification, the rate of non-equilibrium phase transformation is not specified by the free energy change value. Strictly speaking, the rate is a kinetic parameter, which should be calculated from reaction rate theory.

Cellular automata and discrete heat mass transfer models

Cellular automata (CA) have been used to predict the interface evolution during binary alloy solidification due to their high computational efficiency.^87–90 In particular, CA have been successfully combined with PF methods to predict the dendrite growth of multicomponent and multiphase alloys during the solidification process.⁹⁰ The microscale CA model was built to track the dendrite morphology evolution and associated variation of temperature–concentration fields, while PF model was used to calculate the growth kinetics. Cellular automata approach is analogous to the discrete heat mass transfer model, which introduces the discrete time τ and length h scales of the process and gives the transfer equations in the discrete form. According to its definition,^30,41,91,92 the discrete model takes into account both time τ and space h non-locality of transfer processes and can be used directly in the discrete form as highly efficient tool for computer simulation. In the continuum limit and , the discrete model gives hierarchy of parabolic and hyperbolic continuum diffusion equations depending on the scaling law of the continuum limit.^30,41,91,92 If and so that has a finite value, then the continuum limit gives hierarchy of parabolic diffusion equations including classical diffusion equation as a first order approximation. The second order approximation gives the parabolic diffusion equation with two additional terms: second order time derivative and third order mixed derivative, which are responsible for the time and space non-local effects respectively. If and so that has a finite value, then the continuum limit gives hierarchy of hyperbolic diffusion equations including HDE (equation (3)) as a first order approximation. The discrete model for the heat mass transfer processes can be also combined with MD, PF or PFC methods.

Conclusions

The local non-equilibrium diffusion effects play a decisive role in rapid alloy solidification: they lead to a sharp transition from diffusion controlled to diffusionless, thermal controlled solidification when the interface velocity V passes through the critical point . The transition is accompanied by complete solute trapping and a sharp break of velocity–undercooling (or interface temperature–velocity) relationship. The local non-equilibrium diffusion effects and relevant phenomena, such as disorder and density trapping, grain refinement, absolute stability, etc., which have met substantial experimental difficulties in liquid–solid transformations, may be favourably investigated during solid state transformations and colloidal solidifications owing to the small diffusion coefficients and, hence, small diffusion velocities in such systems. The hybrid models, which combine the advantages of the continuum local non-equilibrium modelling with MD, PF and PFC methods, give us a new powerful tool to study real alloy solidification processes under extreme conditions. Molecular dynamics, PF and PFC methods can be used in the very phase transformation region, where active processes of phase transformations take place, whereas the continuum local non-equilibrium models are more efficient in a much wider region affected by the heat mass transfer in the system. The discrete heat mass transfer models and CA, which also take into account the local non-equilibrium effects, are particularly convenient for computer simulations because they do not need to be translated to the language of discrete mathematics and can be also used in combination with MD, PF and PFC methods.

Acknowledgements

The reported study was partially supported by RFBR, research project no. 14-48-03535.

References

Glicksman

M. E.

: ‘Principles of solidification’; 2011, New York, Springer.

Jackson

K. A.

: ‘Kinetic processes: crystal growth, diffusion, and phase transitions in materials’, 2nd edn; 2010, Weinheim, John Wiley & Sons.

Asta

, Beckermann

, Karma

, Kurz

, Napolitano

, Plapp

, Purdy

, Rappaz

and Trivedi

: ‘Solidification microstructures and solid-state parallels: recent developments, future directions’, Acta Mater., 2009, 57, 941–971.

Liu

and Yang

G. C.

: ‘Rapid solidification of highly undercooled bulk liquid superalloy: recent developments, future directions’, Int. Mater. Rev., 2006, 51, 145–170.

Serdyukov

S. I.

: ‘Higher order heat and mass transfer equations and their justification in extended irreversible thermodynamics’, Theor. Found. Chem. Eng., 2013, 47, 89–103.

Wang

G. X.

and Prasad

: ‘Microscale heat and mass transfer and non-equilibrium phase change in rapid solidification’, Mater. Sci. Eng. A, 2000, A292, 142–148.

Willnecker

, Herlach

D. M.

and Feuerbacher

: ‘Evidence of nonequilibrium processes in rapid solidification of undercooled melts’, Phys. Rev. Lett., 1989, 62, 2707–2710.

Willnecker

, Herlach

D. M.

and Feuerbacher

: ‘Grain refinement induced by a critical crystal growth velocity in undercooled melts’, Appl. Phys. Lett., 1990, 56, 324–326.

Eckler

, Cochrae

R. F.

, Herlach

D. M.

and Feuerbacher

: ‘Evidence for a transition from diffusion-controlled to thermally controlled solidification in metallic alloys’, Phys. Rev. B, 1992, 45B, (9), 5019–5022.

10.

Walder

and Ryder

P. L.

: ‘Critical solidification behavior of undercooled Ag–Cu alloys’, J. Appl. Phys., 1993, 74, (10), 6100–6106.

11.

Herlach

D. M.

: ‘Direct measurements of crystal growth velocities in undercooled melts’, Mater. Sci. A, 1994, A179/A180, 147–152.

12.

Barth

, Wei

and Herlach

D. M.

: ‘Crystal growth in undercooled melts of intermetallic compounds FeSi and CoSi’, Phys. Rev. B, 1995, 51B, 3422–3427.

13.

Walder

: ‘Dendritic growth rate in undercooled, dilute Ti–Ni melts’, Mater. Sci. Eng. A, 1997, A229, 156–162.

14.

Herlach

D. M.

: ‘Metastable materials solidified from undercooled melts’, J. Phys. Condens. Matter, 2001, 13, 7737–7751.

15.

Biswas

, Phanikumar

, Herlach

and Chattopadhyay

: ‘Non-equilibrium solidification of concentrated Fe–Ge alloys’, Mater. Sci. Eng. A, 2007, A449–A451, 12–17.

16.

Biswas

, Phanikumar

, Holland-Moritz

, Herlach

and Chattopadhyay

: ‘Disorder trapping and grain refinement during solidification of undercooled Fe–18 at Ge melts’, Philos. Mag., 2007, 87, 3817–3837.

17.

Ahmad

, Cochrane

R. F.

and Mullis

A. M.

: ‘Disorder trapping during the solidification of βNi3Ge from its deeply undercooled melt’, J. Mater. Sci., 2012, 47, 2411–2420.

18.

Sobolev

S. L.

: ‘Local-nonequilibrium model for rapid solidification of undercooled melts’, Phys. Lett. A, 1995, 199A, 383–386.

19.

Sobolev

S. L.

: ‘Effects of local-nonequilibrium solute diffusion on rapid solidification of alloys’, Phys. Status Solidi A, 1996, 156A, 293–303.

20.

Sobolev

S. L.

: ‘Rapid solidification under local nonequilibrium condition’, Phys. Rev. E, 1997, 55E, (6), 6845–6854.

21.

Sobolev

S. L.

: ‘Local nonequilibrium diffusion effects on the kinetic phase boundaries in solidification’, Int. J. Non-Equilib. Proc., 1997, 10, 49–58.

22.

Sobolev

S. L.

: ‘Influence of local nonequilibrium on the rapid solidification of binary alloys’, Tech. Phys., 1998, 43, 307–313.

23.

Sobolev

S. L.

: ‘Rapid colloidal solidifications under local nonequilibrium diffusion conditions’, Phys. Lett. A, 2012, 376A, 3563–3566.

24.

Sobolev

S. L.

: ‘Local non-equilibrium diffusion model for solute trapping during rapid solidification’, Acta Mater., 2012, 60, 2711–2718.

25.

Sobolev

S. L.

: ‘An analytical model for local-nonequilibrium solute trapping during rapid solidification’, Trans. Nonferr. Met. Soc. China, 2012, 22, 2749–2755.

26.

Sobolev

S. L.

: ‘An analytical model for complete solute trapping during rapid solidification of binary alloys’, Mater. Lett., 2012, 89, 191–194.

27.

Sobolev

S. L.

: ‘On the transition from diffusion-limited to kinetic-limited regimes of alloy solidification’, Acta Mater., 2013, 61, 7881–7888.

28.

and Sobolev

S. L.

: ‘Local nonequilibrium solute trapping model for non-planar interface’, J. Cryst. Growth, 2013, 380, 68–71.

29.

Sobolev

S. L.

, Poluyanov

L. V.

and Liu

: ‘An analytical model for solute diffusion in multicomponent alloy solidification’, J. Cryst. Growth, 2014, 395, 46–54.

30.

Sobolev

S. L.

: ‘Nonlocal diffusion models: application to rapid solidification of binary mixtures’, Int. J. Heat Mass Trans., 2014, 71, 295–302.

31.

Sobolev

S. L.

: ‘Diffusion-stress coupling in liquid phase during rapid solidification of binary mixtures’, Phys. Lett. A, 2014, 378A, 475–479.

32.

Sobolev

S. L.

: ‘New interpretation of experimental data on Si–As alloy solidification with planar interface’, Trans. Nonferr. Met. Soc. China, to be published.

33.

Brice

J. C.

: ‘The growth of crystal from the melts’; 1965, Amsterdam, North-Holland.

34.

Aziz

M. J.

and Kaplan

: ‘Continuous growth model for interface motion during alloy solidification’, Acta Metall., 1998, 36, 2335–2347.

35.

Smith

P. M.

and Aziz

M. J.

: ‘Solute trapping in aluminium alloys’, Acta Metall. Mater., 1994, 42, 3515–3525.

36.

Hillert

: ‘Solute drag, solute trapping and diffusional dissipation of Gibbs energy’, Acta Mater., 1999, 47, 4481–4505.

37.

Jackson

K. A.

, Beatty

K. M.

and Gudgel

K. A.

: ‘An analytical model for non-equilibrium segregation during crystallization’, J. Cryst. Growth, 2004, 271, 481–494.

38.

Emmerich

, Löwen

, Wittkowski

, Gruhn

, Tóth

G. I.

, Tegze

and Gránásy

: ‘Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview’, Adv. Phys., 2012, 61, 665–743.

39.

Jou

, Casas-Vazquez

and Lebon

: ‘Extended irreversible thermodynamics’; 1996, Berlin, Springer.

40.

Sobolev

S. L.

: ‘Transport processes and travelling waves in systems with local nonequilibrium’, Sov. Phys. Usp., 1991, 34, (3), 217–229.

41.

Sobolev

S. L.

: ‘Local non-equilibrium transport models’, Phys. Usp., 1997, 40, (10), 1043–1053.

42.

Nettleton

R. E.

and Sobolev

S. L.

: ‘Application of extended thermodynamics to chemical, rheological, and transport processes: a special survey. Part I: approaches and scalar rate processes’, J. Non-Equilibr. Thermodyn., 1995, 20, (3), 205–229.

43.

Nettleton

R. E.

and Sobolev

S. L.

: ‘Application of extended thermodynamics to chemical, rheological, and transport processes: a special survey. Part II: vector transport processes, shear relaxation and rheology’, J. Non-Equilibr. Thermodyn., 1995, 20, (4), 297–331.

44.

Nettleton

R. E.

and Sobolev

S. L.

: ‘Application of extended thermodynamics to chemical, rheological, and transport processes: a special survey. Part III: wave phenomena’, J. Non-Equilibr. Thermodyn., 1996, 21, (1), 1–16.

45.

Wang

H. F.

, Liu

, Chen

, Yang

and Zhou

: ‘Analysis of non-equilibrium dendrite growth in a bulk undercooled alloy melt: model and application’, Acta Mater., 2007, 55, 497–506.

46.

Wang

H. F.

, Liu

, Chen

, Yang

and Zhouet

: ‘Effect of non-linear liquidus and solidus in undercooled dendrite growth: a comparative study in Ni–0.7 at. B and Ni–1 at. Zr systems’, Scr. Mater., 2007, 57, 413–416.

47.

Wang

H. F.

, Liu

, Yang

, Chen

, Yang

and Zhouet

: ‘An extended morphological stability model for a planar interface incorporating the effect of nonlinear liquidus and solidus’, Acta Mater., 2008, 56, 2592–2601.

48.

Wang

H. F.

, Liu

, Wang

and Zhai

H. M.

: ‘Oscillatory morphological stability for rapid directional solidification: effect of non-linear liquidus and solidus’, Acta Mater., 2011, 59, 5859–5867.

49.

Wang

H. F.

, Liu

and Tan

Y. M.

: ‘Modeling grain refinement for undercooled single-phase solid-solution alloys’, Acta Mater., 2011, 59, 4787–4797.

50.

Tan

and Wang

: ‘Modeling constrained dendrite growth in rapidly directional solidification’, J. Mater. Sci., 2012, 47, 5308–5316.

51.

, Zhang

and Wu

: ‘A comparative study on migration of a planar interface during solidification of non-dilute alloys’, J. Cryst. Growth, 2010, 312, 982–988.

52.

, Zhang

and Wu

: ‘Numerical solution and comparison to experiment of solute drag models for binary alloy solidification with a planar phase interface’, Scr. Mater., 2010, 62, 716–719.

53.

Wang

, Liu

, Zhai

and Wang

: ‘Application of the maximal entropy production principle to rapid solidification: a sharp interface model’, Acta Mater., 2012, 60, 1444–1454.

54.

Wang

, Liu

, Yang

, Chen

, Yang

and Zhou

: ‘Solute trapping model incorporating diffusive interface’, Acta Mater., 2008, 56, 746–753.

55.

Wang

, Liu

, Chen

and Yang

: ‘Solute trapping model based on solute drag treatment’, Trans. Nonferr. Met. Soc. China, 2010, 20, 877–881.

56.

Wang

H. F.

, Liu

, Yang

G. C.

and Zhou

Y. H.

: ‘Modeling the overall solidification kinetics for undercooled single-phase solid-solution alloys. I: model derivation’, Acta Mater., 2010, 58, 5402–5410.

57.

, Zhang

and Wu

: ‘Analysis for free dendritic growth model applicable to non-dilute alloys’, Metall. Mater. Trans. A, 2012, 43A, 3748–3754.

58.

Cook

S. J.

and Clancy

: ‘Solute trapping at a rapidly moving solid/liquid interface for a Lennard–Jones alloy’, Mol. Simul., 1990, 5, 99–117.

59.

Cook

S. J.

and Clancy

: ‘Impurity segregation in Lennard–Jones A/AB heterostructures’, J. Chem. Phys., 1993, 99, 2175–2191.

60.

Yang

, Humadi

, Buta

, Laird

B. B.

, Sun

, Hoyt

J. J.

and Asta

: ‘Atomistic simulations of nonequilibrium crystal-growth kinetics from alloy melts’, Phys. Rev. Lett., 2011, 107, 025505-4.

61.

Zhang

, Danilova

E. V.

, Steinbach

, Medvedev

and Galenko

P. K.

: ‘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., 2013, 61, 4155–4168.

62.

Humadi

, Ofori-Opoku

, Provatas

and Hout

J. J.

: ‘Atomistic modeling of solidification phenomena using the phase-field-crystal model’, JOM, 2013, 65, (9), 1103–1110.

63.

Humadi

, Hoyt

J. J.

and Provatas

: ‘Phase-field-crystal study of solute trapping’, Phys. Rev. E, 2013, 87E, 022404-10.

64.

Humadi

, Hoyt

J. J.

and Provatas

: ‘Microscopic treatment of solute trapping and drag’, arXiv:1412.0297v1 [cond-mat.mtrl-sci], 2014, available at: http://arxiv.org/abs/1412.0297.

65.

Tang

, Yu

Y. -M.

, Wang

, Li

, Wang

, Guo

and Zhou

: ‘Phase-field-crystal simulation of nonequilibrium crystal growth’, Phys. Rev. E, 2014, 89E, 012405-6.

66.

Hecht

, Granasy

, Pusztai

, Bottger

, Apel

, Witusiewicz

, Ratke

, De Wild

, Froyen

, Camel

, Drevet

, Faivre

, Fries

S. G.

, Legendre

and Rex

: ‘Multiphase solidification in multicomponent alloys’, Mater. Sci. Eng. R, 2004, R46, 1–49.

67.

Ludwig

: ‘The interface response-functions in multi-componental alloy solidification’, Physica D, 1998, 124D, 271–284.

68.

Hunziker

: ‘Theory of plane front and dendritic growth in multicomponent alloys’, Acta Mater., 2001, 49, 4191–4203.

69.

Ruan

and Dai

F. P.

: ‘Rapid dendrite growth subjected to multi-solute trapping in an undercooled Fe-based quaternary alloy’, Intermetallics, 2012, 25, 80–85.

70.

Wang

, Wang

, Liu

and Zhai

: ‘Modeling rapid solidification of multi-component concentrated alloys’, Acta Mater., 2013, 61, 1359–1372.

71.

Wang

, Wang

H. F.

, Liu

and Zhai

H. M.

: ‘Morphological stability analysis for planar interface during rapidly directional solidification of concentrated multi-component alloys’, Acta Mater., 2014, 67, 220–231.

72.

Jacot

, Sumida

and Kurz

: ‘Solute trapping-free massive transformation at absolute stability’, Acta Mater., 2011, 59, 1716–1724.

73.

Hillert

and Schalin

: ‘Modeling of solute drag in the massive phase transformation’, Acta Mater., 2000, 48, 461–468.

74.

Deville

: ‘Ice templating, freeze casting: Beyond materials processing’, J. Mater. Res., 2013, 28, (17), 2202–2219.

75.

Elliott

J. A. W.

and Peppin

S. S. L.

: ‘Particle trapping and banding in rapid colloidal solidification’, Phys. Rev. Lett., 2011, 107, 168301-5.

76.

Deville

, Maire

, Bernard-Granger

, Lasalle

, Bogner

, Gauthier

, Leloup

and Guizard

: ‘Metastable and unstable cellular solidification of colloidal suspensions’, Nature Mater., 2009, 8, 966–972.

77.

Nozawa

, Uda

, Naradate

, Koizumi

, Fujiwara

, Toyotama

and Yamanaka

: ‘Impurity partitioning during colloidal crystallization’, J. Phys. Chem. B, 2013, 117B, 5289–5295.

78.

Dullens

R. P. A.

, Aarts

D. G. A. L.

and Kegel

W. K.

: ‘Dynamic broadening of the crystal–fluid interface of colloidal hard spheres’, Phys. Rev. Lett., 2006, 97, 228301-4.

79.

Roth

, Sonntag

, Karlin

, Paredes

C. T.

, Sartison

, Kraub

and Trebin

H. -R.

: ‘Molecular dynamics simulations studies of laser ablation in metals’, AIP Conf. Proc., 2012, 1464, 504–523.

80.

Zhigilei

L. V.

, Lin

, Ivanov

D. S.

, Leveugle

, Duff

W. H.

, Thomas

, Sevilla

and Guy

S. J.

: ‘Atomic/molecular-level simulations of laser-materials interactions, Vol. 130, Chap. 3’, in ‘Laser–surface interactions for new materials production: tailoring structure and properties’ ‘Springer series in materials science’ (ed. Miotello

, et al.,., 43–79; 2010, New York, Springer Verlag.

81.

Danilov

and Nestler

: ‘Phase-field modelling of solute trapping during rapid solidification of a Si–As alloy’, Acta Mater., 2006, 54, 4659–4664.

82.

Fan

, Greenwood

, Haataja

and Provatas

: ‘Phase-field simulations of velocity selection in rapidly solidified binary alloys’, Phys. Rev. E, 2006, 74E, 031602-7.

83.

Tegze

, Granasy

, Toth

G. I.

, Douglas

J. F.

and Pusztai

: ‘Tuning the structure of non-equilibrium soft materials by varying the thermodynamic driving force for crystal ordering’, Soft Mater., 2011, 7, 1789–1799.

84.

Bhattacharya

, Upadhyay

C. S.

and Sangal

: ‘Phase-field model for mixed-mode of growth applied to austenite to ferrite transformation’, Metall. Mater. Trans. A, 2015, 46, 926–936.

85.

Wang

, Kuang

, Zhang

and Liu

: ‘A hyperbolic phase-field model for rapid solidification of a binary alloy’, J. Mater. Sci., 2015, 50, 1277–1286.

86.

Tang

, Yu

Y. -M.

, Wang

, Li

, Wang

, Guo

and Zhou

: ‘Phase-field-crystal simulation of nonequilibrium crystal growth’, Phys. Rev. E, 2014, 89E, 012405-6.

87.

, Li

and Zhang

: ‘A cellular automaton technique for modelling of a binary dendritic growth with convection’, Appl. Math. Model., 2007, 31, 971.

88.

Shin

Y. H.

and Hong

C. P.

: ‘Modeling of dendritic growth with convection using a modified cellular automaton model with a diffuse interface’, ISIJ Int., 2002, 42, 359–367.

89.

Zhu

M. F.

, Lee

S. Y.

and Hong

C. P.

: ‘Modified cellular automaton model for the prediction of dendritic growth with melt convection’, Phys. Rev. E, 2004, 69E, 061610-12–061612.

90.

Tan

, Bailey

N. S.

and Shin

Y. C.

: ‘A novel integrated model combining cellular automata and phase field methods for microstructure evolution during solidification of multi-component and multi-phase alloys’, Comp. Mater. Sci., 2011, 50, 2573–2585.

91.

Sobolev

S. L.

: ‘Discrete model for transfer processes’, Phys. Lett. A, 1992, 163A, 101–103.

92.

Sobolev

S. L.

: ‘Two-temperature discrete model for nonlocal heat conduction’, J. Phys. III France, 1993, 3, 2261–2269.

93.

H. L.

, Ghosh

S. K.

, Liu

W. J.

, Lee

K. Y.

and Bhadeshia

H. K. D. H.

: ‘Non-equilibrium solidification and ferrite in δ-TRIP steel’, Mater. Sci. Technol., 2010, 26, 817–823.

94.

Zhang

, Liu

, Xu

J. F.

and Yang

G. C.

: ‘Microstructure refinement of Fe40Ni40B20 alloy in non-equilibrium solidification: possibility of nanostructure formation’, Mater. Sci. Technol., 2012, 28, 844–849.

95.

Castanho

M. A. P.

, Goulart

P. R.

, Brito

, Spinelli

J. E.

, Cheung

and Garcia

: ‘Garcia Steady and unsteady state peritectic solidification’, Mater. Sci. Technol., 2015, 31, 105–114.

96.

Guan

, Jia

, Chen

, Weng

, Ding

and Zhang

: ‘Synergy of Cr concentration and withdraw rate on microstructure evolution of directionally solidified Nb–14Si–24Ti alloys’, Mater. Sci. Technol., 2014, 30, 1359–1366.

97.

X. L.

, Chen

Y. Z.

and Liu

: ‘Study of microstrain in rapidly solidified structures of hypercooled Co80Pd20 alloys’, Mater. Sci. Technol., 2013, 29, 117–120.