考虑热力学和动力学相关性的二元合金凝固界面动力学建模.pdf

Trans. Nonferrous Met. Soc. China 312021 306−316 Interface kinetics modeling of binary alloy solidification by considering correlation between thermodynamics and kinetics Shu LI1,2, Yu-bing ZHANG1, Kang WANG1, Feng LIU1 1. State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China; 2. School of Science, Harbin University of Science and Technology, Harbin 150080, China Received 27 March 2020; accepted 6 August 2020 Abstract By considering collision-limited growth mode and short-range diffusion-limited growth mode simultaneously, an extended kinetic model for solid−liquid interface with varied kinetic prefactor was developed for binary alloys. Four potential correlations arising from effective kinetics coupling the two growth modes were proposed and studied by application to planar interface migration and dendritic solidification, where the linear correlation between the effective thermodynamic driving force and the effective kinetic energy barrier seems physically realistic. A better agreement between the results of free dendritic growth model and the available experiment data for Ni−0.7at.B alloy was obtained based on correlation between the thermodynamics and kinetics. As compared to previous models assuming constant kinetic prefactor, a common phenomenon occurring at relatively low undercoolings, i.e. the interface migration slowdown, can be ascribed to both the thermodynamic and the kinetic factors. By considering universality of the correlation between the thermodynamics and kinetics, it is concluded that the correlation should be considered to model the interface kinetics in alloy solidification. Key words modelling; interface; dendritic solidification; binary alloy; thermodynamics; kinetics; correlation 1 Introduction Interface kinetics, along with thermal and solutal transport and morphological stability, determines the final behavior of solidification for alloy melts [1]. The classical Fick diffusion equation or the extended hyperbolic diffusion equation was used to describe the thermal and solutal transport in liquid ahead of the solid−liquid S/L interface, due to latent heat releasing and solute redistribution at the interface. Taking into account the nonisothermal and nonisosolutal S/L interface boundary conditions, more accurate solutions of the steady state Fick diffusion equation for the solidification front of a paraboloid of revolution were further obtained [2−7]. The marginal stability theory [8−10], the microscopic solvability theory [11−14], and the phase field theory [15,16] have also been developed to describe the morphological stability of S/L interface very well. Turnbull’s collision-limited growth model [17] is commonly used to describe the interface kinetics for both metals and alloys, to treat the relation between interface migration velocity and thermo- dynamic driving force, as follows VV0[1−expΔG/RTi] 1 where V is the interface migration velocity, the kinetic prefactor V0 is assumed to be a constant with a value of sound speed in melts, ΔG is the change of Gibbs free energy of alloy, R is the mole gas constant and Ti is the interfacial temperature. The collision-limited growth regime assumes that the crystallization rate is controlled by the impingement Corresponding author Feng LIU; Tel 86-29-88460374; E-mail liufeng DOI 10.1016/S1003-63262165497-3 1003-6326/ 2021 The Nonferrous Metals Society of China. Published by Elsevier B.V. the varied kinetic prefactor thus reflects the transition of mobility and energy barrier between the two extreme modes. Following the current theoretical framework, the correlations between the kinetic prefactor and the thermodynamic driving force were proposed and studied by application to planar interface migration and dendritic solidification for binary alloys. A linear correlation between the effective thermo- dynamic driving force and the effective energy barrier seems physically realistic, by comparison with the available experiment data of Ni−B alloys. On this basis, universality of the correlation was discussed. It is finally concluded that the correlation should be taken into account to model the interface kinetics in alloy solidification. 2 Model derivation 2.1 Interface kinetics 2.1.1 Effective energy barrier and effective thermodynamic driving force For alloy melts under sufficiently high undercooling conditions, V is so fast that the solute partitioning is suppressed and the complete solute trapping occurs. Then, the interface kinetics upon solidification is similar to that for pure metals, which can be described by Eq. 1. Under this condition, the solidification process is mainly controlled by the thermal transport and the mechanism can be regarded as thermal-controlled, so Eq. 1, with an effective energy barrier QeffQT, can be rewritten as follows VV0exp−QT/RTi[1−expΔG/RTi] 3 where QT is the activation energy for thermal diffusion, which can be approximately considered as negligible. For alloy melts under sufficiently low undercooling conditions, the solidification behavior is mainly determined by the interdiffusion between solute and solvent atoms. The short-range diffusion-limited growth regime can be regarded as 万方数据 Shu LI, et al/Trans. Nonferrous Met. Soc. China 312021 306−316 308 solute-controlled, which, as a thermally activated process, reduces the interface mobility and also the kinetic prefactor. Thus, the kinetic prefactor VDI, i.e. the solutal diffusion velocity at the interface, can be defined by [30] VDIV0exp−QD/RTi 4 It then follows that Eq. 2 with an effective energy barrier QeffQD, can be rewritten as follows VV0exp−QD/RTi[1−expΔG/RTi] 5 where QD is the activation energy for solutal diffusion, which, in contrast with QT0, cannot be ignored here. The value of V0 is normally three orders of magnitude greater than that of VDI. At intermediate undercoolings, there must be a transition between the solute-controlled and the thermal-controlled modes. This implies that the interface kinetics should not be solely determined by Eq. 3 or Eq. 5, but be correlated with both. In order to describe the interface kinetics using one equation suitable for the entire undercooling conditions, an effective energy barrier Qeff is thus introduced, which is defined by QeffηηQD1−ηQT 6 where the key parameter η0,1 represents a typically kinetic state of solidification, reflecting different contributions from thermal- and solutal- controlled mechanisms. Then, a unified equation for the interface kinetics is given by VV0exp−Qeff/RTi[1−expΔG/RTi] 7 Define an effective kinetic prefactor eff 0 V as eff 00effi exp[/]VVQRT 8 Then Eq. 7 can be rewritten as follows eff 0i [1 exp/]V VGRTη∆ 9 where eff 0 Vη can be written as eff 00DI0 /VV VV η 10 or eff 00Di [exp/]VVQRT η  11 At sufficiently high undercoolings, the parameter η equals zero and eff 0 V equals V0, so Eq. 7 reduces to Eq. 3 or Eq. 1, representing the collision-limited growth regime. Then, a continuously increased η with decreasing undercooling indicates a transition of solidification mechanism from the thermal-controlled growth to the solute-controlled growth. At sufficiently low undercoolings, η tends to be 1 and eff 0 V reduces to VDI, so Eq. 7 reduces to Eq. 5 or Eq. 2, representing the short-range diffusion-limited growth regime. Note that, in order to study the effect of solute drag ΔG in Eq. 9 is also replaced by ΔGeff, as done in previous models [23−27] ΔGeffΔG−βΔGD 12 where ΔGD is the solute drag free energy and β is the solute drag factor. 2.1.2 Correlations between thermodynamics and kinetics Suppose that the complete solute trapping corresponds to the critical state marked by * eff G∆, i.e. the solute partition coefficient k1, η0 and VVD, in contrast with the state of negligible ΔGeff, i.e. kke, η→1 and V→0. With increasing ΔGeff from zero to * eff G∆, the solidification mechanism is changed from the solute-controlled growth to the thermal-controlled growth and the parameter η varies continuously from 1 to 0. In order to derive the correlation between Qeff and ΔGeff , a functional relation between η and ΔGeff must be specified according to Eq. 6, and Mode 1 is proposed by assuming a linear relationship between η and ΔGeff in the range of [0, * eff G∆], i.e. at VVD, * Mode 1effeff 1/GGη ∆∆ 13 where * eff G∆ corresponds to the critical under- cooling ΔT* with VVD; for V≥VD, η holds constant as zero, indicating the collision-limited growth regime. Based on Mode 1, a linear correlation between Qeff and ΔGeff can be given as QeffQD− * effeff /GG∆∆QD−QT. Similarly, Mode 2 assumes an exponential relation of η with ΔGeff, at ΔGeff * eff G∆, * effeffi Mode 2 * effi exp[/] exp[/] 1 exp[/] i GRTGRT GRT η ∆∆ ∆ 14 where, for ΔGeff≥ * eff G∆, η≡0 always holds; under the condition that eff 0 /V Vtends to zero, ΔGeff tends to zero and Mode 2 reduces to Mode 1. Considering possible relations between η and other thermodynamic parameters, another two potential modes are still available. Assuming a linear relation between η and k, i.e. for kke, η1; for k1, η0 and for kek1, the following relation holds 万方数据 Shu LI, et al/Trans. Nonferrous Met. Soc. China 312021 306−316 309 Mode 3 e 1 1 k k η   15 With the assumption of a linear relation between η and the difference ** LS CC, at V0, **eqeq LSLS CCCCand η1; at VVD, ** LS 0CC and η0; and η can be described as follows, under the condition that ** LS CC takes other values, *** LSL Mode 4 eqeqeq LSLe 1 1 CCCk CCCk η   16 where * L C , * S C , eq L C and eq S C stand for the solute concentrations at the S/L interface; and the superscript “eq” represents the equilibrium values. Incorporating Eqs. 13−16 into Eqs. 6 and 9, different correlations between thermodynamics and kinetics can be obtained. 2.2 Solidification with planar interface In steady state solidification with planar interface for binary alloy melts, * S C is a constant and equals the equilibrium value eq S C as well as nominal composition of alloys C0 [17]. Under this condition, Mode 4 defined by Eq. 16 reduces to the following expression e Mode 4 e 1 1 kk kk η   17 For non-dilute alloys, ΔGeff can be calculated numerically and thermodynamically by using subregular solution model based on CALPHAD [23,33]. For dilute alloys, Henry’s law and Baker and Cahn’s approximation for the chemical potentials of solute and solvent lead to an analytical expression as follows [24] eqeq* effiSLL {[11 GRT CCCkkk β∆  2 eD ln 1]} kV k kV  18 where β is the solute drag factor, which is introduced to unify the two treatments with β1 and without β0 solute drag. For linear liquidus and solidus within the composition range of interest, Eq. 18 can be rewritten as *e effieMLLiL 1 /GRTkTC m VTm∆ 19 where TM is the melting temperature of solvent, e L m is the slope of equilibrium liquidus and mLV is regarded as kinetic liquidus slope defined by e 2 L L eeD [11 ln 1] 1 mkV m Vkkk βk kkV    20 Equations 18 and 20 can be further simplified at V≥VD, due to the complete solute trapping with k1. For small velocities relative to eff 0 Vη, Eq. 9 is further approximated by eff 0effi /VVGRTη∆ 21 Substituting Eq. 19 into Eq. 21, the interface temperature Ti can be described as e * L iMLL eff e0 1 mV TTC m V k Vη  22 For dilute binary alloys, considering the relaxation effect of local nonequilibrium solute diffusion in bulk liquid, the solute partition coefficient k is given by SOBOLEV as [34] 22 DeDI 22 DDI 1// 1// VVkV V k VVV V   , VVD 23a k1, V≥VD 23b Combining Eqs. 20, 22, 23a and 23b with Modes 1−4, one can describe the final behavior of solidification with planar interface. 2.3 Free dendritic growth model For further modeling the free dendritic growth, another physical quantity i.e. curvature radius r should be introduced to denote the tip morphology of dendrite curvature radius. Based on the marginal stability theory, the curvature radius r is described as [9] * * fLL ttcc 22 D / 2 1 1/ p r Hm V Ck PP cVV  σ ξξ ∆  , VVD 24a * ttf / / p r PHc  σ ξ ∆ , V≥VD 24b where Γ is the Gibbs−Thompson coefficient; σ* is the stability constant σ*≈1/4π2; Pt[rV/2α] is the thermal Pclet number; Pc[rV/2D] is the solute Pclet number; α and D are respectively the thermal diffusivity and solute diffusion coefficient in the liquid; ΔHf is the latent heat of fusion; cp is the heat capacity of liquid alloy; and the parameters ξt and ξc are defined by 万方数据 Shu LI, et al/Trans. Nonferrous Met. Soc. China 312021 306−316 310 t *2 t 1 1 1 1/P ξ σ  25 c 22*2 Dc 2 1 211 1// k kVVP ξ σ    , VVD 26a ξc0, V≥VD 26b From the current interface kinetics, Eqs. 21, 24a and 24b, the interface response function assuming linear liquidus and solidus can be modified as e *L iMLL eff e0 2 1 mV TTC m V krV  η      27 From this equation, the total bath undercooling ΔT is described as e* L0LL 2 [ ]Tm Cm V C r  ∆  e Lf t eff e0 Iv 1 p mHV P kcVη ∆    28 where Iv is the Ivantsov function. In the right- hand side of Eq. 28, the four terms represent the constitutional undercooling, the curvature undercooling, the kinetic undercooling and the thermal undercooling i.e. obtained from the solution of thermal transport equation in the liquid region, in order. ΔT is defined by ΔTTM e L m C0− T∞, where T∞ is the temperature in liquid far from the interface. Another result by solving the solute diffusion equation in the liquid region gives the description of liquid solute concentration at the interface * L C [21] *0 L c 1 1Iv C C kP  29 Assuming linear liquidus and solidus, the solute trapping model is also described by Eqs. 23a and 23b. Up to now, the extended kinetic model considering the correlation between thermo- dynamics and kinetics i.e. the effective kinetic prefactor eff 0 Vη has been determined. If the correlations between thermodynamics and kinetics were ignored, i.e. eff 00 VVη≡ holds with all values of η in Eqs. 22, 27 and 28, the present model would reduce to previous ones [21], in wh