Application of non-equilibrium dendrite growth model considering thermo-kinetic correlation in twin-roll casting

doi:10.1016/j.jmst.2019.09.042

Abstract

Abstract:

Upon non-equilibrium solidifications, dendrite growth, generally as precursor of as-solidified structures, has severe effects on subsequent phase transformations. Considering synergy of thermodynamics and kinetics controlling interface migration and following conservation of heat flux in solid temperature field, a more flexible modeling for the dendrite growth is herein developed for multi-component alloys, where, two inherent problems, i.e. correlation between thermodynamics and kinetics (i.e. the thermo-kinetic correlation), and theoretical connection between dendrite growth model and practical processing, have been successfully solved. Accordingly, both the thermodynamic driving force ΔG and the effective kinetic energy barrier Qeff have been found to control quantitatively the dendrite growth (i.e. especially the growth velocity, V), as reflected by the thermo-kinetic trade-off. Compared with previous models, it is the thermo-kinetic correlation that guarantees quantitative connection between the practical processing parameters and the current theoretical framework, as well as more reasonable description for kinetic behaviors involved. Applied to the vertical twin-roll casting (VTC), the present model, realizes a good prediction for kissing points, which influences significantly alloy design and processing optimization. This work deduces quantitatively the thermo-kinetic correlation controlling the dendrite growth, and by proposing the parameter-triplets (i.e. ΔG - Qeff - V), further opens a new beginning for connecting solidification theories with industrial applications, such as the VTC.

Key words: Dendrite growth, Multi-component alloys, Thermo-kinetic correlation, Vertical twin-roll casting

Yubing Zhang, Jinglian Du, Kang Wang, Huiyuan Wang, Shu Li, Feng Liu. Application of non-equilibrium dendrite growth model considering thermo-kinetic correlation in twin-roll casting[J]. J. Mater. Sci. Technol., 2020, 44: 209-222.

Figures/Tables 14

Table 1 Evolution of dendritic growth model arising from deviation between the classical theoretical hypothesis and the practical processing path.

Non-equilibrium level	Liquidus & solids slope	Solution approximation	Alloy component	Thermal transportation	Model parameters	Thermodynamics & kinetics
Local-equilibrium	Linear liquidus & solidus	Ideal solution	Binary & dilute	Liquid temperature fields	Alloy component	Independence
Local non-equilibrium	Non-linear liquidus & solidus	Thermodynamic database	Multi-components & concentrated	Liquid & solid temperature fields	Alloy component & practical processing parameters	Correlation

Fig. 1. Modular illustration for the present dendrite growth model, where, Section 2.1 corresponds to module 1 for the kinetics of the planar interface, Sections 2.2 and 2.3 to module 2 for the governing equations, while Section 2.4 to module 3 for the kinetics of the perturbed interface. Accordingly, a specified description for the dendrite growth involved in the VTC can be practicalized, together with the theoretical connection between dendrite growth model and practical processing.

Fig. 2. Schematic diagram for the planar interface (?ΩS/L) upon dendrite growth, which separates the solid (S) and the liquid (L) and moves toward the liquid with the velocity V along the direction normal to the L/S interface. Note that a single closed system Ω is considered because there is no solute fluxes at the interface of the solid (?ΩS) and the liquid (?ΩL), which is different from the inner-interface (?ΩS/L), the bulk solid (S) and the bulk liquid (L) with solute fluxes (Jks,l,il,is), concentration (Xks,l,il,is) and chemical potential (μks,l,il,is).

Fig. 3. Graphic illustration for the typical VTC process solidified along the direction normal to the L/S interface (i.e., Z direction). The geometric conditions described by the three different positions for the kissing point, which corresponds to Case1, 2 and 3, represent the squeezing casting, the stable casting and the breakout, respectively. The height of melt feeding (Hmelt) determines the start angle (θstrat) of solidification, and the necessary rotation angle of the kissing point is reflected by the rotation angle (θroll).

Table 2 Physical parameters for Ni-18 at.%Cu-18 at.%Co alloys used for calculations of dendrite growth model considering the thermo-kinetic correlation.

Definition/Symbol	Value	Refs.
Thermal diffusion coefficients, al and as(m²/s)	1.5 × 10^-5	[26]
Gibbs-Thomson coefficient, Γ(K m)	1.3 × 10^-7	[26]
Diffusion coefficient of Ni in the melt, D11 (m²/s)	2.68 × 10^-9	[32]
Diffusion coefficient of Cu in the melt, D22 (m²/s)	3 × 10^-9	[26]
Diffusion coefficient of Co in the melt, D33 (m²/s)	3.46 × 10^-9	[32]
Mobilities for the rapid interface migration, M0 (m/s)	290	[32]
Upper limit of interface velocity, V₀ (m/s)	1000	Present model (PM)
Diffusion speed at interface, V_DI (m/s)	19	[26]
Diffusion speed in the bulk liquid, V_DL (m/s)	20	[26]

Fig. 4. Evolution of the dendrite tip velocity (V) with the bath undercooling (ΔT) of the undercooled Ni-18 at.%Cu-18 at.%Co alloy during free dendrite growth, where the full line denotes the predicted results by the present model (PM), the discrete points denotes the experimental results [32], and the dot line denotes the values predicted by Wang et al [32]. Insert shows the close-up of an intersection between the two predicted results of PM and Wang at ΔT=62.5K.

Fig. 5. Variation of the thermodynamic driving force ΔG and the effective kinetic energy barrier Qeff with the bath undercooling ΔT of the undercooled Ni-18 at.%Cu-18 at.%Co alloy during free solidification.

Fig. 6. Evolution of the interface behaviors at the dendrite tip with the bath undercooling (ΔT) of the undercooled Ni-18 at.%Cu-18 at.%Co alloy during free solidification predicted by the present model (PM), compared with those calculated by Wang et al. [32]: (a) dendrite tip velocity V and absolute solute stability velocity Vc; (b) dendrite tip temperature Ti and dendrite tip radius R; (c) dendrite tip concentrations Xkil,Xkis, and (d) solute partition coefficient K, evolving with the bath undercooling ΔT.

Table 3 Processing parameters and physical parameters used in the VTC process of Al-2 at.%Mg-1.5 at.%Zn for calculations of dendrite growth model (present model PM) considering the thermo-kinetic correlation.

Definition/Symbol	Value	Refs.
Outer roll radius of copper rolls, R_roll (m)	0.215	[4]
Inner roll radius of copper roll, r_roll (m)	0.200	[4]
Height of molten pool, H_melt(m)	0.040 (0.025-0.075)	[4]
Roll gap, d_gap (m)	0.003 (0.0015-0.0055)	[4]
Density of coolant water, ρ_water (kg/m³)	1 × 10³	[69]
Average flow of coolant water, V_water (m³/s)	2 × 10^-8	PM
Heat capacity of coolant water,C_water (J/(kg K))	4.18 × 10³	[69]
Thermal conductivity in copper, K_M (W/(m K))	324	[70]
Thermal conductivity in α-Al,K_s (W/(m K))	300 (209-420)	[4]
Thermal conductivity in liquid melt,K_l (W/(m K))	78 (75-80)	[4]
Average heat resistance in the interface of roll/solid, Rˉinterface (m² K/W)	2 × 10^-5	[71]
Temperature of cooling-water, Tw (K)	298	[4]
Thermal diffusion coefficients, al and as (m²/s)	1.5 × 10^-5	[31]
Thermal diffusion coefficients, al and as (m²/s)	8.3761 × 10^-5	[31]
Gibbs-Thomson coefficient, Γ (K m)	2.41 × 10^-7	[31]
Diffusion coefficient of Al in the melt, D11 (m²/s)	Variables	[31]
Diffusion coefficient of Mg in the melt, D22 (m²/s)	Variables	[31]
Diffusion coefficient of Zn in the melt, D33 (m²/s)	Variables	[31]
Mobilities for the rapid interface migration, M0 (m/s)	290	[32]
Upper limit of interface velocity, V₀ (m/s)	1000	[31]
Diffusion speed at interface, V_DI (m/s)	1	[31]
Diffusion speed in the bulk liquid, V_DL (m/s)	10	[31]

Fig. 7. Absolute value of the temperature gradient in the solid (|GS |) and liquid (|GL |) versus the coolant water flow (Vwater) during the overall VTC process of the Al-2 at.%Mg-1.5 at.%Zn alloy.

Fig. 8. Evolution of the kissing point Hkiss with the roll speed ωroll during the overall VTC process of the Al-0.7 wt%Mg-1.1 wt%Si alloy, predicted by the present model (PM full line) and the finite element calculations (dot line) reported by Stolbchenko et al [4].

Table 4 Processing parameters and physical parameters used in the VTC process of Al-0.7 wt.%Mg-1.1 wt.%Si for calculations of dendrite growth model (present model PM) considering the thermo-kinetic correlation.

Definition/Symbol	Value	Refs.
Outer roll radius of copper rolls, R_roll (m)	0.215	[4]
Inner roll radius of copper roll, r_roll (m)	0.200	[4]
Height of molten pool, H_melt (m)	0.040 (0.025-0.075)	[4]
Roll gap, d_gap (m)	0.003 (0.0015-0.0055)	[4]
Density of coolant water, ρ_water (kg/m³)	1 × 10³	[69]
Average flow of coolant water, V_water (m³/s)	3 × 10^-8	PM
Heat capacity of coolant water,C_water (J/(kg K))	4.18 × 10³	[69]
Thermal conductivity in copper, K_M(W/(m K))	324	[70]
Thermal conductivity in α-Al,K_S (W/(m K))	300 (209-420)	[4]
Thermal conductivity in liquid melt,Kl (W/(m K))	78 (75-80)	[4]
Average heat resistance in the interface of roll/solid, Rˉinterface (m² K/W)	2 × 10^-5	[71]
Temperature of cooling-water, Tw (K)	298	[4]
Thermal diffusion coefficients, al and as (m²/s)	1.5 × 10^-5	[31]
Thermal diffusion coefficients, al and as (m²/s)	8.3761 × 10^-5	[31]
Gibbs-Thomson coefficient, Γ (K m)	2.4 × 10^-7	[67]
Diffusion coefficient of Al in the melt, D11 (m²/s)	1.79 × 10^-7	[68]
Diffusion coefficient of Mg in the melt, D22 (m²/s)	9.7 × 10^-9	[68]
Diffusion coefficient of Zn in the melt, D33 (m²/s)	2.56 × 10^-8	[68]
Mobilities for the rapid interface migration, M0 (m/s)	290	[32]
Upper limit of interface velocity, V₀ (m/s)	1000	[31]
Diffusion speed at interface, V_DI (m/s)	0.001	[22]
Diffusion speed in the bulk liquid, V_DL (m/s)	10	[31]

Fig. 9. Evolution of the thermodynamic driving force ΔG and the effective kinetic energy barrier Qeff with the dendrite tip velocity V for (a) free solidifications of the undercooled Ni-18 at.%Cu-18 at.%Co alloy, and (b) constrained solidifications of the VTC Al-2 at.%Mg-1.5 at.%Zn alloy.

Fig. 10. Evolution of the parameter-triplets (ΔG - Qeff - V) reflected in the process of VTC for the Al-Mg-Zn alloys: (a) different alloy compositions (XMg0 (at.%): 1, 2, 3 and XZn0 (at.%): 0.5, 1.5, 2.5.) but with the same apparatus parameters; (b) different sets of apparatus parameters with the same composition of Al-2 at.%Mg-1.5 at.%Zn, where the yellow surface is the so-called thermo-kinetic trade-off surface, existing five conditions with different roll radius and height of kissing points, i.e., Rroll (mm): 215, 415, 615, 815, 1000 and Hkiss (mm): 8.7, 19.4, 9.3, 39.4, 30.1.

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