Thermodynamics of rapid solidification and crystal growth kinetics in glass-forming alloys

Abstract

Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given.

This article is part of the theme issue ‘Heterogeneous materials: metastable and non- ergodic internal structures’.

1. Introduction

Rapid solidification can be initiated by fast quenching or slow cooling to avoid premature nucleation and to attain deep undercoolings. Cooling rates may reach 10⁶ (K s⁻¹) for melt spinning, or atomization by gas, impulse or in a drop tube; temperature gradients can reach 10⁹ (K m⁻¹) in laser annealing, and undercoolings of up to 300–500 (K) can be achieved (see refs. [1–6] for details of techniques and measurements). Such large driving forces lead to solidification rates in the order of several tens of metres per second [5,7]. Considering that the characteristic diffusion speed is in the order of several metres or tens of metres per second in metallic alloys and of several centimetres or tens of centimetres per second in semiconductors [8], it becomes obvious that the solid/liquid interface may exhibit velocities in the order of or even larger than the diffusion speed. In such a case, the rapid solidification front may undergo diffusionless (chemically partitionless) transformation which may proceed in a wide or narrow interval of driving forces [3,4,9,10].

Progress in the formulation of the thermodynamical basis [11–16] has led to an extended description [17] going beyond the hypothesis of local equilibrium that is stipulated by broken ergodicity in rapidly solidifying samples [18]. Using a local non-equilibrium model of dendritic growth [19], the kinetics of rapid solidification in slowly and deeply undercooled samples has been described consistently using experimental data on the solidification kinetics of a number of interstitial and substitutional solution phases during dendritic growth [20–23]. In these studies, it was shown that the interface velocity monotonically increases with undercooling (even though specific peculiarities in the transition from diffusion limited to thermally controlled growth of dendrites were found). Rapid kinetics with an unusually steep dendrite velocity/undercooling relationship has also been obtained in intermetallic alloys with an order–disorder transition [24,25]. However, the velocity of dendrites in melts of intermetallic phases also exhibited a monotonic increase with increasing undercooling. Another type of velocity/undercooling relationship can be experimentally found in the kinetics of dendritic solidification of glass-forming metals and alloy melts. For glass-forming samples produced from metals, alloys, oxides, organics or chalcogenides [26–28], the growth rate has a maximum value at a defined undercooling that lies between the glass-transition undercooling and zero undercooling directly at the melting (or liquidus) temperature. Moreover, for the alloy Cu₅₀Zr₅₀ (concentrations in %), a transition from purely thermodynamic to kinetically controlled growth has been experimentally observed for the first time [27]. Attempts to describe such transitions using common dendrite growth theory [29] lead to a smooth and gradual behaviour of velocity versus undercooling, describing the main part of the experimental data well, but overestimating velocities at the highest undercooling where the so-called ‘abrupt drop in kinetics’ occurs. The thermodynamic and kinetic analysis of the solidification behaviour in glass-forming samples is the main goal of the present paper.

The article is divided into two main parts. In the first part (§§2 and 3), we analyse thermodynamic driving forces and growth rates in dendritic rapid solidification and compare them with previously verified theoretical predictions. Here the solidification can be considered a rapid process if the solid/liquid interface moves with a velocity comparable to the diffusion speed in the bulk liquid. In this case, the relaxation time of the solute diffusion flux is not negligible and should be included in the model equations that characterize deviations from local equilibrium and ergodicity breaking [18]. The second part of the article is devoted to the analysis of dendritic solidification of congruently melting alloys in which the chemical composition does not influence the growth velocity due to the absence of chemical segregation. A dendrite growth model for predicting the solidification kinetics of glass-forming alloys is formulated in §4. The model consists of a system of equations describing (i) the undercooling balance at the dendritic surface, (ii) the stability condition for the dendrite tip at arbitrary growth Péclet numbers, and (iii) the kinetic contribution following on from the travelling wave solution of the phase field model (see appendices). In this case, we also deal with rapid solidification considering that gradient flow relaxation is used in the model equations describing the deviation from local equilibrium and ergodicity breaking [18].

2. Entropy change in rapid solidification

Consider a solidifying sample of a volume v₀ of a binary alloy. The concentration distribution of solute is given by C(x, t), and at the interface the concentrations are C*_L in the liquid phase and C*_S in the solid phase. The temperature T of the sample is constant and the total pressure tensor P is related to the viscous pressure P^ν by P = pU + P^ν, where p is the equilibrium (static) pressure and U the identity tensor. As usual, P^ν is split into two parts, a bulk pressure $p^{\nu }=\frac{1}{3}\mathrm{trace}\hspace{0.17em}{\mathit{P}}^{\nu }$ and a deviatoric part ${\hat{\mathit{P}}}^{\nu }$, so that

$$\mathit{P}=p\mathit{U}+p^{\nu }\mathit{U}+{\hat{\mathit{P}}}^{\nu }.$$ 2.1

The necessity of the pressure introduction is dictated by (i) accumulated stresses around rapidly moving interface and (ii) the difference in density of the liquid and solid leading to shrinkage effects. For example, an increasing dislocation density in the microstructure of alloys with increasing solidification rate [30] directly testifies that rapid solidification results in an accumulation and relaxation of stresses.

(a). The bulk entropy

The temporal evolution of the concentration field follows from the formalism of extended irreversible thermodynamics (EIT) [31,32]. The total entropy of the entire system of the volume v_v is described by

$$-TS(t)={\int }_{v_v}[{\mu }_A{C}_A+{\mu }_B{C}_B+{\alpha }_A{J}_A^2+{\alpha }_B{J}_B^2+{\alpha }_{\mathbf{P}}:\mathbf{P}]\hspace{0.17em}\mathrm{d}{v}_v,$$ 2.2

where C_A is the concentration of the A-atoms having the chemical potential μ_A, C_B is the concentration of the B-atoms having the chemical potential μ_B, J_A and J_B are the diffusion fluxes of the A- and B-atoms, respectively. The thermodynamic coefficients α_A and α_B are defined by

$${\alpha }_A=\frac{{\tau }_D^{(A)}}{D^{(A)}}\frac{\mathrm{\partial }{\mu }_A}{\mathrm{\partial }{C}_A}=\frac{1}{{(V_D^{(A)})}^2}\frac{\mathrm{\partial }{\mu }_A}{\mathrm{\partial }{C}_A},{\alpha }_B=\frac{{\tau }_D^{(B)}}{D^{(B)}}\frac{\mathrm{\partial }{\mu }_B}{\mathrm{\partial }{C}_B}=\frac{1}{{(V_D^{(B)})}^2}\frac{\mathrm{\partial }{\mu }_B}{\mathrm{\partial }{C}_B},$$ 2.3

where τ^(A)_D and τ^(B)_D are the relaxation times of the diffusion fluxes, J_A and J_B, respectively, D^(A) and D^(B) the diffusion coefficients of the A- and B-atoms, respectively, and V^(A)_D = (D^(A)/τ^(A)_D)^1/2 and V^(B)_D = (D^(B)/τ^(B)_D)^1/2 are the maximum speeds for diffusion of the A- and B-atoms, respectively.

Using the expression for the pressure (2.1) and the respective expression for α_P (see [31,32]), the entropy (2.2) can be re-written as

$$-TS(t)={\int }_{v_v}[{\mu }_A{C}_A+{\mu }_B{C}_B+{\alpha }_A{J}_A^2+{\alpha }_B{J}_B^2-pv+v{\alpha }_0{(p^{\nu })}^2+v{\alpha }_2{\hat{\mathbf{P}}}^{\nu }:{\hat{\mathbf{P}}}^{\nu }]\hspace{0.17em}\mathrm{d}{v}_v,$$ 2.4

where the thermodynamic coefficients α₀ = ζ⁻¹_B and α₂ = (2η_S)⁻¹ are defined by the bulk viscosity ζ_B and shear viscosity η_S, respectively.

We use the concentration definitions C = C_B = 1 − C_A together with the fluxes J_D =  − J_A = J_B, J²_D = J²_A = J²_B. Then, considering further only a semi-infinite one-dimensional solidification domain, equation (2.2) simplifies to

$$S(x,t)={\int }_0^{\mathrm{\infty }}[s_{\mathrm{eq}}(x,t)+s_{\mathrm{ne}}(x,t)]\hspace{0.17em}\mathrm{d}x,$$ 2.5

where the local equilibrium part of the entropy density is

$$-T{s}_{\mathrm{eq}}(x,t)={\mu }_A(1-C)+{\mu }_{B}C-pv,$$ 2.6

and the local non-equilibrium part of the entropy density is

$$-T{s}_{\mathrm{ne}}(x,t)={\alpha }_D{J}_D^2+v{\alpha }_0{(p^{\nu })}^2+v{\alpha }_2{(P_{xx}^{\nu })}^2,$$ 2.7

with α_D = α_A + α_B and P^ν_xx the component of viscous pressure along the x-axis.

(b). Entropy produced by the moving interface

Neglecting the diffusion processes in the solid phase, J_D(x, t) = 0:x < x*(t), where x*(t) is the position of the solid/liquid interface, consider now the change of entropy in the entire system caused by a small movement ε of the interface in a small step of time Δt, corresponding to an interface velocity of V = ε/Δt (figure 1). In the area swept by the interface, x = x*(t), entropy is produced by the moving interface. The local equilibrium part of entropy increase is described by

$$\begin{array}{rl}-{\mathrm{\Delta }}_{\mathrm{\Delta }t}(T{S}_{\mathrm{eq}})& =-[T{S}_{\mathrm{eq}}(t+\mathrm{\Delta }t)-T{S}_{\mathrm{eq}}(t)]\\ & =\epsilon [(1-C_S^{\ast })({\mu }_A^S-{\mu }_A^L)+C_S^{\ast }({\mu }_B^S-{\mu }_B^L)-p\mathrm{\Delta }v],\end{array}$$ 2.8

where Δv is the volume change and, for isobaric conditions, equation (2.8) is equivalent to the expression given in refs. [11,16]. In this occasion, all diffusion fluxes have vanished in the area swept by the interface, yielding a total entropy increase as

$$\begin{array}{rl}{\mathrm{\Delta }}_{\mathrm{\Delta }t}(T{S}_{\mathrm{ne}})& =T{S}_{\mathrm{ne}}(t+\mathrm{\Delta }t)-T{S}_{\mathrm{ne}}(t)\\ & ={\int }_{x^{\ast }(t_0)}^{x^{\ast }(t_0+\epsilon )}[{\alpha }_D{J}_D^2(x,t_0)+v{\alpha }_0{(p^{\nu })}^2+v{\alpha }_2{(P_{xx}^{\nu })}^2]\hspace{0.17em}\mathrm{d}x.\end{array}$$ 2.9

Figure 1.

Two snapshots of the concentration distribution at a solidification front, separated by a small time step Δt, where the interface has the position x* and moves with a constant velocity V to the right of the figure. (Online version in colour.)

The entropy production is evaluated by applying the differential quotient in the vanishing limit and applying the relation V = ε/Δt by

$$\begin{array}{rl}-\frac{\mathrm{d}}{\mathrm{d}t}{(TS)}_{\mathrm{e}}& =-\underset{\mathrm{\Delta }t\to 0}{lim}\frac{T{S}_{\mathrm{e}}(t+\mathrm{\Delta }t)-T{S}_{\mathrm{e}}(t)}{\mathrm{\Delta }t}\\ & =\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\epsilon }{\mathrm{\Delta }t}[(1-C_S^{\ast })({\mu }_A^S-{\mu }_A^L)+C_S^{\ast }({\mu }_B^S-{\mu }_B^L)-p\mathrm{\Delta }v]\\ & =V[(1-C_S^{\ast })({\mu }_A^S-{\mu }_A^L)+C_S^{\ast }({\mu }_B^S-{\mu }_B^L)-p\mathrm{\Delta }v].\end{array}$$ 2.10

With no change of volume, Δv = 0, equation (2.10) becomes identical to the classical result obtained in [11,16]. For the local non-equilibrium part, equation (2.9), we simplify the analysis by neglecting the bulk pressure, i.e. p^ν → 0. Then, a Taylor series for the square flux and the square of the deviatoric part of the viscous pressure near the interface is introduced by

$$\begin{array}{rl}& J_D^2(x^{\ast }+\epsilon ,t_0)=J_D^2(x^{\ast },t_0)+\epsilon \frac{\mathrm{d}}{\mathrm{d}x}{J}_D^2(x^{\ast },t_0)+\cdots \\ \mathrm{and}& {(P_{xx}^{\nu })}^2(x^{\ast }+\epsilon ,t_0)={\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\epsilon \frac{\mathrm{d}}{\mathrm{d}x}{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\cdots \end{array}\}$$ 2.11

Using equation (2.11), the integral in (2.9) can be evaluated as

$$\begin{array}{rl}& {\int }_{x^{\ast }(t_0)}^{x^{\ast }(t_0+\epsilon )}[{\alpha }_D{J}_D^2(x,t_0)+v{\alpha }_2{\left(P_{xx}^{\nu }\right)}^2]\hspace{0.17em}\mathrm{d}x\\ & ={\alpha }_D[\epsilon J_D^2(x^{\ast },t_0)+\frac{{\epsilon }^2}{2}\frac{\mathrm{d}}{\mathrm{d}x}{J}_D^2(x^{\ast },t_0)+\cdots ]\\ & +v{\alpha }_2[\epsilon {\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\frac{{\epsilon }^2}{2}\frac{\mathrm{d}}{\mathrm{d}x}{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\cdots ].\end{array}$$ 2.12

This leads to the following expression for the local non-equilibrium part in the entropy production

$$\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}{(TS)}_{\mathrm{ne}}& =\underset{\mathrm{\Delta }t\to 0}{lim}\frac{T{S}_{\mathrm{ne}}(t+\mathrm{\Delta }t)-T{S}_{\mathrm{ne}}(t)}{\mathrm{\Delta }t}\\ & ={\alpha }_D\underset{\mathrm{\Delta }t\to 0}{lim}[\frac{\epsilon }{\mathrm{\Delta }t}{J}_D^2(x^{\ast },t_0)+\frac{{\epsilon }^2}{2\mathrm{\Delta }t}\frac{\mathrm{d}}{\mathrm{d}x}{J}_D^2(x^{\ast },t_0)+\cdots ]\\ & +v{\alpha }_2\underset{\mathrm{\Delta }t\to 0}{lim}[\frac{\epsilon }{\mathrm{\Delta }t}{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\frac{{\epsilon }^2}{2\mathrm{\Delta }t}\frac{\mathrm{d}}{\mathrm{d}x}{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\cdots ]\\ & ={\alpha }_{D}V{J}_D^2(x^{\ast },t_0)+{\alpha }_D[\underset{\mathrm{\Delta }t\to 0}{lim}\mathrm{\Delta }t{V}^2\frac{\mathrm{d}}{\mathrm{d}x}{J}_D^2(x^{\ast },t_0)+\cdots ]\\ & +v{\alpha }_{2}V{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+v{\alpha }_2[\underset{\mathrm{\Delta }t\to 0}{lim}\mathrm{\Delta }t{V}^2\frac{\mathrm{d}}{\mathrm{d}x}{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0)+\cdots ]\\ & \approx {\alpha }_{D}V{J}_D^2(x^{\ast },t_0)+v{\alpha }_{2}V{\left(P_{xx}^{\nu }\right)}^2(x^{\ast },t_0),\end{array}$$ 2.13

where P^ν_xx(x*, t₀) is the deviator of the viscous pressure at the interface and J_D(x*, t₀) is the diffusion flux away from the interface. Note that this flux generally does not vanish (except in the special case of diffusionless solidification in which C*_L = C*_S), leading to the local non-equilibrium contribution for interface velocities V smaller than the solute diffusion speed V_D in bulk liquid, i.e. with V < V_D, where V_D≡V^(B)_D is the maximum speed for the diffusion of B-atoms.

Finally, using equations (2.10) and (2.13), the total entropy production per unit time at the interface is given by

$$\begin{array}{rl}-\frac{\mathrm{d}{(TS)}_{\mathrm{Interface}}}{\mathrm{d}t}& =V\cdot [(1-C_S^{\ast })({\mu }_A^S-{\mu }_A^L)+C_S^{\ast }({\mu }_B^S-{\mu }_B^L)+{\alpha }_D{J}_D^2(x^{\ast },t_0)]\\ & -V\cdot [p\mathrm{\Delta }v-v{\alpha }_2{(P_{xx}^{\nu })}^2(x^{\ast },t_0)].\end{array}$$ 2.14

(c). Gibbs free energy change on rapid solidification

Using the established system of thermodynamic relations [14,16,17], one obtains

$$\begin{array}{rl}& -\frac{\mathrm{d}{(TS)}_{\mathrm{Interface}}}{\mathrm{d}t}=\frac{\mathrm{d}G}{\mathrm{d}t}=-v_m^{-1}V\mathrm{\Delta }G\\ \mathrm{and}& \frac{\mathrm{d}G}{\mathrm{d}t}=J_D{X}_D+J_C{X}_C+J_P{X}_P,\end{array}\}$$ 2.15

where G and ΔG are the Gibbs free energy and the Gibbs free energy change on solidification, respectively. Taking into account that the change in Gibbs free energy ΔG proceeds at a constant volume, i.e. at Δv = 0, then, as follows from equation (2.14), the diffusion flux J_D and the crystallization flux J_C together with their conjugate driving forces, X_D and X_C, respectively, are defined in equation (2.15) by

$$\begin{array}{rl}& J_D=-v_m^{-1}V(C_L^{\ast }-C_S^{\ast }),X_D=\mathrm{\Delta }{\mu }_A-\mathrm{\Delta }{\mu }_B+{\alpha }_D(C_L^{\ast }-C_S^{\ast })V^2,\\ & J_C=-v_m^{-1}V,X_C=(1-C_L^{\ast })\mathrm{\Delta }{\mu }_A+C_L^{\ast }\mathrm{\Delta }{\mu }_B\\ \mathrm{and}& J_P=-v_m^{-1}V,X_P=v{\alpha }_2{(P_{xx}^{\nu })}^2,\end{array}\}$$ 2.16

where Δμ_A = μ^S_A − μ^L_A and Δμ_B = μ^S_B − μ^L_B are the differences in the chemical potentials for A- and B-atoms, respectively. In the linear approximation, J_D∝X_D, J_C∝X_C and J_P∝X_P one can obtain the following relations between fluxes and conjugate driving forces:

$$\begin{array}{rl}& (C_L^{\ast }-C_S^{\ast })\frac{V}{v_m}=L_D[\mathrm{\Delta }{\mu }_A-\mathrm{\Delta }{\mu }_B+{\alpha }_D(C_L^{\ast }-C_S^{\ast })V^2],\\ & \frac{V}{v_m}=L_C[(1-C_L^{\ast })\mathrm{\Delta }{\mu }_A+C_L^{\ast }\mathrm{\Delta }{\mu }_B]\\ \mathrm{and}& \frac{V}{v_m}=L_{P}v{\alpha }_2{\left(P_{xx}^{\nu }\right)}^2,\end{array}\}$$ 2.17

where L_D, L_C and L_P are the kinetic coefficients for diffusion and crystal growth due to the driving force for atom attachment and due to the elastic and plastic relaxation, respectively. Several limiting cases can be outlined from equation (2.17). First, for the diffusionless (chemically partitionless) solidification, the analytical solution [33] gives naturally C*_L = C*_S at V ≥V_D. Then, the first equation gives the expression Δμ_A = Δμ_B, which indicates the equality of the chemical potential differences for rapid diffisionless solidification. From this follows the second case, which describes the growth kinetics as for a pure one-component substance upon the diffusionless solidification given by the second equation from equation (2.17): V = v_mL_CΔμ_A with V ≥V_D. The third equation of equation (2.17) predicts that the viscous relaxation of stresses may cause the motion of the solid/liquid interface.

Using the relations from equation (2.16) in equation (2.15), one obtains

$$\mathrm{\Delta }G=(1-C_S^{\ast })\mathrm{\Delta }{\mu }_A+C_S^{\ast }\mathrm{\Delta }{\mu }_B+{\alpha }_D{(C_L^{\ast }-C_S^{\ast })}^2{V}^2+v{\alpha }_2{(P_{xx}^{\nu })}^2.$$ 2.18

This expression represents the Gibbs free energy change at the solid/liquid interface and describes the driving force for interface motion. Neglecting relaxation of the diffusion flux α_D = 0, and relaxation of viscous stresses α₂ = 0, equation (2.18) takes its standard form given and analysed in refs. [11,13,14,16]. The additionally included effects in equation (2.18), which may accompany rapid solidification, give the following contribution to ΔG (see also refs. [34,35]): the diffusion flux relaxation ∝V², and the viscous stress relaxation ∝(P^ν_xx)² should increase the driving force for interface motion.

The established thermodynamic relation for chemical diffusion in rapid solidification (2.18) introduces a pressure term, ∝(P^ν_xx)², that may have the potential to advance the current understanding of non-equilibrium solidification. More specifically, it may bring about:

• —new insights into undercooled solidification under hydrostatic pressures or in centrifugal casting [36];
• —the analysis of dendrite growth on the inner core of Earth [37];
• —influence of (elastic and plastic) stresses and dislocation motion on crystal growth kinetics [38].

The quantitative effect of the pressure term ∝(P^ν_xx)² on solidification will be clarified in the future works.

Finally, one should note that a thermodynamic analysis was used to describe the liquidus slope in kinetic phase diagram with the appropriate methods described in detail in refs. [12,13,15,17]. Using these methods and neglecting the viscous stresses, P^ν_xx = 0, one can, in particular, find from the driving force, equation (2.18), the velocity-dependent liquidus slope, m_v(V ), as [34,35]

$$m_v=\{\begin{array}{l}\frac{m_e}{1-k_e}\left\{1-k_v+\ln \left(\frac{k_v}{k_e}\right)+{(1-k_v)}^2\frac{V^2}{V_D^2}\right\},V<V_D,\\ \frac{m_e\ln k_e}{k_e-1}\equiv \mathrm{const},V\ge V_D,\end{array}$$ 2.19

where m_e and k_e are the equilibrium liquidus slope and solute partition coefficient (which can be obtained from the phase diagram or calculated using CALPHAD), respectively, and k_v≡C*_S/C*_L is the velocity-dependent partition coefficient. The function (2.19) differs slightly from the one derived for the approximation of dilute mixtures [17]. For finite concentrations, the liquidus slope, equation (2.19), depends on the square of the interface velocity, m_v∝(V/V_D)², explicitly.¹ In the theory of diluted mixtures/alloys, the kinetic liquidus was obtained as linearly proportional to the velocity [17], i.e. m_v∝V/V_D. However, as we show in the following section, the predictions of the crystal growth kinetics are quantitatively valid using equation (2.19) and ref. [17] due to the fact that the strongest dependence of m_v on V is dictated by the nonlinear behaviour of the atomic distribution function at the interface k_v(V ). An estimation of the pressure influence (with P^ν_xx≠0) on the velocity-dependent kinetic liquidus should be specially given. On the one hand, the pressure plays an insignificant role in the crystal growth under usual circumstances (see Chapter 2 in the monograph of Glicksman [39]). On the other hand, the growth of dendrites might be essentially influenced on the inner core of the Earth from the magma's side [37], where the pressure and temperature are important parameters for dynamical equilibrium at the solid/liquid interface. How important the effect of viscous pressure relaxation on the rapidly moving solid/liquid interface is should be clarified in forthcoming works.

3. Dendrite growth model

In the following, we consider two models for rapid dendritic growth. The first model was formulated to describe primary growth developed for dendrites of interstitial and substitutional solutions. The model was tested in numerous cases of binary and ternary alloys solidifying from undercooled melts [19–23]. We formulate this model for two reasons. First, we compare kinetic curves for two cases of liquidus line slopes as formulated in §2 and as derived in [17]. The second model appears as an extended advanced first model to describe kinetic curves in glass-forming alloys exhibiting a maximum in the dendrite growth velocity/melt undercooling relationship.

(a). Undercooling balance and stability criterion

Equations of a sharp interface model were formulated for the analysis of rapid solidification taking into account deviations from equilibrium at the solid/liquid interface and in bulk liquid [19,20,33,40,41]. At the solid/liquid interface, a deviation from local thermodynamic equilibrium may arise due to atom attachment kinetics and solute trapping. A deviation from thermodynamic equilibrium in the liquid may exist in the diffusion field of solutes, which has no time to relax to local chemical equilibrium due to the rapidly growing dendrite. This model combines a selection criterion for stable dendritic growth mode and the balance of different undercooling contributions at the dendritic tip.

The total undercooling, ΔT = T_m + m_eC₀ − T₀, represents the undercooling balance at the dendrite tip as

$$\mathrm{\Delta }T=\mathrm{\Delta }{T}_T+\mathrm{\Delta }{T}_C+\mathrm{\Delta }{T}_N+\mathrm{\Delta }{T}_R+\mathrm{\Delta }{T}_K,$$ 3.1

where T_m is the melting temperature of the solvent, C₀ and T₀ are the initial (i.e. nominal or far-field) composition and temperature of the liquid, respectively. The balance includes the thermal undercooling ΔT_T

$$\mathrm{\Delta }{T}_T=T_Q\mathrm{Iv}(P_T),$$ 3.2

the undercooling ΔT_C due to solute redistribution at the solid–liquid interface

$$\mathrm{\Delta }{T}_C=\{\begin{array}{ll}{k}_v{\mathrm{\Delta }}_v\frac{\mathrm{Iv}(P_C)}{1-(1-k_v)\mathrm{Iv}(P_C)},& V<V_D,\\ 0,& V\ge V_D,\end{array}$$ 3.3

with the Ivantsov functions Iv_T(C)(P_T(C)) defined by

$${\mathrm{Iv}}_{T(C)}(P_{T(C)})=\{\begin{array}{ll}{\mathrm{P}}_{\mathrm{T}(\mathrm{C})}\exp ({\mathrm{P}}_{\mathrm{T}(\mathrm{C})})E_1({\mathrm{P}}_{\mathrm{T}(\mathrm{C})}),& \mathrm{in the }3\mathrm{D space},\\ \sqrt{\pi {\mathrm{P}}_{\mathrm{T}(\mathrm{C})}}\exp ({\mathrm{P}}_{\mathrm{T}(\mathrm{C})})\mathrm{erfc}\sqrt{P_{T(C)}},& \mathrm{in the }2\mathrm{D space},\end{array}$$ 3.4

exponential integral function E₁(P_T(C)) of the first kind and the thermal/chemical Peclét numbers P_T(C),

$$E_1(P_{T(C)})\equiv {\int }_{P_{T(C)}}^{\mathrm{\infty }}\frac{\exp (-u)}{u}\hspace{0.17em}\mathrm{d}u,P_T=\frac{VR}{D_T},P_C=\frac{VR}{D_L},$$ 3.5

where T_Q = ΔH_f/c_p is the adiabatic temperature of solidification (hypercooling), ΔH_f is the enthalpy of melting, c_p is the heat capacity, D_T and D_L are the thermal and chemical diffusion coefficients, respectively, and V is the velocity of the dendrite tip with radius R.

The velocity-dependent non-equilibrium interval Δ_v of solidification in equation (3.3) is given by

$${\mathrm{\Delta }}_v=\{\begin{array}{ll}\frac{m_v{C}_0(1-k_v)}{k_v},& V<V_D,\\ 0,& V\ge V_D.\end{array}$$ 3.6

Note that if the dendrite tip velocity V is equal to or greater than the solute diffusion speed V_D, the constitutional undercooling ΔT_C is equal to zero, corresponding to the transition to the diffusionless regime of solidification.

Finally, the undercooling ΔT_N arising due to the shift of the equilibrium liquidus line from its equilibrium position in the kinetic phase diagram of steady-state solidification is given by

$$\mathrm{\Delta }{T}_N=(m_e-m_v)C_0.$$ 3.7

The curvature undercooling ΔT_R due to the Gibbs–Thomson effect and the kinetic undercooling ΔT_K that change the driving force for interface movement due to the atomic attachment from the liquid to the solid phase are defined as

$$\mathrm{\Delta }{T}_R=\frac{2d_0{T}_Q}{R}\mathrm{and}\mathrm{\Delta }{T}_K=\frac{V}{{\mu }_k},$$ 3.8

where d₀ is the capillary length and μ_k is the interface kinetic coefficient.

The velocity-dependent liquidus slope is described by [17]

$$m_v=\{\begin{array}{ll}\frac{m_e}{1-k_e}\left\{1-k_v+\ln \left(\frac{k_v}{k_e}\right)+{(1-k_v)}^2\frac{V}{V_D}\right\},& V<V_D,\\ \frac{m_e\ln k_e}{k_e-1}\equiv \mathrm{const}.,& V\ge V_D,\end{array}$$ 3.9

and the velocity-dependent partition coefficient k_v≡C*_S/C*_L is given by [33]

$$k_v=\{\begin{array}{ll}\frac{(1-V^2/V_D^2)k_e+V/V_{DI}}{(1-V^2/V_D^2)[1-(1-k_e)C_L^{\ast }]+V/V_{DI}},& V<V_D,\\ 1,& V\ge V_D,\end{array}$$ 3.10

where C*_S and C*_L are solute concentrations of the solid and liquid at the dendrite tip, respectively, with

$$C_L^{\ast }(P_C)=\{\begin{array}{ll}\frac{C_0}{1-(1-k_v){\mathrm{Iv}}_C(P_C)},& V<V_D,\\ C_0,& V\ge V_D,\end{array}$$ 3.11

Here m_e is the slope of the liquidus line in the phase diagram of coexisting phases, and k_e is the equilibrium partitioning coefficient of solute atoms.

Because equation (3.1) is the only equation for two variables, V and R, at the given (and experimentally measurable) undercooling ΔT, a second equation is required to close the problem, and it is given by the stability condition [41]

$$R=\{\begin{array}{ll}\frac{1}{{\sigma }^{\ast }}\cdot \frac{d_0{\mathrm{\Delta }}_0}{T_Q{P}_T{\xi }_T-2{\mathrm{\Delta }}_v{P}_C{\xi }_C},& V<V_D,\\ \frac{1}{{\sigma }^{\ast }}\cdot \frac{d_0{\mathrm{\Delta }}_0}{T_Q{P}_T{\xi }_T},& V\ge V_D,\end{array}$$ 3.12

with Δ₀ the temperature interval of solidification in the phase state defined by

$${\mathrm{\Delta }}_0=\{\begin{array}{ll}\frac{m_e(k_e-1)C_0}{k_e},& \mathrm{for alloys},\\ T_Q,& \mathrm{for pure substance},\hspace{0.17em}{C}_0=0.\end{array}$$ 3.13

Condition (3.12) defines the dendrite tip growing in the stable mode due to the anisotropy ε_c of the interfacial energy and the stability parameter σ*∝ε^7/4_c [42–45]. The thermal stability function for rapid solidification is given by

$${\xi }_T(P_T)={[1+a_1\sqrt{{\alpha }_d}{P}_T(1+\frac{a_0{D}_T{\beta }_0}{d_0})]}^{-2},$$ 3.14

where a₀ and a₁ are the asymptotic coefficients of sewing the large thermal Peclét numbers regime and growth kinetics regime, respectively, β₀ = 1/(μ_kT_Q) is the kinetic parameter of growth and α_d = 15ε_c is the stiffness. The chemical stability function for rapid solidification is described by

$${\xi }_C(P_C,W)=\{\begin{array}{ll}{[1+a_2\sqrt{{\alpha }_d}{P}_C^{\ast }(1+\frac{a_0{D}_L{\beta }_0}{d_{0\mathrm{CD}}})]}^{-2},& V<V_D,\\ 0,& V\ge V_D,\end{array}$$ 3.15

where a₂ is the asymptotic coefficients of sewing the large thermal Peclét numbers regime, d_0CD is the chemical capillary length and $P_C^{\ast }=P_C/\sqrt{1-V^2/V_D^2}$. With the local equilibrium limit, V_D → ∞, the stability growth mode by equations (3.12)–(3.15) describes the regime of Fickean diffusion taken into account by Trivedi & Kurz [46] for rapid dendritic growth [47]. Within the limit of small Peclét numbers, P_T≪1 and P*_C≪1, and with the local equilibrium limits V_DI → ∞ and V_D → ∞, equations (3.12)–(3.15) transform to the previously obtained conditions of Ben Amar and Pelce [43]. The selection criteria (3.12)–(3.15) are written for a fourfold symmetry of the crystal lattices. It can be generalized to other crystalline symmetries [48,49].

(b). Predictions of the model: dendrite velocity and tip radius

The model equations (3.1)–(3.15) describe (i) solute diffusion limited growth of dendrites (i.e. growth of ‘solutal’ dendrites at low undercoolings), (ii) solute diffusion limited and thermally controlled growth of dendrites (i.e. growth of solutal and thermal dendrites in the intermediate range of undercoolings), and (iii) purely thermally controlled dendritic solidification at high undercoolings. These regimes are shown in figure 2 and were previously described for alloy dendrites (see [22,41,50] and references therein).

Figure 2.

Predictions of the model given by two main equations (3.1) and (3.12). Calculations were made using the parameters for Ni–0.7 at.%B taken from ref. [41] for a dendrite velocity V (a) and a tip radius R (b). Different growth regimes are separated from each other by the critical undercoolings, ΔT*₁ and ΔT*₂ such that the solute diffusion limited growth of dendrites proceeds for ΔT < ΔT*₁ (range I), the intermediate stage consisting of solute diffusion limited and thermally controlled growth exists in the range ΔT*₁ < ΔT < ΔT*₂ (range II), and purely thermally controlled dendritic solidification occurs at ΔT > ΔT*₂ (range III). The solid line was obtained for the kinetic liquidus slope given by equation (3.9) and the dashed line was calculated using the kinetic liquidus slope given by equation (2.19). The difference between both curves (dashed and solid) are nearly negligible and they are both consistent with experimental data in a whole range of measurable undercoolings and dendrite velocities within the error bars of experiment, see [41].

The transition from solute diffusion limited growth to the thermally controlled regime of growth in the range ΔT*₁ < ΔT < ΔT*₂ is characterized by the change from slow growth at ΔT < ΔT*₁ to the increase of the interface velocity, figure 2a, and increase of the dendrite tip radius up to its maximal value followed by a decrease at further increasing velocities (figure 2b). Such nonlinear behaviour in V and R exists due to the change in the characteristic spatial lengths defining the dendrite velocity and tip radius, particularly the solute diffusion length and the thermal diffusion length.

At the critical undercooling, ΔT = ΔT*₂, the velocity V = V_D leads to complete solute trapping, C*_L = C*_S = C₀, for the dendrite stems [51]. Beyond this critical point, dendrite growth occurs without solute redistribution and with the initial chemical composition, i.e. in the diffusionless (chemically partitionless) regime. Thus, in the range ΔT≥ΔT*₂, the dendrite growth is thermally controlled only. This result has a clear physical meaning: a source of concentrational disturbances at the solid/liquid interface moving at a velocity equal to or higher than the maximum speed of these disturbances cannot disturb the liquid phase ahead of itself. Therefore, when the interface velocity V passes through the critical point V = V_D, the solidification mechanism changes qualitatively and the undercooling in liquid will not depend on the solute distribution during diffusionless growth. In a microscopic interpretation, the atoms have no time for diffusion jumps if V ≥V_D; they are all stochastically attaching to the interface and instantaneously captured by the solid phase.

Finally note that the transition to diffusionless solidification proceeds sharply at V = V_D and ΔT = ΔT*₂ with a steep change of slope of the ‘velocity versus undercooling’ and ‘tip radius versus undercooling’ curves (figure 2). Such sharp transition in the growth kinetics at some critical value of non-equilibrium governing parameter (undercooling, supersaturation) is related to the ‘kinetic phase transitions’, as defined by Alexander Chernov [52], that was observed in experiments on rapid solidification [7,8,22] and analysed using different models (see ref. [25] and references therein).

4. Dendritic solidification in glass-forming metals and alloys

(a). Model equations

The curves shown in figure 2a exhibit a monotonic increase of the dendrite tip velocity with the undercooling that is typically observed in experiments on solidifying interstitial and substitutional alloys [7,8,13]. However, there is a wide class of alloys exhibiting a maximum of the dendrite velocity that is situated between liquidus temperature and glass temperature (at which an amorphous phase forms and the building of primary crystalline structures is impossible) [26,28]. This class includes glass-forming alloys and, quite frequently, congruently melting alloys. To describe such behaviour, the model of dendritic solidification in equations (3.1)–(3.15) can be reformulated and extended to describe the kinetic curves in glass-forming alloys.

Consider the solidification of a glass-forming alloy which is congruently melting without chemical segregation. In this case, the system of equations (3.1)–(3.15) is reduced to the balance of undercoolings as

$$\mathrm{\Delta }T=\mathrm{\Delta }{T}_T+\mathrm{\Delta }{T}_R+\mathrm{\Delta }{T}_K,$$ 4.1

with the thermal, capillary/curvature and kinetic contributions

$$\mathrm{\Delta }{T}_T=T_Q\mathrm{Iv}(P_T),\mathrm{\Delta }{T}_R=\frac{2d_0{T}_Q}{R},\mathrm{\Delta }{T}_K=\frac{V}{{\mu }_k(\mathrm{\Delta }T)},$$ 4.2

where the kinetic coefficient μ_k(ΔT) is a strongly dependent function of undercooling ΔT. The stable growth mode of the dendrite tip is defined from equations (3.12)–(3.15) as

$$\begin{array}{rl}& R=\frac{1}{{\sigma }^{\ast }}\cdot \frac{d_0{\mathrm{\Delta }}_0}{T_Q{P}_T{\xi }_T},{\sigma }^{\ast }={\sigma }_0{\epsilon }_c^{7/4}\\ \mathrm{and}& {\xi }_T={[1+a_1\sqrt{{\alpha }_d}{P}_T(1+\frac{a_0{D}_T{\beta }_0}{d_0})]}^{-2}.\end{array}\}$$ 4.3

All the coefficients and functions of equations (4.1)–(4.3) are defined in §3.

(b). Kinetic undercooling

The crystal growth models based on the rate kinetic theory give excellent predictions of the molecular dynamics simulation data at small undercoolings (see appendix A, figure 5a). With the increasing undercooling, the crystal growth velocity exhibits clear nonlinearity which may not be predicted by the models based on the rate kinetic theory (see appendix A, figure 5b).

So far, the nonlinearity in the velocity versus undercooling relationships has been obtained similarly to curves with saturation, which are limited by the maximum velocity of the phase field (see [53] as well as the solid curve with saturation in figure 5b). To predict the nonlinear curves with the velocity maximum [54–56] exhibiting one of the distinct characteristics of the crystallization of glass-forming metals and alloys [26,28], we use the travelling wave solution (B 17) of the kinetic phase field model for obtaining the dependence of the kinetic undercooling on the interface velocity.

Using the approximations ΔG≪k_BT and the simplest possible expression for the driving force [57]

$$\mathrm{\Delta }G=\frac{\mathrm{\Delta }{H}_f(-\mathrm{\Delta }{T}_K)}{T_m},\mathrm{\Delta }{T}_K=T_m-T,$$ 4.4

equation (B 17) takes the form (see appendix B)

$$V={\beta }_K^{(\mathrm{PFM})}\mathrm{\Delta }{T}_K,$$ 4.5

which is consistent with the approximations of the kinetic equations (A 4) and (A 5) and for which the kinetic coefficient of growth depends on the kinetic undercooling ΔT_K as

$${\beta }_K^{(\mathrm{PFM})}=\frac{D_{\varphi }(\mathrm{\Delta }{T}_K)\mathrm{\Delta }{H}_f}{\gamma T_m\sqrt{1+{[(D_{\varphi }(\mathrm{\Delta }{T}_K)\mathrm{\Delta }{H}_f/\gamma T_m{V}_{\varphi }(\mathrm{\Delta }{T}_K))\mathrm{\Delta }{T}_K]}^2}}.$$ 4.6

The maximum propagation speed of the phase field depends on the undercooling through the diffusion of the phase field as

$$V_{\varphi }(\mathrm{\Delta }{T}_K)=\sqrt{\frac{D_{\varphi }(\mathrm{\Delta }{T}_K)}{{\tau }_{\varphi }}},$$ 4.7

where τ_ϕ is the relaxation time of the gradient flow, ∂ϕ/∂t, of the phase field (which is taken in the present analysis as a constant, independent of temperature). The diffusion coefficient of the phase field is taken in the form

$$D_{\varphi }(\mathrm{\Delta }{T}_K)=D_{\varphi }^0\exp (-\frac{E_A}{T_m-\mathrm{\Delta }{T}_K-T_A}),$$ 4.8

where the diffusion factor D⁰_ϕ, the temperature T_A and the energetic barrier E_A are the parameters of the phase field propagation. Particularly, T_A controls the temperature at which a drastic change in the crystal growth kinetics may occur.

Figure 3 shows the dependence of the phase field diffusion coefficient, equation (4.8), on the undercooling (ΔT_K > 0) and the superheating (ΔT_K < 0). As the figure shows, the change in the barrier E_A can lead to a qualitative change in the behaviour of the phase field diffusion. The higher value of E_A (see the curve the curve E⁽²⁾_A) gives a gradual change of the phase field diffusion around the equilibrium melting point and a steep decrease of the phase field diffusion at larger undercoolings far from the melting point. By contrast, for the smaller value of E_A the curve E⁽³⁾_A exhibits a linear change of the phase field diffusion with the change of ΔT_K up to its zero value, D_ϕ = 0.

Figure 3.

Phase field diffusion coefficient versus kinetic undercooling. Curve E⁽¹⁾_A corresponds to E_A = 42.11 K, fitted to the experimental data on crystallization of a Cu₅₀Zr₅₀ alloy (figure 4). Curve E⁽²⁾_A corresponds to higher E_A = 126.34 K; E⁽³⁾_A corresponds to E_A = 12.63 K. The coefficients D⁰_ϕ = 3.35 × 10⁻¹⁰ (m² s⁻¹) and T_A = 940 K are the same for all calculations.

One has to finally note that the form of the diffusion coefficient, equation (4.8), is similar to the form of the phase field mobility M_ϕ(T) given by Novokreshchenova & Lebedev [58]. This form is inversely proportional to the kinematic viscosity described by the Vogel–Fulcher–Tamman expression [59] which predicts a strongly anomalous increasing viscosity as the temperature begins to approach the glass temperature. Function (4.8) shows inverse behaviour: as soon as the undercooling approaches its critical value, the phase field diffusion begins its steep decrease down to its zero value exactly at the critical undercooling.

(c). Kinetics of dendritic growth in Cu₅₀Zr₅₀

Cu₅₀Zr₅₀ is a glass-forming alloy which has been investigated in detail by heating, melting, cooling and solidifying droplets processed in an electrostatic levitation facility [27,60]. The dendrite growth velocity was measured for undercoolings of up to ΔT = 311 K. It was found that the velocity first increases and then decreases as the total undercooling increases. The congruently melting Cu₅₀Zr₅₀ alloy solidifies without chemical segregation and, together with the first measurements on Ag [61], it was the metallic system for which the transition from thermodynamically to kinetically controlled growth was firstly experimentally observed [27].

Wang et al. [29] suggested a model of dendritic growth in which the kinetic undercooling was defined via a dendrite growth coefficient that is dependent on the viscosity behaviour. They tried to describe the experiments of ref. [27] in the whole range of undercoolings ΔT(K) ≤ 311. The dash-dotted line in figure 4 shows that the experimental data are well described up to an undercooling of approximately ΔT = 290K. The interval 290 < ΔT (K) ≤ 311 shows a drastic decrease of the velocity with its a further abrupt drop at the critical undercooling ΔT = 311K that is not described by the smooth and gradually changing V (ΔT)-curve of Wang et al. (figure 4). The same abrupt behaviour has also been obtained for the solidification of Cu–Zr-based alloys, particularly Zr₅₀Cu₃₀Ni₂₀ and Zr₅₀Cu₄₅Ni₅ alloys [62]. Therefore, the model presented here, equations (4.1)–(4.8), has been applied to describe the experimental data on Cu–Zr-based alloys exhibiting a maximum of a velocity and an abrupt drop behaviour at the highest critical undercooling.

Figure 4.

Experimental data on the dendrite growth velocity of Cu₅₀Zr₅₀ measured by Q. Wang et al. (open squares, [27]), R. Kobold et al. (open circles, [63]) and P. Paul (solid diamonds, [60]). The theoretical curves display: (1) solid curve, complete model (4.1)–(4.8); (2) dotted curve, model (4.1)–(4.8) with instant relaxation of the gradient flow, τ_ϕ = 0; (3) dash-dotted curve, model of Wang et al. [29]. The insert shows how the theoretical prediction may lie far from experimental data if the gradient flow is not included in the model as an additional thermodynamic variable (see appendix B).

Using the material parameters of the Cu₅₀Zr₅₀ alloy from table 1, the dendrite growth velocity was calculated by the model, equations (4.1)–(4.8). Note, to obtain parameters for the phase field diffusion and relaxation time, we supposed that a part of the velocity versus undercooling curve is described mainly by the kinetic undercooling ΔT_K. Therefore, just to obtain optimal fitting for τ_ϕ, D⁰_ϕ, E_A and T_A we have used the ‘kinetic undercooling shift’, ΔT^Δ_k = 70 (K), which is a shift of the curve V − ΔT from the origin along the ΔT-axis.

Table 1.

Material parameters of the Cu₅₀Zr₅₀ alloy used in the present calculations.

parameter	value	source
Tm (K), melting temperature	1208	[27]
ΔHf (J kg−1), enthalpy of fusion	8.78 × 108	[27]
TQ (K), adiabatic temperature	204.87	[27]
d0 (m), capillary length	6.83 × 10−10	[27]
γ (J m−2), interface energy	0.6	[27]
εc (–), anisotropy of interface energy	0.15	present work
σ0 (–), constant of selection criterion	12.8 × 10−4	present work
τϕ (s), relaxation time	3.41 × 10−7	present work
D0ϕ (m2 s−1), diffusion factor	3.35 × 10−10	present work
EA (K), energetic barrier	42.11	present work
TA (K), pseudo-glass temperature	940	present work

Figure 4 shows the comparison of the present model with the predictions of Wang et al. and experimental data obtained from samples processed by an electromagnetic levitator on the Ground [27,63] and under microgravity generated during parabolic flights [60,62]. It can be seen that the solid line calculated by the full model (4.1)–(4.8) describes the whole set of experimental data including the abrupt drop in the velocity at the critical undercooling ΔT ≈ 311 K. If we exclude the local non-equilibrium effect of the relaxation of the gradient flow, i.e. τ_ϕ → 0 and V_ϕ → ∞ in equations (4.7) and (4.8), the square root will yield unity in equation (4.5) and we get the velocity as plotted by the dotted line in figure 4. More specifically, it is shown in the insert of figure 4 how the predicted velocity lies far from the experimental data if the local non-equilibrium effect in the phase field dynamics is not taken into account.

The necessity to introduce the gradient flow of the phase field as an additional thermodynamic variable (see appendix B) lies in the fact that with the increase of undercooling in glass-forming alloy one can find a transition from single atoms and single clusters to chains of connected clusters (see data of molecular dynamics simulations in [64]). Such chains represent long molecules which lead to a delay in the kinetic processes in the bulk of the liquid phase and at the solid/liquid interface. In the present model, this transition is phenomenologically expressed by the special form of the phase field diffusion coefficient (4.8) which describes the delay in the phase field propagation up to an interface velocity of zero. The critical undercooling at which one may obtain this interface velocity characterizes the impossibility of the phase field propagating due to the specific structure of the undercooled liquid represented by the very long chains of connected clusters. Behind this critical undercooling the remaining liquid may transform into the amorphous phase (see ref. [60] and references therein). Therefore, glass forms at undercoolings larger than the critical undercooling for the zero velocity of the solid/liquid interface. Details and numerical values for these critical undercoolings can be clarified in an atomistic and phenomenological investigation of crystallization and amorphization processes.

Finally, even though we describe experimental data on solidification kinetics of the Cu₅₀Zr₅₀ melt quite well, figure 4, a couple of important issues should be noted as well. First, the primary phase which may solidify from the undercooled Cu₅₀Zr₅₀ melt is the intermetallic phase of B2-type. Solidification of intermetallic compounds may be drastically influenced by the order–disorder transition at the rapidly moving interface [65–69], therefore, this effect leading to pronounced disorder trapping should be introduced into the model of the Cu₅₀Zr₅₀ solidification. Second, intensive investigations of the liquid structure [70,71] and crystallizing phase [72] by in-situ high-energy synchrotron X-ray diffraction on electrostatically levitated samples show that Cu₅₀Zr₅₀ always solidifies as the B2-phase over the whole accessible undercooling range up to 310K. However, periodicity of such measurements was about 5 s [72], which might overcome the duration of decomposition of a primary metastable phase. Indeed, the primary crystalline phase can appear as a new metastable phase at high undercooling with its lifetime smaller than the second phase [62]. Therefore, phase selection and sequence of phase precipitation may also affect the solidification kinetics, which has to be considered as a further advancement of theoretical models for dendrite growth in glass-forming melts.

Appendix A. Appendix A. Models based on the rate kinetic theory

The rate kinetic theory (or thermally activated growth theory [73]) in application to melting/crystallization compares two atomic fluxes at the moving crystal/liquid interface [9]: the first flux is from the liquid to the crystal per unit time at a single kink or atomic micro-roughness, and the second one is from the crystal to the liquid. This results in the non-zero interface velocity

$$V=k_T^{(M)}(T)[1-\exp \left(\frac{-\mathrm{\Delta }G}{k_{B}T}\right)],$$ A 1

where k_B is the Boltzmann constant and T is the absolute temperature. The temperature-dependent kinetic coefficient k^(M)_T in equation (A 1) can be found from various atomistic theories adapted to the concrete growth mechanism of crystals.

In the present work, we refer to theories describing the collision limited mechanism as formulated by Broughton, Gilmer and Jackson [74] and the diffusion limited mechanism as analysed by Wilson & Frenkel [75,76]. These theories summarize the kinetic coefficients as

$$k_T^{(M)}(T)=\{\begin{array}{l}{k}_T^{(\mathrm{CLT})}=\frac{a}{\mathrm{\lambda }}\sqrt{\frac{3k_{B}T}{m}}{f}_0,\\ k_T^{(\mathrm{DLT})}=\frac{D(T)a}{{\mathrm{\lambda }}^2}{f}_0\exp \left(\frac{-\mathrm{\Delta }{H}_f}{k_{B}T}\right),\end{array}$$ A 2

with the superscript ‘M’ corresponding to ‘CLT’ in the collision limited theory or ‘DLT’ in the diffusion limited theory, λ is the displacement during crystallization which is proportional to the lattice parameter a, m is the atomic mass, f₀ the fraction of liquid atom collisions with the solid which result in a crystallization event, ΔH_f the melting enthalpy, D(T) the diffusion coefficient taken as [9] $D(T)=a^2\tilde{\nu }\exp [-\mathrm{\Delta }{E}_B/(k_{B}T)]$, $\tilde{\nu }$ the frequency of thermal vibrations of an atom (assumed as equal in the crystal and the liquid for the sake of simplicity), E_B the activation energy for diffusion, $\exp [-\mathrm{\Delta }{H}_f/(k_{B}T)]$ the probability of finding an atom of the liquid in the immediate vicinity of the kink on the crystal surface in the most advantageous activation complex corresponding to the barrier E_B.

Using the approximation ΔG≪k_BT

$$\mathrm{\Delta }G=\frac{\mathrm{\Delta }{H}_f(-\mathrm{\Delta }{T}_K)}{T_m},\mathrm{\Delta }{T}_K=T_m-T,$$ A 3

which is adopted for a pure substance, equation (A 1) is reduced to its linear form as

$$V={\beta }_K^{(M)}(T)\mathrm{\Delta }{T}_K,$$ A 4

with the temperature-dependent coefficients (A 2) transforming to the kinetic coefficients of crystal growth as

$${\beta }_K^{(M)}=\{\begin{array}{l}{\beta }_K^{(\mathrm{CLT})}=\frac{a}{\mathrm{\lambda }}{f}_0\sqrt{\frac{3k_{B}T}{m}}\frac{\mathrm{\Delta }{H}_f}{k_B{T}_m^2},\\ {\beta }_K^{(\mathrm{DLT})}=\frac{D(T)a}{{\mathrm{\lambda }}^2}\frac{\mathrm{\Delta }{H}_f}{k_B{T}_m^2}{f}_0\exp \left(\frac{-\mathrm{\Delta }{H}_f}{k_{B}T}\right),\end{array}$$ A 5

where T_m is the melting point and ΔT_K is the undercooling.

In equations (A 1)–(A 5), the diffusion limited theory takes into account the diffusion transport coefficient D(T) of the bulk supercooled liquid; therefore, k^(DLT)_T(T) (as well as β^(DLT)_K) exhibits a strong temperature dependence which is generally associated with thermally activated processes. Contrary to this, the collision limited theory predicts a kinetic coefficient proportional to the mean particle velocity, that is, $k_T^{(\mathrm{CLT})}\propto \sqrt{T}$ (and also ${\beta }_K^{(\mathrm{CLT})}\propto \sqrt{T})$ presenting the barrierless mechanism of growth. Quantitative estimations of kinetic growth coefficients for various crystal growth models can be found in [77,78] as obtained from atomistic simulations and in [79] as obtained from experimental measurements. However, the collision limited mechanism fails in many cases of crystal growth (see, e.g. supplementary material of ref. [54]). The diffusion limited theory also has difficulties quantitatively describing the growth kinetics in a wide range of temperatures [55,77]. Figure 5 shows the results of calculations by equations (A 1) and (A 2) for Ni crystals in comparison with data of molecular dynamics simulations [78,80] (all data for the computations are given in ref. [53]). In the relatively small range of undercoolings, 0 < ΔT_K (K) < 50, and superheatings, −55 < ΔT_K (K) < 0, figure 5a shows that MD data exhibit a linear behaviour of the interface velocity, both in melting and in crystallization, that is well described by the diffusion limited theory. For the large range of undercoolings, 0 < ΔT_K (K) < 710, figure 5b shows that the equation following from the diffusion limited theory leads to inconsistent behaviour in comparison with MD data qualitatively and, as a consequence, quantitatively.

Figure 5.

Velocity of crystal growth versus kinetic undercooling at a planar solid/liquid interface [53]. (a) Comparison of the predicted interface velocity V by equations (A 1) and (A 2) (dotted line, curve 2) of the diffusion limited growth (‘M=DLT’) and by equation (B 17) (solid line, curve 1) of the Phase Field Model (PFM) with data of molecular dynamics simulations (full circles) obtained by Mendelev et al. [78] for Ni in the case of low undercoolings 0 < ΔT (K) < 50 and superheatings −55 < ΔT (K) < 0, respectively. (b) Comparison of the predicted interface velocity V by equations (A 1) and (A 2) (dashed line, curve 2) of diffusion limited growth (‘M=DLT’) and by equation (B 17) (solid line, curve 1) of the PFM with data of molecular dynamics simulations (full circles) after Hoyt et al. [80] for Ni in the case of large undercoolings 0 < ΔT (K) < 710. The solid curve is limited by the maximum speed of the phase field propagation (dash-dotted line) which is V = V_ϕ = 185 (m s⁻¹) for growing Ni crystals.

Appendix B. Kinetic phase field model

Traditional phase field models based on the hypothesis of local thermodynamic equilibrium predict linear behaviour for the velocity, V ∝ΔG, describe data of atomistic simulations at relatively small interface velocity and clearly disagree with these data for large values of driving forces [81]. Owing to today's experimentally reachable large driving forces and high growth velocities [8], a phase field model going beyond the hypothesis of local equilibrium has been formulated [82,83] which predicts linear behaviour at small driving forces and a nonlinear behaviour at large driving forces [58,84]. We reformulate and further develop this model to compare its predictions with the data of atomistic simulations exhibiting a maximum in the velocity/undercooling relationship.

In the present derivation, the so-called ‘kinetic energy approach’ [82,85] as defined in [86] is used. This model describes non-equilibrium effects appearing in rapid solidification, in particular, non-equilibrium solute trapping with a transition to diffussionless crystal growth kinetics [87].

Consider a binary mixture consisting of solvent and solute undergoing a phase transition, particularly solidification/melting, from the undercooled/overheated state, for which the free energy is described by

$$\mathcal{G}={\int }_{v_0}[\frac{{\epsilon }_{\varphi }^2}{2}|\mathbf{\nabla }\varphi {|}^2+G(T,C,\varphi ,\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t})]\mathrm{d}{v}_0,$$ B 1

where v₀ is the volume of the mixture, ε²_ϕ is the gradient energy coefficient related to the interface energy γ, ϕ is the phase field variable defined as ϕ = 0 in the liquid phase and ϕ = 1 in the solid phase, ∂ϕ/∂t is the gradient flow for the phase field and C is the solute concentration. Introducing the phase field ϕ and gradient flow ∂ϕ/∂t as independent thermodynamic variables in the Gibbs potential, G(T, C, ϕ, ∂ϕ/∂t) can be considered to be fully analogous to Newtonian mechanics where the initial position and velocity of a particle must be specified to determine their evolution and velocity. Indeed, if inertial effects are sufficiently low in comparison with dissipative effects during phase field propagation, ∂ϕ/∂t will be directly determined by a dynamical equation in terms of ϕ and its gradient. Otherwise, ϕ and ∂ϕ/∂t will be independent and an equation for ∂²ϕ/∂t² must be found [82].

The Gibbs potential G(T, C, ϕ, ∂ϕ/∂t) is given by [88]

$$G(T,C,\varphi )=G_{\mathrm{eq}}(T,C,\varphi )+G_{\mathrm{neq}}(T,\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}),$$ B 2

with the local equilibrium contribution

$$G_{\mathrm{eq}}(T,C,\varphi )=[1-p(\varphi )]G_l(T,C)+p(\varphi )G_s(T,C)+W_{\varphi }(T,C)g(\varphi ),$$ B 3

and the local non-equilibrium contribution

$$G_{\mathrm{neq}}(T,\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t})=\frac{{\alpha }_{\varphi }(T)}{2}{\left(\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}\right)}^2.$$ B 4

The contributions equations (B 3) and (B 4) include the interpolation function p(ϕ) and the double-well function g(ϕ) defined by [89]

$$p(\varphi )=(3-2\varphi ){\varphi }^2\mathrm{and}g(\varphi )={(1-\varphi )}^2{\varphi }^2,$$ B 5

the barrier W_ϕ(T, C) between the phases and the phenomenological coefficients α_ϕ(T) being proportional to the relaxation time τ_ϕ of the gradient flow ∂ϕ/∂t. The local non-equilibrium contribution equation (B 4) can be considered a kinetic energy term added to the Gibbs potential in the traditional phase field theory. Attempts to introduce such a kinetic term was made earlier in the theory of adsorption of sound in liquids [90] or in the theory of kink propagation [91]. Using the formalism of EIT [32], the gradient flow ∂ϕ/∂t is introduced as an independent thermodynamic variable that yields the kinetic energy equation (B 4) naturally [92].

A stable evolution of the entire system is given by the Lyapunov condition of a non-positive change of the total Gibbs free energy with time. For the functional equation (B 1), this condition is given by the non-strict inequality [88]

$$\frac{\mathrm{d}\mathcal{G}}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{v_0}[\frac{{\epsilon }_{\varphi }^2}{2}|\mathbf{\nabla }\varphi {|}^2+G(T,C,\varphi ,\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t})]\mathrm{d}{v}_0\le 0,$$ B 6

from which one finds the following phase field equation [84]

$${\tau }_{\varphi }\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{t}^2}+\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}=D_{\varphi }{\mathrm{\nabla }}^2\varphi -M_{\varphi }[\mathrm{\Delta }G\frac{\mathrm{d}p(\varphi )}{\mathrm{d}\varphi }+W_{\varphi }(T,C)\frac{\mathrm{d}g(\varphi )}{\mathrm{d}\varphi }],$$ B 7

where the Gibbs free energy difference ΔG is given by

$$\mathrm{\Delta }G=G_s(T,C)-G_l(T,C)\{\begin{array}{ll}<0,& \mathrm{solidification},\\ >0,& \mathrm{melting}.\end{array}$$ B 8

The variety of transformations are obtained for different temperature approximations [57], dilute mixtures [93] and thermodynamic functions from data bases [94,95]. The diffusion coefficient of the phase field

$$D_{\varphi }(T)={\epsilon }_{\varphi }^2{M}_{\varphi }(T)$$ B 9

essentially depends on the temperature, if the phase transition is considered in a wide temperature range [58].

The hyperbolic equation (B 7) describes the relaxation of two variables: relaxation of the slow ϕ-field by the first time derivative and relaxation of the gradient flow ∂ϕ/∂t by the second time derivative. In this sense, due to introducing the relaxation of ∂ϕ/∂t, equation (B 7) describes the evolution of the local non-equilibrium system. In a general case, this equation should be solved numerically using specially developed algorithms [88,96,97]. Further, we show the importance of using the gradient flow relaxation and, consequently, the role of local non-equilibrium in rapid crystal growth kinetics using a travelling wave analytical solution.

In equilibrium, ΔG = 0, G_s(T, C) = G_l(T, C), equation (B 7) allows a single dimensionally steady solution at ∂ϕ/∂t = 0 along the spatial x-axis:

$$\varphi (x)=\frac{1}{2}[1-\tanh \left(\frac{x}{{\delta }_I}\right)]\mathrm{within}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{diffuse}\hspace{0.17em}\mathrm{interface}\hspace{0.17em}-\frac{{\delta }_I}{2}<x<+\frac{{\delta }_I}{2},$$ B 10

with the stationary width of diffuse interface

$${\delta }_I=\frac{\sqrt{2}{\epsilon }_{\varphi }}{\sqrt{W_{\varphi }(T,C)}},$$ B 11

and with the boundary conditions ϕ = 1 as x ≤ δ_I and ϕ = 0 as x≥δ_I. Then, the surface energy γ of the crystal/liquid interface is given by

$$\gamma ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[\frac{{\epsilon }_{\varphi }^2}{2}{\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}^2+W_{\varphi }(T,C)g(\varphi )]\mathrm{d}x=\frac{{\epsilon }_{\varphi }\sqrt{W_{\varphi }(T,C)}}{3\sqrt{2}}=\frac{{\delta }_I{W}_{\varphi }(T,C)}{6}.$$ B 12

In the dynamics, ΔG = G_s(T, C) − G_l(T, C)≠0, equation (B 7) has one dimensional travelling wave solution [84,87]

$$\varphi (x,t)=\frac{1}{2}[1-\tanh \left(\frac{x-Vt}{\ell }\right)],$$ B 13

with the boundary conditions ϕ → 1 as x − V t → − ∞ and ϕ → 0 as x − V t → + ∞, with the constant velocity V limited by V_ϕ as a maximum speed of phase field propagation

$$V=\frac{{\mu }_k}{\mathrm{\Delta }{H}_f}\frac{-\mathrm{\Delta }G}{\sqrt{1+{\displaystyle {\left({\mu }_k\mathrm{\Delta }G/\mathrm{\Delta }{H}_f{V}_{\varphi }\right)}^2}}},$$ B 14

the velocity-corrected effective interface thickness

$$\ell =\frac{2{\delta }_I}{3}{[1-\frac{V^2}{V_{\varphi }^2}]}^{1/2},$$ B 15

and the mobility M_ϕ related to the interface kinetic coefficient μ_k as

$${\mu }_k=\frac{18\gamma }{W_{\varphi }(T,C)}{M}_{\varphi }(T)\mathrm{\Delta }{H}_f.$$ B 16

First, the particular solution equation (B 13) with the hyperbolic tangent function follows from the general set of analytical solutions of Allen–Cahn-type equations [98,99] which is given in the present model by equation (B 7). Second, the interface velocity, V , cannot exceed the maximum speed of disturbance propagation in the phase field, because the phase field itself dictates the interface shape and its velocity, i.e. V < V_ϕ in the solutions equations (B 13)–(B 15). Third, with regard to the effective interface thickness (B 15), one has to note two important issues:

• (i)with increasing interface velocity, ℓ should become smaller than the constant interface width δ_I that has been chosen as a reference for the interface thickness in equilibrium state, equation (B 11);
• (ii)within the limit $V\to V_{\varphi }$, one gets ℓ → 0, therefore, the phase field variation will be steeper with the tendency to build up a sharp interface as the velocity increases.

Using equations (B 12) and (B 16), the crystal growth velocity (B 14) can be re-written as

$$V=\frac{-D_{\varphi }(\mathrm{\Delta }{T}_K)\mathrm{\Delta }G(\mathrm{\Delta }{T}_K)}{\gamma \sqrt{1+{\displaystyle {[(D_{\varphi }(\mathrm{\Delta }{T}_K)/\gamma V_{\varphi }(\mathrm{\Delta }{T}_K))\mathrm{\Delta }G(\mathrm{\Delta }{T}_K)]}^2}}},$$ B 17

where the diffusive motion of the phase field is dictated by the temperature-dependent coefficient equation (B 9). At small and moderate driving forces, D_ϕΔG/γ≪V_ϕ, the interface velocity is linearly proportional to the difference of the free energy, i.e. V ∝ΔG. At large driving forces, when D_ϕΔG/γ is of the order of V_ϕ, the square root $\sqrt{1+{[D_{\varphi }\mathrm{\Delta }G/(\gamma V_{\varphi })]}^2}$ in equation (B 17) causes nonlinearity of the velocity. This square root appears in equation (B 17) due to taking into account the local non-equilibrium effect in the form of the relaxation of the gradient flow, ∂ϕ/∂t, which results in the second derivative ∂²ϕ/∂t² in the dynamic equation (B 7).

The quantitative comparison, figure 5a, shows that in the range of small undercoolings, 0 < ΔT_K(K) < 50, and superheating, −55 < ΔT_K(K) < 0, equation (B 17) predicts linear kinetics for growing/melting Ni crystals, V ∝ΔT_K. This is well modelled by the molecular dynamics simulations in [78] and is in confluence with the kinetic behaviour described by the diffusion limited theory, equations (A 1)–(A 5) with ‘M=DLT’.

For the large range of undercoolings, 0 < ΔT_K (K) < 710, figure 5b shows that equation (B 17) predicts a gradual deviation of the velocity V from the linear behaviour as the undercooling increases. Such behaviour in crystal growth kinetics has been found by molecular dynamics simulations of elemental systems [80,100], qualitatively confirmed in [84] and quantitatively confirmed in [53]. These qualitative and quantitative deviations are due to the existence of the square root $\sqrt{1+{[D_{\varphi }\mathrm{\Delta }G/(\gamma V_{\varphi })]}^2}$ in equation (B 17). Because this square root appears as a consequence of the local relaxation to equilibrium, the quantitative comparison shows the importance and usefulness of the local non-equilibrium effect in the dynamics of a phase field at large driving forces.

5. Conclusion

The thermodynamic driving force for rapid solidification that takes into account both the difference in chemical potential and the relaxation of stresses has been derived and analysed. The model for rapid dendritic growth including the transition to diffusionless solidification has been tested and quantified. In particular, we compared the influence of the kinetic liquidus derived for the binary diluted and concentrated mixtures on the growth kinetics of alloy dendrites. On the basis of this model, the description of dendritic solidification in glass-forming alloys of alloy systems with congruent melting phases is possible. The solidification kinetics of this class of alloys is well described by the theory which includes local non-equilibrium effects in the form of relaxation of the gradient flow in the phase field. Good comparison with experimental data confirms the initial theoretical assumption about the predominant influence of local non-equilibrium effects in solidification under large driving forces.

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Authors' contributions

All authors contributed equally to the present review paper.

Competing interests

The authors declare that they have no competing interests.

Funding

This work was supported by the Russian Science Foundation (grant no. 16-11-10095) and the German Space Center Space Management under contract no. 50WM1541.