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

Journal Pre-proof Application of non-equilibrium dendrite growth model considering thermo-kinetic correlation in twin-roll casting Yubing Zhang, Jinglian Du, Kang Wang, Huiyuan Wang, Shu Li, Feng Liu

PII:

S1005-0302(20)30043-8

DOI:

https://doi.org/10.1016/j.jmst.2019.09.042

Reference:

JMST 1910

To appear in:

Journal of Materials Science & Technology

Received Date:

10 August 2019

Revised Date:

14 September 2019

Accepted Date:

15 September 2019

Please cite this article as: Zhang Y, Du J, Wang K, Wang H, Li S, Liu F, Application of non-equilibrium dendrite growth model considering thermo-kinetic correlation in twin-roll casting, Journal of Materials Science and amp; Technology (2020), doi: https://doi.org/10.1016/j.jmst.2019.09.042

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Research Article Application of non-equilibrium dendrite growth model considering thermo-kinetic correlation in twin-roll casting Yubing Zhang 1, Jinglian Du 1, Kang Wang 1, Huiyuan Wang 2, Shu Li 3, Feng Liu 1,4,* State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China Key Laboratory of Automobile Materials of Ministry of Education and Department of Materials Science and Engineering, Jilin University, Changchun 130025, China 3 School of Science, Harbin University of Science and Technology, Harbin 150080, China 4 Analytical & Testing Center, Northwestern Polytechnical University, Xi’an 710072, China 1

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E-mail address: [email protected] (Feng Liu).

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*Corresponding author. Tel.: +86 29 88460374; Fax: +86 29 88491484.

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[Received 10 August 2019; Received in revised form 14 September 2019; Accepted 15 September 2019]

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Abstract Upon non-equilibrium solidifications, dendrite growth, generally as precursor of

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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

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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 ∆𝐺 and the effective kinetic energy barrier 𝑄eff have been found to control

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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 1

parameter-triplets (i.e. ∆𝐺-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

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1. Introduction 1.1. From twin-roll casting to dendrite growth

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In the vertical twin-roll casting (VTC) process, liquid metal is directly poured through a tundish

between two counter-rotating rolls, which, usually made of conducting metals and cooled internally by

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coolant water, provide the cooling condition essential for solidification and contribute to the strip with

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reduced thickness within short time duration (see Fig. S1 in Supplementary material) [1,2]. This revolutionary technology, arising from near-net-shape casting and sub-rapid solidification, is

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considered to most likely change traditional manufacturing for metallic material and solve problems of plurality of steps, high energy consumption and high emission. As an integration of continuous casting and rolling, solid shells of a given alloy is produced on each of the rotating rolls which are squeezed

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together upon kissing point (Fig. S1), so that a wide range of knowledges such as solidification, solid-state phase transformation (PT), plastic deformation (PD) are involved in improving the VTC techniques to grasp inter-relations among practical processing parameters, microstructure and performance. As shown in Fig. S1, the kissing point, theoretically corresponding to the contact of

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oppositely growing dendrites, determines the initiating point of deformation and ultimately influences the subsequent microstructural evolution, as the integrated reflection of casting and rolling parameters [1,3]. In order to optimize the kissing point, a lot of research work has been carried out [3-6], however, most of which are concerned about technical details without referring to an inherent question, dendrite growth. From the perspective of solidification, as far as the authors believe, it is thus one of the most important targets to develop a dendritic growth model, which can be used to quantitatively determine

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the kissing point, or to optimize the processing parameters for a fixed kissing point. 1.2. Theoretical development for interface stability and dendrite growth Since the crystal microstructure, named ‘dendrite’ is observed in 1897, extensive research have been performed, particularly including two mile-stones correlated with planar interface stability, e.g. the theory of constitutional supercooling proposed in 1953 [7] and the linear stability analysis proposed in 1964, i.e. M-S (Mullins and Sekerka) criterion [8]. In 1985, Trivedi and Kurz extended

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the M-S criterion into cases of high thermal Pelect numbers, and proposed a more general criterion for planar interface stability [9]. The development of interface stability criterion gave rise to an open

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question, i.e. how to relate the dendrite growth to the interface evolution? In debt to distinguished work by Langer and Mülller-Krumbhaar (LMK) [10], the dendrite tip radius is related to the

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marginally stable wavelength, which is predicted by the linear stability analysis of planar interface: 𝜆 = 𝑅dendritic

(1)

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Validity of Eq. (1) was confirmed by experimental studies on dendrite growth [11], which further produced Marginal Stability Criterion (MSC) to express analytically the dendrite tip radius. Thereafter,

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a complete model for dendrite growth (MSI, Marginal Stability Criterion & Ivantsov function) integrating MSC and simultaneous solution of Ivantsov function [12,13] predicted reasonably relations

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among 𝑉 (velocity), 𝑅 (radius), Ti (temperature) and 𝑋𝑘il and 𝑋𝑘is (liquid and solid concentration of component k at liquid/solid (L/S) interface) of dendrite tip. Actually, the MSC is a good approximation while microscopic solvability is fundamentally correct[14], by which, quantitative information for the dendrite growth, free of adjustable parameters, can be predicted [15]. Therefore,

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the MSC and the MSI models are still commonly applicable nowadays. Arising from interface kinetics and MSC, the evolution of MSI models is always accompanied by

the development of temperature/concentration fields involved in the dendrite growth (Table 1). Regarding sufficiently fast thermal diffusion, the concentration fields controlled by solutal diffusion have become much more focused upon solidification. In the classical models of interface kinetics and morphological stability, local equilibrium diffusion (LED) is assumed at the interface, so that the interfacial concentrations are consistent with the equilibrium phase diagram [16-18]. With increasing 3

bath undercooling (∆𝑇), the L/S interface velocity 𝑉 will reach the same order of magnitude as the interface diffusive speed (𝑉DI = 𝐷𝑘I ⁄𝑎0, with 𝐷𝑘I as the diffusion coefficient of component k and 𝑎0 as atomic jump distance at the interface) introduced by the chemical rate theory [19], thus resulting in local non-equilibrium diffusion effect (LNDE) at the interface but a partial differential equation of parabolic-type for bulk liquid assuming LED, 𝜕𝐽𝑘l 𝜕𝑡

= 𝐷𝑘L ∇2𝑋𝑘l

(2)

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On this basis, the model proposed by Boettinger, Coriell and Trivedi (BCT) [16] had become the most widely accepted one due to its effectiveness and mathematical simplicity. Unfortunately, BCT model

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is limited by assumptions of binary dilute alloy and LED in bulk liquid. By introducing relaxation

1⁄2

,

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effect [20-23], the interface velocity will catch up the bulk-liquid diffusive speed (𝑉DL = (𝐷𝑘L⁄𝜏D ) where, 𝜏D is the relaxation time of atoms (molecules, particles) to their equilibrium state in local

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volume of alloy, and 𝐷𝑘L the diffusion coefficient of component k in bulk liquid), thus resulting in LNDE at both the interface and the bulk liquid, and meanwhile, a partial differential equation of

𝜕𝐽𝑘l 𝜕𝑡

+ 𝜏𝐷

𝜕 2 𝐽𝑘l 𝜕𝑡 2

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hyperbolic-type for bulk liquid diffusion, = 𝐷𝑘L∇2 𝑋l𝑘

(3)

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Accordingly, a linear stability analysis [24] and a modified dendrite growth model were proposed by Sobolev and Galenko et al [25,26]. Compared with BCT model, Sobolev and Galenko’s model overcame the limitation of LED, but the assumption of dilute binary alloy was still prevalent. Applying maximal entropy production principle (MEPP) [27,28] and extended irreversible

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thermodynamics (EIT) [29], Wang et al. [30] considered LNDE at the interface and the bulk phases for concentrated multi-component alloys, in combination with non-linear L/S phase boundary, and derived a self-consistent model for sharp interface kinetics as: 𝑉 = −𝑀V ∆𝐺c

𝑉

(4) 2

∆𝐺c = ∑𝑛𝑘=1 ( 2m 𝛼𝑘is (𝐽𝑘is ) −

𝑉m 2

2

𝛼𝑘il (𝐽𝑘il ) ) − ∑𝑛𝑘=1 𝑉𝛼𝑘is 𝐽𝑘is (𝑋𝑘il − 𝑋𝑘is ) + ∑𝑛𝑘=1 𝑋𝑘il ∆𝜇𝑘

𝐽𝑘il = 𝑀𝑘D [(∆𝜇𝑘 + 𝑉𝛼𝑘il 𝐽𝑘il − 𝑉𝛼𝑘is 𝐽𝑘is ) −

is is D il il ∑𝑛 𝑖=1 𝑀𝑖 (∆𝜇𝑖 +𝑉𝛼𝑖 𝐽𝑖 −𝑉𝛼𝑖 𝐽𝑖 ) D ∑𝑛 𝑗=1 𝑀𝑗

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]

(5) (6)

𝑉

𝐷𝑘I

where 𝑀V = 𝑅 0𝑇 , 𝑀𝑘D = 𝑎 g i

0 𝑉m

(

𝜕𝜇𝑘il

𝜕𝑋𝑘il

−1

) , 𝛼𝑘il =

𝑉m

2 (𝑉𝑘L )

(

𝜕𝜇𝑘il

𝜕𝑋𝑘il

) and 𝛼𝑘is =

𝑉m

2 (𝑉𝑘S )

(

𝜕𝜇𝑘is

𝜕𝑋𝑘is

); for details see

Section 2. On this basis, the morphological stability analysis for planar interface and MSI models were also extended for concentrated multi-component alloys, and the highlighted description involved in MSC [31] and dendrite growth [32] can be modified as, ⋯ −𝑀𝑛l ⋯ 𝑁2𝑛 ⋱ ⋮ ⋯ 𝑁𝑛𝑛 ] ⋯ −𝑀𝑛l ⋯ 𝑁2𝑛 ⋱ ⋮ ⋯ 𝑁𝑛𝑛 ]

(7)

of

Sn (𝜔) = −Γ𝜔2 − [ 𝐾L 𝐺Lin 𝜉L + 𝐾𝑆 𝐺Sin 𝜉S ] +

0 −𝑀2l 𝑑𝑒𝑡 −𝜍2 𝑁22 ⋮ ⋮ [ −𝜍𝑛 𝑁𝑛2 1 −𝑀2l 𝑑𝑒𝑡 1 𝑁22 ⋮ ⋮ [ 1 𝑁𝑛2

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where 𝐾L (𝐺Lin ) and 𝐾S (𝐺Sin ) are the thermal conductivities (i.e. temperature gradients, TGs) at the L/S interface. As far as the authors summarize, the developed dendrite growth model has been always

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accompanied by the evolved equations governing the concentration fields by relaxing deviations between classical theoretical hypothesis and practical processing path (see also Table 1), but without

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modifying the governing equations of temperature fields. 1.3. Inherent problems for dendrite growth theories

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The dendrite growth model from Wang et al. [32] can be considered as one of the most improved MSI theories so far, which, however, if applied to the VTC, is inevitably subjected to sub-rapid

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solidification (i.e. mixed effect of thermal and solutal diffusion; Section 2.1) and positive TG (i.e. heat transportation mainly evolving in the solid; Section 2.3), thus resulting in two inherent flaws as follows:

All the first-order PTs can be considered as typical nucleation-growth processes, which are

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accompanied with, assisted, and even initiated by thermodynamics and kinetics, i.e. the kinetic processes controlled by the thermodynamics. The correlation between thermodynamics and kinetics can be summarized as: the increased thermodynamic driving force ∆𝐺 is always accompanied with the decreased effective kinetic energy barrier, i.e. activation energy 𝑄eff , and vice versa [33-37]. As for the dendrite growth, the kinetic evolution with increasing the solidification velocity (i.e. 𝑉), generally experiences a series of assumptions, such as equilibrium, local equilibrium, local non-equilibrium and global non-equilibrium for the solutal diffusion in bulk liquid, while always 5

assuming LED for the thermal transportation. This evolution by continuously increased ∆𝐺 corresponds to continuously decreased 𝑄eff for diffusion (i.e. contributed by both thermal and solutal transport) [32,38]. In order to benefit from the thermo-kinetic correlation, however, we need to establish a theoretical connection between the dendrite growth model and the practical processing. Comparing the prevalent negative TG and the rapid solidification assumed in most dendrite growth models, a question

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arises as, how to deal with the dendrite growth in sub-rapid solidification controlled by the prevalent positive TG, e.g. in the VTC. Definitively, a smooth transition from TGs evolving in the liquid (𝐺L ) to

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the solid (𝐺S ) is required. Assuming the negative TG, actually, it is the thermal transportation in the

liquid that mainly controls the whole dendrite growth, whereas, in the VTC assuming the positive TG,

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the thermal transportation in the solid, related to the practical processing, must be incorporated into the interface stability analysis, to realize the theoretical connection by calculations of thermal

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diffusion in solid and liquid [9,39]. The thermo-kinetic correlation, in combination with the theoretical connection in between, will guarantee the man-made adjustment by relations among thermodynamics,

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kinetics and practical (i.e. including apparatus and processing) parameters; see Sections 4.1 and 4.2. A new dendrite growth model is proposed and applied in the VTC, where, the thermodynamic

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extremal principle (TEP) and the thermo-kinetic correlation are adopted, in combination with the positive TG evolving in bulk solid and LNDE assumed for both interface and bulk liquid, to realize theoretical connection between the dendrite growth and the practical processing (Section 2). Then, validity of the present dendrite growth model is tested by application into undercooled Ni-18

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at.%Cu-18 at.%Co alloys (Section 3), where the thermo-kinetic correlation is demonstrated and analyzed. Application of the current theoretical framework is performed for solidifications of Al-2 at.%Mg-1.5 at.%Zn and Al-0.7 wt%Mg-1.1 wt%Si alloys involved in the VTC, where optimization of parameters, e.g. the kissing point, is discussed following the thermo-kinetic correlation (Section 4). Finally, conclusions are summarized in section 5.

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2. Model derivation As mentioned in Section 1, the evolution of dendrite growth model is always accompanied by relaxing the deviations between classical assumptions and practical processing, as reflected by several theoretical comparisons (Table 1). Accordingly, the current modeling can be performed as follows: (1) thermo-kinetic correlation upon non-equilibrium solidification, (2) smooth transition from temperature mainly changing in the liquid to the solid, (3) connection with practical processing parameters, and (4)

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linear stability analysis basing on the thermo-kinetic correlation; see Fig. 1. In the following sections,

these four steps. 2.1. Thermo-kinetic correlation upon dendrite growth

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universality, typicality and steerability of the current modeling can be thoroughly demonstrated by

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2.1.1. Inherently thermo-kinetic correlation in interface migration equation

kinetics is generally expressed as,

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Following the classical transitional theory, such as Wilson-Frankel equation [40], the interface

(8)

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𝑉(𝑇i (𝑡)) = 𝑉0 exp(−𝑄/𝑅g 𝑇i (𝑡))(1 − exp(∆𝐺/(𝑅g 𝑇i(𝑡)))

where the kinetic pre-factor 𝑉0 is assumed to be a constant value as the upper limit of migration velocity, 𝑅g the gas constant and 𝑇i (𝑡) the interface temperature with respect to time. Analogously,

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as for the interface kinetics upon solidification, Turnbull’s collision-limited growth model [41] is commonly used to treat the relationship between 𝑉 and ∆𝐺, as follows, 𝑉 = 𝑉0 (1 − exp(∆𝐺/(𝑅g 𝑇i )))

(9)

where 𝑉0 is assumed to be a constant with a value of sound speed in melts and 𝑇i the temperature at

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the L/S interface. Strictly speaking, Eq. (9) is only suitable for solidification subjected to sufficiently high ∆𝑇, where solute trapping occurs and solute redistribution disappears at the L/S interface. As for solidification subjected to low ∆𝑇, however, solute partitioning is required for alloy solidification, which can only be accomplished by inter-diffusions between solute and solvent atoms at the interface. This thermally activated process, as Aziz and Boettinger proposed [42], is generally controlled by short-range diffusion-limited growth, which is described as, 7

𝑉 = 𝑉DI (1 − exp(∆𝐺/(𝑅g 𝑇i )))

(10)

where 𝑉DI is the diffusive speed at the interface. Actually, the solidification is simultaneously controlled by thermal (with relative small Q) and solutal (with relative large Q) transport processes, so that a unified equation including an effective kinetic energy barrier, 𝑄eff , is given as, 𝑉 = 𝑉0 exp(− 𝑄eff ⁄(𝑅g 𝑇i )) (1 − exp (∆𝐺⁄(𝑅g 𝑇i )))

(11)

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By assuming physically variable value for 𝑄eff , Eq. (11) becomes different from Eqs. (9) and (10). For sufficiently high ∆𝑇 Eq. (11) reduces to Eq. (9), representing the collision-limited growth regime;

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and for sufficiently low ∆𝑇 Eq. (11) reduces to Eq. (10), representing the short-range

diffusion-limited growth regime. At intermediate ∆𝑇, there must be a transition between the

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solute-controlled and the thermal-controlled models. This implies that the interface kinetics should not be solely determined by Eq. (9) or (10), but be correlated with both, i.e. from Eq. (11), the

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thermo-kinetic trade-off, as revealed experimentally [37], does exist. Arising from this thermo-kinetic

transformation, step by step.

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correlation, it is timely to deal with the L/S interface kinetics, the dendrite growth and the overall

2.1.2. Interface kinetics reflecting thermo-kinetic correlation

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Most of the industrial alloys contain more than two components, so that modeling solidification in multi-component system is becoming more and more inevitable. As for the formulation of multi-component system, the TEP has been proven as a handy tool [43-48], whose application addresses several keypoints about interface kinetics as follows.

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System description and assumption

Upon dendrite growth, we focus on an arbitrary system of planar solidification with solid bulk (S),

liquid bulk (L) as well as interfaces of dendrite tip (∂𝛺L/S ), solid (∂𝛺S) and liquid (∂𝛺L ), which is regarded as a single closed system Ω; see Fig. 2. Accordingly, migration velocity of the planar interface 𝑉 is determined following the direction normal to the L/S interface, which, due to the planar interface assumption, is omitted in the following derivations. Both bulk solid and liquid contain

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n-component concentrations and solute fluxes, however, no solute flux in the solid is assumed. Thus 𝑋𝑘s , 𝐽𝑘s = 0, 𝑋𝑘is , 𝐽𝑘is (𝑘 = 1, 2, 3 ⋯ 𝑛) respectively denote the solid concentration, no solute fluxes in the bulk solid, the interface concentration in the solid, and the solute fluxes at the dendrite tip interface in the solid, and analogously, 𝑋𝑘l , 𝐽𝑘l , 𝑋𝑘il , 𝐽𝑘il (𝑘 = 1, 2, 3 ⋯ 𝑛) , the liquid concentration, the solute fluxes in the bulk liquid, the interface concentration in the liquid and the solute fluxes at the dendrite tip interface in the liquid. Commonly, summation of the concentrations equivalent to unity:

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∑𝑛𝑘=1 𝑋𝑘s,l,il,is = 1 is assumed. Since the atoms are propagated by the exchanging mechanism; while the sources, the sinks and the diffusion flux of vacancies are neglected, so that the diffusion fluxes are

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constrained by ∑𝑛𝑘=1 𝐽𝑘s,l,is,il = 0(𝑘 = 1, 2, 3 ⋯ 𝑛). Gibbs free energy and dissipation of system

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As for a single closed system including bulk and interface, the total Gibbs energy, 𝐺, consists of,

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solid bulk (𝑔S ), liquid bulk (𝑔L) and L/S interface (𝑔𝜕𝛺S/L ) with independent variables as solid concentration, liquid concentration and dissipation flux. Considering mass conservation law 𝑉 Vm

(𝑋𝑘il − 𝑋𝑘is )), the change rate of total Gibbs energy at the L/S interface can be

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(𝐽𝑘is = 𝐽𝑘il −

according to EIT written as [20, 29]:

Vm 2

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𝐺∂Ω̇ L/S = 𝑔𝜕𝛺̇ S/L = ∫ ∑𝑛𝑘=1 𝐽𝑘il (𝜇𝑘il − 𝜇𝑘is )𝑑𝐴 +

𝑉

Vm

Vm

∫ ∑𝑛𝑘=1 [

2

2

𝛼𝑘is (𝐽𝑘is ) − 𝑋𝑘il (𝜇𝑘il − 𝜇𝑘is ) −

2

𝛼𝑘il (𝐽𝑘il ) ] 𝑑𝐴

where the coefficients follow [20]: 𝛼𝑘is =

(12) Vm

2 (𝑉𝑘S )

(

𝜕𝜇𝑘is

𝜕𝑋𝑘is

) and 𝛼𝑘il =

Vm

2 (𝑉𝑘L )

(

𝜕𝜇𝑘il

𝜕𝑋𝑘il

), with 𝜇𝑘is and 𝜇𝑘il as the

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chemical potential of solid and liquid at the L/S interface, respectively. If assume linear diffusive processes occurring in the system, the total Gibbs energy dissipation at the L/S interface will follow a definite positive quadratic form of independent fluxes [43-49], where, the dissipations by sharp interface migration (𝐽V =

𝑉 𝑉m

) and trans-diffusion at the interface (𝐽𝑘in = 𝐽𝑘il ) hold in consistent with

Ref. [50]. Therefore, the total Gibbs energy dissipation at the L/S interface can be rewritten as [30]:

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Q ∂ΩL/S

= ∫ ∑𝑛𝑘=1

where 𝑀𝐽il = 𝑘

2

(𝐽𝑘in ) 𝑀

𝐷𝑘I

𝐽in 𝑘

d𝑉 + ∫

𝜕𝜇𝑘il

(

(𝐽V )2 𝑀V

−1

d𝑉 ≈ ∫ ∑𝑛𝑘=1

and 𝑀V = il )

a0 𝑉m 𝜕𝑥𝑘

𝑀0 Rg 𝑇i

2

(𝐽𝑘il ) 𝑀

𝐽il 𝑘

d𝐴 + ∫

(𝑉)2 𝑉m 𝑀V

d𝐴

(13)

. The first term in the right-hand side of Eq. (13) can be

perceived as the trans-diffusion and the second term as the interface migration. Meanwhile, the mass conservation law at the interface holds as, 𝐽𝑘il = 𝐽𝑘is +

𝑉(𝑋𝑘il −𝑋𝑘is )

(14)

𝑉m

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which is regarded as an additional constraint in the current system that should be considered during

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the following TEP treatment. Evolution of planer L/S interface

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Following the maximum energy dissipation rate along the evolution path of interface ∂ΩL/S , the evolution equations for characteristic parameters (i.e. the diffusion fluxes at the interface 𝐽𝑘il , 𝐽𝑘is and

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the interface velocity 𝑉) are determined to derive a new response function for dendrite interface under il

is

𝑉(𝑋𝑘 −𝑋𝑘 ) sub-rapid solidification, with the requirements as 𝐺∂Ω̇ L/S + 𝑄∂ΩL/S = 0, 𝐽𝑘il − 𝐽𝑘is = and 𝑉 m

𝑄∂ΩL/S 2

+ ∫ ∑𝑛𝑘=1 𝜆𝑘 [𝐽𝑘il − 𝐽𝑘is −

𝑉(𝑋𝑘il −𝑋𝑘is )

] d𝐴 + 𝛽 ∫ ∑𝑛𝑘=1 𝐽𝑘il d𝐴} = 0

𝑉m

(15)

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𝛿 {𝐺∂Ω̇ L/S +

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∑𝑛𝑘=1 𝐽𝑘il = 0. Accordingly, the variation at the interface follows [30],

with 𝜆𝑘 and 𝛽 as the Lagrange multiplier. Actually speaking, independent treatment for the dissipation paths is assumed from the linear relations between the flux and the driving force, so that thermodynamics and kinetics for the interface phenomenon must be considered separately. This is not compatible with the physically realistic process where the interface trans-diffusion and the interface

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migration are closely intertwined, subjected to essential interactions in between. On this basis, it is timely to introduce fundamentally the thermo-kinetic correlation, to quantitatively describe the interactions arising from different dissipation paths. Combining the physical essence of Eq. (14) (i.e. considering interactions among dissipation paths at the interface such as 𝐽𝑘il , 𝐽𝑘is and 𝐽V ) with Eq. (13), the total energy dissipation 𝑄∂ΩL/S at the interface can be rewritten as,

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𝑄∂ΩL/S =

∫ ∑𝑛𝑘=1

[𝐽𝑘is +

2 𝑉 (𝑋 il −𝑋𝑘is )] 𝑉m 𝑘

𝑀

(𝑉)2

d𝐴 + ∫ 𝑉

m 𝑀V

𝐽il 𝑘

d𝐴

(16)

where the first term at the right-hand side of Eq. (16) introduces the dissipation flux by the interface migration to that by the trans-diffusion. Following Lagrange Multiplier Method [51,52], substituting Eqs. (12) and (16) into Eq. (15) and performing the variation over 𝑉 at the interface give, 𝜕𝑉

=

1

Vm

𝑉m

∫ ∑𝑛𝑘=1

∫ ∑𝑛𝑘=1 [

2

2

𝛼𝑘is (𝐽𝑘is ) − 𝑋𝑘il (𝜇𝑘il − 𝜇𝑘is ) −

𝐽is 𝑘 (𝑋 il −𝑋 is )+ 𝑉 (𝑋 il −𝑋 is )2 𝑘 𝑘 𝑘 Vm 𝑘 V2 m

𝑀

d𝐴 + ∫ 𝑉

𝑉

m 𝑀V

𝐽il 𝑘

𝑉m 2

2

𝛼𝑘il (𝐽𝑘il ) ] d𝐴 + 1

d𝐴 + ∫ ∑𝑛𝑘=1 𝜆𝑘 [− 𝑉 (𝑋𝑘il − 𝑋𝑘is )] d𝐴 = 0

（17-a）

of

𝜕𝐹

m

ro

where the second integral at the right-hand side of Eq. (17-a) is introduced to consider interactions arising from different dissipation paths. Further for Eq. (15), taking the variation of 𝐽𝑘il and

= ∫ ∑𝑛𝑘=1(𝜇𝑘il − 𝜇𝑘is )d𝐴 + ∫ ∑𝑛𝑘=1[−𝑉𝛼𝑘il 𝐽𝑘il ]d𝐴 + ∫ ∑𝑛𝑘=1

and, 𝜕𝐹

𝐽il 𝑘

d𝐴 + ∫ ∑𝑛𝑘=1 𝜆𝑘 d𝐴 + ∫ ∑𝑛𝑘=1 𝛽d𝐴 （17-b）

= ∫ ∑𝑛𝑘=1[𝑉𝛼𝑘is 𝐽𝑘is ]d𝐴 + ∫ ∑𝑛𝑘=1 −𝜆𝑘 d𝐴

ur na

𝜕𝐽𝑘is

𝐽𝑘il

𝑀

re

𝜕𝐽𝑘il

lP

𝜕𝐹

-p

𝐽𝑘is , respectively, gives,

（17-c）

Integrating Eqs. (17-a)-(17-c) yields a universal equation for the interface kinetics, which, analogous to Wang et al [53,32], can be expressed as: （18）

Jo

𝑉 = −𝑀eff ∆𝐺c

𝐽𝑘il

=

𝑀𝐽il {[(𝜇𝑘is 𝑘

−

𝜇𝑘il )

+

𝑉𝛼𝑘il 𝐽𝑘il

−

𝑉𝛼𝑘is 𝐽𝑘is ] +

∑𝑛 𝑖=1 𝑀

[(𝜇𝑖il −𝜇𝑖is )+𝑉𝛼𝑖is 𝐽𝑖is −𝑉𝛼𝑖il 𝐽𝑖il ] 𝐽il 𝑖 ∑𝑛 𝑗=1 𝑀𝐽il 𝑗

}

（19）

but with,

2

𝑀eff =

1 (𝑋 il −𝑋 is ) [∑𝑛𝑘=1 𝑉 𝑘𝑀 𝑘 m il 𝐽𝑘

1

−1

+𝑀 ] V

2

=

a (𝑋 il −𝑋 is ) [∑𝑛𝑘=1 0 𝑘𝐷I 𝑘 𝑘

and, 11

(

𝜕𝜇𝑘il

𝜕𝑋𝑘il

)+

𝑅g 𝑇i 𝑀0

−1

]

（20）

𝑉m

∆𝐺c = ∑𝑛𝑘=1 [

2

2

𝛼𝑘is (𝐽𝑘is ) − 𝑋𝑘il (𝜇𝑘il − 𝜇𝑘is ) −

𝑉m 2

𝐽𝑘is (𝑋𝑘il −𝑋𝑘is )

2

𝛼𝑘il (𝐽𝑘il ) − 𝑉𝛼𝑘is 𝐽𝑘is (𝑋𝑘il − 𝑋𝑘is )] + ∑𝑛𝑘=1

𝑀

𝐽il 𝑘

（21） 𝑉m

If assuming 𝐽𝑘is = 0 and 𝛼𝑘il =

2 (𝑉𝑗L )

(

𝜕𝜇𝑘il

𝜕𝑋𝑘il

)=

𝑉m

2 (𝑉𝑗L )

𝜇𝑘il −𝜇𝑘is

𝑋𝑘il −𝑋𝑘is

𝑉

, Eq. (14) then reduces to 𝐽𝑘il = 𝑉 (𝑋𝑘il − m

𝑋𝑘is ), and in turn, Eq. (21) reduces to Eq. (5) in consistence with Wang et al [32]: 2

2(𝑉𝑗L )

(𝜇𝑘il − 𝜇𝑘is )(𝑋𝑘il − 𝑋𝑘is )]

（22）

of

𝑉2

∆𝐺c = ∑𝑛𝑘=1 [−𝑋𝑘il (𝜇𝑘il − 𝜇𝑘is ) −

𝑉m

(𝑋𝑘il − 𝑋𝑘is ) = 𝑀𝐽il [(𝜇𝑘is − 𝜇𝑘il )𝜓𝑘 −

∑𝑛 𝑖=1 𝑀

(𝜇𝑖il −𝜇𝑖is )𝜓𝑖 𝐽il 𝑖 ∑𝑛 𝑗=1 𝑀𝐽il 𝑗

𝑘

]

-p

𝑉

ro

and similarly, Eq. (19), the interface trans-diffusion reduces to [32]:

（23）

2

where 𝑀𝐽il is the mobilities for the trans-interface diffusion and 𝜓𝑘 (= 1 − 𝑉 2⁄(𝑉𝑗L ) ) the 𝑘

re

non-equilibrium coefficient, which shows the difference between the interface velocity and the

lP

reference value (i.e. 𝑉𝑗L as the diffusion speed in the liquid), reflecting the non-equilibrium degree. On this basis, taking the first order Taylor expansion for Eq. (11) leads to: ∆𝐺c

𝑅g 𝑇i

ur na

𝑉 = −𝑉0 exp(− 𝑄eff ⁄(𝑅g 𝑇i ))

（24）

Following the thermo-kinetic correlation, 𝑄eff is related with ∆𝐺c and deduced by integrating Eqs. (18) and (24) as:

𝑄eff = 𝑅g 𝑇i ln [∑𝑛𝑘=1

2

𝑎0 𝑉0 (𝑋𝑘il −𝑋𝑘is ) 𝑅g 𝑇i 𝐷𝑘I

(

𝜕𝜇𝑘il

𝜕𝑋𝑘il

𝑉

（25）

) + 𝑀0 ] 0

Jo

where, as a value of sound speed in melts, 𝑉0 should be a constant with the same magnitude (1000 m/s [54]) as assumed in Eq. (9). So far, a new model for interface kinetics considering the thermo-kinetic correlation has been

obtained by combination of Eqs. (18) and (19), which, in association with morphological stability analysis for the planar interface, leads to a more flexible and universal dendrite growth model, applicable for non-equilibrium solidification of multi-component concentrated alloy systems; see 12

Section 3. Analogous to previous models assuming Eq. (4), both solute-controlled and thermal-controlled mechanisms prevails upon solidifications, but different from Eq. (4), the evolved 𝑄eff with ∆𝐺 (Eq. (18) or Eq. (11)) dominates arising from the proposed thermo-kinetic correlation, in contrast with a fixed 𝑄eff following Turnbull’s collision-limited growth model and Aziz’s short-range diffusion-limited growth model; see Section 2.1.1. This superiority of the present dendrite growth model becomes strengthened during sub-rapid solidifications such as the VTC, as reflected

of

drastically by the thermo-kinetic correlation in medium and low ∆𝑇 ranges. As V becomes close to 𝑉DL , the efficient mobility 𝑀eff in Eq. (18) reduces to 𝑀V in Eq. (4), the present model reduces to

ro

previous ones by solute trapping. Provided if typical processing condition is given (Fig. 1 module 2), the present dendrite growth can then be located, particularly to industrial application; see Section 4.

-p

2.2. Smooth transition from TG evolving in the liquid to the solid

As compared to previous models focusing on solidification of undercooled melts while ignoring

re

solid temperature field, the present modeling, in combination with the practical processing of VTC, must be confronted with an urgent problem, i.e. the transportation path for thermal diffusion. In

lP

previous modeling, the practical processing cannot be effectively connected with the theoretical framework, due mainly to the thermal transportation along the undercooled liquid. Currently, the

ur na

sub-rapid solidification involved in the VTC is a specific process whose thermal transportation is performed along the solid, which is controlled by various processing parameters; see Fig. 3. Following the M-S criterion [55], the interface instability is accelerated by the negative TG in the undercooled liquid, which is generally not compatible with applying the current model to the current VTC. Under

Jo

this circumstance, |𝐺L | < |𝐺S | holds and then, the TG in the liquid will not be the major controlling factor, i.e. the TG with the larger absolute value dominates the sub-rapid solidification, independent of solid or liquid and positive or negative (Sections 2.3 and 4). That is to say, the interface stability or the dendrite growth is not decided by the positive or negative TG, but by the thermal transportation mainly through the liquid or the solid (Section 2.4). Accordingly, smooth transition of TG evolving in the liquid to the solid can be realized by the current modeling through a theoretical connection with the practical processing [56]; see module 2 in Fig. 1 and supplementary Fig. S2. 13

2.3. Theoretical connection with practical processing conditions To determine the kissing point, the relation between the kissing point and the dendrite growth must be obtained in advance. Then it will be meaningful to build a theoretical connection between the dendrite growth model and the practical processing conditions which guarantees quantitative connections between the kissing point and the dendrite growth; see module 2 in Fig. 1. 2.3.1. Geometric model of kissing point

of

As shown in Fig. 3, the VTC is a complicated process determined by numerous processing parameters, such as melt feeding height (𝐻melt), roll speed (𝜔roll ), outer roll radius (𝑅roll), inner roll

ro

radius( 𝑟roll), roll gap (𝑑gap ), amount of coolant (𝑉water), etc. Regarding the narrow solidification

region from the nozzle tip to the kissing point (or roll nip), totally three cases can happen [1], i.e. the

-p

case 1 corresponding to too high separation force [57], the case 3 corresponding to breakout, and the case 2 corresponding to stable casting; see Fig. 3. In order to quantitatively control the kissing point,

re

therefore, a geometric model must be innovatively designed to reveal the relation between the dendrite growth velocity and the geometrical parameters. Upon solidification in the VTC, the dendrite grows

lP

along the roll normal (z direction at the interface) since the coolant maximizes the normal temperature gradient. On this basis, if the start point 𝜃strat is given, then the angle of rotation 𝜃roll analytically

ur na

corresponds to the location of kissing point. The whole process can be mathematically express as: ∆𝐿geo =

𝑅roll +𝑑gap ⁄2 𝜋 2

cos( −𝜃start −𝜃roll )

（26）

− 𝑅roll

（27）

∆𝐿act = 𝑉dendrite × 𝑡time

Jo

where 𝑅roll is the outer radius of the roll and 𝑑gap the overall geometry of the roll gap. Regarding the value of 𝜃roll realized within a time duration of 𝑡time, the roll speed can be expressed as 𝜔roll (= 𝜃roll /𝑡time). Following both the solidification and the geometric conditions, ∆𝐿geo= ∆𝐿act, if 𝑉dendrite can be extracted by the present dendrite growth model, the theoretical connection with the practical processing parameters will be realized.

14

2.3.2. Connection with practical processing parameters In order to accurately extract the 𝑉dendrite, the thermal transportation in the solid (see supplementary Fig. S3) must be performed, where, the L/S interface moves toward the liquid with a constant velocity 𝑉 and the temperature governing equations, respectively for solid and liquid phase, in the moving one-dimensional coordinate system, are given as [9]: 𝜕2𝑇

（28-a）

𝑎s 𝜕𝑍2s = 0 𝜕2𝑇

𝜕𝑇

（28-b）

of

−𝑉 𝜕𝑍l = 𝑎l 𝜕𝑍2l

ro

where 𝑇l , 𝑇s , 𝑎l and 𝑎s are the temperature and the thermal diffusivities in liquid and solid,

respectively. For a steady solidification, the temperature fields only change along the Z direction, so

-p

that typically linear solid temperature field and non-linear liquid temperature field can be according to Eqs. (28-a) and (28-b) obtained as:

𝐺Lin 𝑎l 𝑉

[1 − exp(−

𝑉𝑍 𝑎l

)]

lP

𝑇l = 𝑇i +

re

𝑇s = 𝑇i + 𝐺sin 𝑍

（29-a） （29-b）

where 𝐺Sin and 𝐺Lin are the TGs at planar interface. Unfortunately, no information about practical

ur na

processing parameters is reflected in Eqs. (29-a)-(29-b), so that the man-made adjustment will not be permitted, unless a concrete form of temperature field is provided. This can be realized by considering both solutal and thermal conservation at the L/S interface, which is controlled by various processing parameters in the VTC [56]; see supplementary Fig. S3. Accordingly, the heat flux conservation in the

Jo

L/S interface must be satisfied: i.e. for liquid and solid: 𝑞 = 𝐾s 𝐺sin = 𝐾l 𝐺Lin = 𝐾R 𝐺tR = 𝑞interface = 𝜌water 𝐶water 𝑉water ∆𝑇water

（30）

and further,

𝑇 −𝑇 R−s

i s 𝑞 = 𝐾s 𝑉×𝑡

time

R 𝑇 R−s −𝑇w

= 𝐾l 𝐺Lin = 𝐾R 𝑅 R

roll −𝑟roll

=

𝑇sR−s −𝑇RR−s 𝑅̅interface

= 1⁄𝜌

R −𝑇 𝑇w w

water 𝐶water 𝑉water

（31）

with 𝑇sR−s , 𝑇RR−s , 𝑇wR , and 𝑇w respectively as the temperature for the roll/solid interface on the solid 15

side, the roll/solid interface on the roll side, the roll/coolant interface, and the coolant; 𝐾s , 𝐾l and 𝐾R the thermal conductivity of solid, liquid and roll; 𝜌water , 𝐶water and 𝑉water the density, the heat capacity and the average flow of coolant water. Omitting the heat resistance of L/S and roll/coolant interfaces but considering the heat resistance of roll/solid, 𝑅̅interface, the larger absolute value of solid TG dominates the overall process, so that a new liquid TG, controlled by essential parameters above mentioned, can be given as: 𝑇i −𝑇w 𝑉dendrite ×𝑡time 𝑅 −𝑟 1 )𝐾l +𝑅̅interface + roll roll+ 𝐾s 𝐾R 𝜌water 𝐶water𝑉water

（32）

(

ro

and then new functions governing the temperature transport are rewritten as:

𝑇i −𝑇w ]𝑎l 𝑉 ×𝑡time 𝑅roll −𝑟roll 1 ̅ ( dendrite +𝑅 + +𝜌 )𝐾 interface 𝐾s 𝐾R water 𝐶water 𝑉water l

𝑉

𝑉𝑍

[1 − exp(− 𝑎 )] l

（33-a）

（33-b）

re

𝑇l = 𝑇i +

-p

𝑇s = 𝑇i +𝐺sin 𝑍

[

of

𝐺Lin =

So far, a concrete form of TG evolving in the VTC has been built up, which guarantees the connection

lP

between dendritic growth models with practical processing parameters. In view of most industrial alloy demanded, Hunziker et al. focused on concentration gradient (CG)

ur na

in the M-S theory and proposed a new model considering concentrated multi-component alloys and LNED both at the interface and in the bulk liquid, diffusional interaction with concentration field [58], and particularly, realizing the connection between the dendrite growth model and the practical processing. On this basis, a similar concentration governing equation without lateral diffusion (diffusion in X perpendicular to the Z direction; see Fig. 3.) also holds for the current modeling [32], 𝑉𝑍

Jo

𝑋𝑘l = 𝑋𝑘il + ∑𝑛𝑖=1 𝐿𝑘𝑖 [1 − exp ( 𝐵 )]

（34）

𝑖

where 𝐵𝑖 𝐿𝑘𝑖 = ∑𝑛𝑗=1 𝐷𝑘𝑗 𝜓𝑗 𝐿𝑗𝑖 with 𝑋𝑘l and 𝑋𝑘il respectively as the concentration of component k at the liquid side of the L/S interface and in the bulk liquid, and 𝐷𝑘𝑗 the interaction diffusion coefficient between the component k and j. From now on, the governing equations coupled with many practical processing parameters such as, 16

alloy concentrations, melt feeding height (𝐻melt), roll speed (𝜔roll ), outer roll radius (𝑅roll), inner roll radius( 𝑟roll), roll gap (𝑑gap ) and amount of coolant (𝑉water), should be integrated with the planar interface kinetics (section 2.2; see module 1 in Fig. 1.) to perform the perturbed interface stability analysis (section 2.4; see module 3 in Fig. 1). 2.4. Linear stability analysis basing on thermo-kinetic correlation Following the normal procedure for dealing with the stability of planar interface by involving an

the amplitude, 𝜔 (=

2𝜋 𝜆

of

infinitesimal perturbance 𝜙 = 𝛿sin𝜔𝑋 in X direction (perpendicular to the Z direction), where 𝛿 is ) the wave number, and 𝜆 the wavelength, the planar interface subjected to

ro

the perturbance should be described as [9,32,55]: il𝜙

-p

𝑋𝑘 = 𝑋𝑘il + 𝑏𝑘 𝜙 𝜙

𝑇i = 𝑇i + 𝑎𝜙

（35-b） （35-c）

re

𝑉 𝜙 = 𝑉 + 𝜙̇

（35-a）

lP

where 𝑏𝑘 and 𝑎 are constant coefficients, which can be adopted to determine the linear correction due to infinitesimal perturbance 𝜙. Then, combing Eqs. (28), (33), (34) and (35) leads to the

ur na

perturbance field in bulk: [

𝜙

𝑇l = 𝑇i +

𝑇i −𝑇w ]𝑎l 𝑉 ×𝑡time 𝑅roll−𝑟roll 1 ̅ ( dendrite +𝑅 + +𝜌 )𝐾 interface 𝐾s 𝐾R water 𝐶water 𝑉water l

𝑉

Jo

𝑉𝑍

il 𝑋𝑘l = 𝑋𝑘il + ∑𝑛𝑖=1 𝐿𝑘𝑖 [1 − exp ( 𝐵 )] + 𝜙(𝑏𝑘 − 𝐺𝑘c )exp(−𝜔𝑘c 𝑍) 𝑖

2

𝑉

𝑎l

)] + (𝑎 −

（37）

𝑇s = 𝑇i + 𝐺Sin (𝑍 + 𝜙)

𝑉

𝑉𝑍

（36）

𝐺Lin )e−𝜔l𝑍 𝜙 𝜙

[1 − exp(−

𝑉

2

𝑉

（38）

𝜔2

where 𝜔l = 2𝑎 + √(2𝑎 ) + 𝜔 2 , 𝜔𝑘c = 2𝐵 + √(2𝐵 ) + 𝜓 , and 𝐵𝑖 is the eigenvalue of the l

l

𝑘

𝑘

𝑘

il effective diffusion matrix (𝐷𝑘𝑗 𝜓𝑗 ) and 𝐿𝑘𝑖 the coefficient satisfying the relationship, 𝐺𝑘c ≡

17

𝜕𝑋𝑘l

𝑉

𝜕𝑍 𝑍→0

= ∑𝑛𝑖=1 𝐿𝑘𝑖 𝐵 [32]. As for the perturbed interface, the evolution kinetics must consider the 𝑖

capillarity condition (∆𝑇R = 𝛤𝜔2 𝜙) [59]: 𝜙

il𝜙

𝑇i = 𝑇m + ∑2𝑘=1 𝑀𝑘l 𝑋𝑘 −

where

𝑀𝑘l

𝜕∆𝐺c 𝜕𝑋il 𝑘 𝜕∆𝐺c 𝑉 [ ] + 𝜕𝑇 𝑇i 𝑀eff

−

=

,

𝑉𝜙 𝜇∗

1

=

𝜇∗

（39-a）

− 𝛤𝜔2 𝜙 𝑅g 𝑇i 𝜕∆𝐺c + V0 𝑀eff 𝜕𝑉 𝜕∆𝐺c 𝑉 + 𝜕𝑇 𝑇i 𝑀eff

, and the conservation relationship of perturbed L/S

𝜙

𝜕𝑇s

𝜕𝑍 𝑍→𝜙

− 𝐾l

𝜙

𝜕𝑇l

𝜕𝑍 𝑍→𝜙

（39-b）

)

ro

1

𝑉 𝜙 = Δ𝐻 (𝐾s

of

interface follows [9,32,55]:

∑𝑛𝑗=1 𝐷𝑘𝑗 𝜓𝑗𝜙 𝐺𝑗cil𝜙 = V 𝜙 𝑋𝑘il𝜙 (𝑘𝑘𝜙 − 1)

（39-c）

-p

Neglecting the solid diffusion, all the governing equations have been obtained until now. On this basis,

and Eq. (39-c), respectively lead to [32]: 1 𝜙̇

𝜙̇ 𝜙

1

lP

𝑎 = ∑2𝑘=1 𝑀𝑘l 𝑏𝑘 − 𝜇∗ 𝜙 − 𝛤𝜔2

re

the substitutions of Eq. (35-b), Eqs. (36) and (37), and Eqs. (35-a)-(35-c) into Eq. (39-a), Eq. (39-b)

𝑉

= Δ𝐻 𝐾l 𝜔l [𝐺Lin (𝑎 𝜔 − 1) + 𝑎] l

l

𝜙̇

ur na

−𝜁𝑘 − 𝜔2 Γ = 𝑎 + 𝜐𝑘 + ∑𝑛𝑘=2 𝑁𝑘𝑗 𝑏𝑗

（40） （41） （42）

𝜙

where, some essential parameters can be expressed as [32]: 𝑉 𝐵𝑖 𝜕𝑘 𝑉𝑋𝑘il 𝑘 𝜕𝑇i

2

il c 𝐷𝑘𝑗 𝜓𝑗 [∑𝑛 𝑖=1[𝐿𝑗𝑖 ( ) ]−𝐺𝑗c 𝜔𝑗 ]

Jo

𝜁𝑘 = ∑𝑛𝑗=1 {

𝑋𝑘il (𝑘𝑘−1)+𝑉𝑋𝑘il

𝜐𝑘 =

[

（42-a）

}

𝜕𝑘𝑘 il 2𝑉 +∑𝑛 𝑗=1(𝐷𝑘𝑗 𝐺𝑗c L 2 ) 𝜕𝑉 (𝑉𝑗 ) 𝜕𝑘𝑘 il 𝑉𝑋𝑘 𝜕𝑇i

（42-b） ]

and,

18

∑𝑛𝑘=2 𝑁𝑘𝑗 𝑏𝑗

=

c il c 𝑉𝑏𝑘 (𝑘𝑘 −1)+∑𝑛 𝑗=2(𝐷𝑘𝑗 𝜓𝑗 𝜔𝑗 −𝐷𝑘1 𝜓1 𝜔1 +𝑉𝑋𝑘

𝑉𝑋𝑘il

𝜕𝑘𝑘 )𝑏𝑗 𝜕𝑋il 𝑗

（42-c）

𝜕𝑘𝑘 𝜕𝑇i

Integrating Eqs. (40)-(42) uniquely determines the evolution of

𝜙̇ 𝜙

with the evolved solidification

condition, which can be simplified as:

𝑇i −𝑇w (

𝑉dendrite ×𝑡time 𝐾s

𝑅roll −𝑟roll

+𝑅̅interface +

𝐾R

+

1

𝜌water 𝐶water 𝑉water

)𝐾l

] (1 −

𝑉 𝑎l 𝜔l

)

… −𝑀𝑛l … 𝑁2𝑛 | ⋮ ⋮ … 𝑁𝑛𝑛 … −𝑀𝑛l … 𝑁2𝑛 | ⋮ ⋮ … 𝑁𝑛𝑛

（43）

of

𝑆n (𝜔) = −Γ𝜔2 − [

0 −𝑀2l −𝜁 𝑁22 | 2 ⋮ ⋮ −𝜁 𝑁 + 1 𝑘 −𝑀𝑛2 l 2 |1 𝑁22 ⋮ ⋮ 1 𝑁𝑛2

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So far, a couple of Eqs. (18) and (43) predicts the thermodynamic and kinetic information for the dendrite growth involved in the VTC, which integrates three modules in Fig. 1 and can be

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theoretically connected with the practical processing; see Figs. 1-3 and supplementary Fig. S3. However, three limitations of this model should be further clarified and deserved our attentions, i.e.

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the conservation of momentum (i.e. convection) is not considered; stress accumulation upon sub-rapid solidification and its influence upon determination of kissing point have not been reflected yet; and the

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microstructure formation cannot be analytically and directly described using the current model. Present work aims to build up a theoretical framework, the above-mentioned areas will be improved in

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the future studies.

3. Model demonstration and analysis 3.1. Details for calculations

Application of the present model considering the thermo-kinetic correlation is performed for free

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dendrite growth of undercooled Ni-18 at.%Cu-18 at.%Co alloy, which was originally employed for testing the dendrite growth model proposed for multi-component alloys [32]. To realize the calculations, thermodynamic database (i.e. Gibbs energy description for solution and liquid of pure elements, binary and ternary systems), and diffusion coefficients of Ni, Cu and Co in the liquid are available from Wang et al. [32]. Analogously, values for diffusion speed at the L/S interface (𝑉DI ) and in the bulk liquid (𝑉DL ) are chosen as those applied in Ref. [32]. As explained in section 2.1, however, 19

the parameter 𝑀0 (=290 m/s) used currently does correspond physically to the parameter V0 applied by Wang et al [32], and a higher value for 𝑉0 (=1000 m/s), arising from the thermo-kinetic correlation considered currently, should be according to Ref. [54] assumed. Physical parameters for calculations are summarized in Table 2. Given an initial value for 𝑉, using Newton iteration method, the interface temperature and the interface concentration (Eq. (23)) can be calculated by the present model, which subsequently produces information required for analyzing the dendrite growth; see Eqs. (18) and (43).

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3.2. Dendrite growth of undercooled Ni-Cu-Co alloy melt 3.2.1. Model demonstration

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For ∆𝑇 = 0 K − 330 K, the evolution of 𝑉 with ∆𝑇 is calculated by the present model

considering the thermo-kinetic correlation and shown in Fig. 4, where, Wang’s model calculations [32]

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are also provided for comparison. Analogous to the previous prediction (i.e. by Wang et al [32]), 𝑉 is continuously increased with increasing ∆𝑇, in perfectly consistent with the experimentally observed

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upward tendency, but with a sharp slope change at a sufficiently high ∆𝑇. As compared to Wang’s model, however, different dendrite growth velocities are observed in the medium and low ∆𝑇 ranges,

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i.e. the value of 𝑉 predicted by the present model seems lager than that by the previous model for ∆𝑇 < 62.5 K, but smaller for ∆𝑇 > 62.5 K as shown in Fig. 4. As far as the authors understand, the

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above phenomenon implies the thermo-kinetic correlation (Section 2.1.1) can be reasonably ignored only for sufficiently rapid solidification but not for sub-rapid solidification considered here. From the only comparison made between experiment and simulation shown in Fig. 4, it is difficult to judge the advantages or disadvantages for Wang’s model and present model, as both models showed good

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agreement with experimental results. Actually, further improvements as compared to Wang’s model should be highlighted. Firstly, the present model can be adopted to describe both free (i.e. by negative TG) and constrained (i.e. by positive TG) solidification, in contrast with Wang’s model limited to negative TG (section 3.2). Secondly, the present model also can be connected with the real processing, which will open a new window for industrial application of solidification theories (Section 4.2). Thirdly, the present model considers the thermo-kinetic correlation, in contrast with the thermo-kinetic independence in Wang’s model, which will lead to physically more reasonable and precise 20

simulations. 3.2.2. Thermo-kinetic correlation and characteristic undercoolings Generally, the thermo-kinetic correlation is deduced from first-scale PTs, e.g. non-equilibrium solidifications [60,61], gamma-alpha transformations of Fe-based alloys [62] and PTs involved in microstructure optimization of low-alloyed steels [63], and recently, has been proved quantitatively by Liu et al in Martensitic transformation [36], precipitation in Al-Cu [35], grain boundary migration [37]

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and nano-scale grain growth [64]. Analogous to the well-known strength-ductility trade-off originated physically from the PDs, the thermo-kinetic correlation generally controlling the PTs can also be

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defined as the thermo-kinetic trade-off, i.e. a parameter-couple composed of increased ∆𝐺 and simultaneously decreased 𝑄eff .

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Upon free dendrite growth controlled by the negative TG in bulk liquid, the thermo-kinetic trade-off is according to Eqs. (22) and (25) calculated and illustrated in Fig. 5, where, the continuously

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increased ∆𝐺 or 𝑉 or ∆𝑇 will not be accompanied with the constant 𝑄eff represented by 𝑉0 (section 2.1), but corresponds to the continuously decreased 𝑄eff (Fig. 5), arising from the contribution from thermal diffusion, as reflected by the sharp slope change

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continuously increased

for 𝑄eff at a sufficiently high ∆𝑇 = 187.2 K in Fig. 5. To prove quantitatively it, three characteristic

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velocities [38], such as ∆𝑇(𝑉C ), ∆𝑇(𝑉Rmax ) and ∆𝑇(𝑉DL ) are adopted to discuss the evolving prevalent mechanisms controlling the dendrite growth. Following Ref. [9], 𝑉C is defined as the critical velocity of absolute solute stability, coinciding with the end of purely solute controlled stage without any thermal effect; see Eq. (43) with Δ𝐻 = 0.

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In combination with Section 2.1.1 and Eq. (25), the purely solute-controlled stage ∆𝑇 < ∆𝑇(𝑉C ) is not fully consistent with the thermo-kinetic trade-off mixed by thermal and solutal diffusions (Fig. 5), which does exist but is arbitrarily neglected. As compared to the pure solute effect, the combined thermal-solutal effect will increase 𝑉, so that the absolute solute stability cannot be guaranteed at the original ∆𝑇(𝑉C ) due solely to the solutal effect, and a reduced ∆𝑇(𝑉C ) must be needed. This is reflected by Fig. 6a, where the smaller ∆𝑇(𝑉C ) (=34.9 K by the symbol “◆” in the purple-dashed

21

line and corresponding to the symbol “◆” in the black-dashed line of Fig. 6(b)) is currently deduced, as compared to the previous one (=42.5 K by the symbol “★” in the blue-dashed line and corresponding to the symbol “★” in the red-dashed line of Fig. 6(b)) without considering the thermo-kinetic correlation. Accordingly, the interface instability is according to Eq. (43) enhanced, as reflected by the smaller ∆𝑇 essential for the minimal dendrite tip radius (Fig. 6(b)), as compared to the previous model predictions. Within this typical ∆𝑇 regime, the increased 𝑉 is mainly caused by

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the increased 𝑇i (Eq. (18) and Fig. 6(b)), which further strengthens the atomic diffusion and partition at the L/S interface (Fig. 6(c)), as also reflected by the increased solute partition coefficient (Fig. 6(d)).

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As compared to the transitional point of ∆𝑇 = 62.5 K (Figs. 4 and 6(a)) showing differently evolved velocities subjected to thermo-kinetic correlation; see Eqs. (4-6) and (22-25), 𝑉Rmax is

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defined as the velocity where the maximal dendrite tip radius is achieved, which stands for the

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transition from mainly solute-controlled growth to mainly thermal-controlled growth [54], which is reflected by the same values for ∆𝑇(𝑉Rmax ) = 70.5 K predicted by both the current and the previous

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models (Fig. 6(b)), no matter whether the thermo-kinetic correlation is considered or not. Furthermore, 𝑉DL is defined as the velocity where the global solute trapping is completely achieved, before which the mainly thermal-controlled growth prevails [32,38], in consistent with the thermo-kinetic

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correlation; see section 2.1.1. As compared to the previous model predictions, the decreased 𝑉 (Fig. 6(a)) is ascribed to the increased 𝑄eff (Eq. (25)), as reflected by the decreased 𝑇i (Fig. 6(b)) and the suppressed atomic diffusion (Fig. 6(c)) or increased solute partition coefficient (Fig. 6(d)), respectively.

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With increasing ∆𝑇, the continuously increased contribution from thermal diffusion will

definitively decrease 𝑄eff , which implies the thermo-kinetic correlation (Section 2.1) can be reasonably ignored for sufficiently rapid solidification. As such, the current model prediction considering the thermo-kinetic correlation gives a similar ∆𝑇(𝑉DL ) = 187.29 K to ∆𝑇(𝑉DL ) = 189K predicted by the previous model without considering the thermo-kinetic correlation but still assuming combined thermal-solutal effect [41]. For ∆𝑇 > ∆𝑇(𝑉DL ), the global solute trapping occurs, and the 22

purely thermal controlling mechanism corresponds to a sharp slop change of ∆𝐺 − 𝑄eff (Fig. 5), as reflected by reduction of Eq. (18) to Eq. (4).

4. Application of current model in VTC processing 4.1. Comparison between model and experimental results 4.1.1. Details of calculations In this section, the current theoretical framework considering the thermo-kinetic correlation will be

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used to predict the kissing point involved in the VTC for Al-2 at.%Mg-1.5 at.%Zn and Al-0.7 wt.%Mg-1.1 wt.%Si alloys, where, the dendrite growth model is successfully coupled with the

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practical processing parameters. Thermodynamic database and interaction coefficients between atoms come from Ref. [65], and diffusion coefficient in liquid Al with reference to Ref. [66]. Other physical

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parameters such as diffusion velocity in bulk liquid and Gibbs-Thomson coefficient are obtained from

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Wang et al [31]. Analogous calculations to that performed for undercooled Ni-18 at.%Cu-18 at.%Co alloy will be applied for the VTC processing, where, the geometric information (i.e. the apparatus

4.1.2. Practicability analysis

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parameters) and the processing parameters must be additionally considered (Table 3).

To realize a theoretical connection between the current modeling with the practical processing, as

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discussed in Sections 2.2 and 2.4, the smooth transition of TG evolving in the liquid to the solid have to be permitted, i.e. the absolute value of TG in the solid decided by the processing parameters must be larger than that in the liquid, and thus dominates the overall VTC process. This has been evidenced by Fig. 7, where the TG in the solid seems far higher than that in the liquid and thus becomes the

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major factor controlling the dendrite growth. Then combining the geometric condition (Eqs. (26) and (27)) with heat fluxes at the interface (Eq. (30)), a series of values for 𝑉 can be obtained by adjusting the apparatus and the processing parameters (e.g. 𝐻melt, 𝑅roll and 𝑟roll, 𝑉water, 𝑇w ). Generally, the most important processing parameter, 𝑉water is fixed to guarantee a given 𝑉, so that the relations between kissing point and other apparatus parameters can be determined by analyzing the interface stability theory (Eqs. (18) and (43) in Section 2.4). The kissing point prediction by the current model

23

is performed for Al-2 at.%Mg-1.5 at.%Zn alloy system, where, unfortunately, a comparison with experimental data is not provided; see supplementary Fig. S4. As a substitution, the current model is applied to describe the Al-0.7 wt%Mg-1.1 wt%Si alloy [4], where, a good predication for the kissing point (approximately equivalent to the ending point of solidification [6]) with the evolved 𝜔roll is obtained and shown in Fig. 8; in consistent with Stolbchenko et al [4], the kissing point moves upwards along with the decreased 𝜔roll. Some physical parameters [22, 31, 32, 67-71] and apparatus

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parameters [4] applied for Al-0.7 wt%Mg-1.1 wt%Si alloy dendrite growth are chosen from published literatures and fixed in advance (listed in Table 4), so that the comparison between model prediction

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and experimental measurement without any fitting parameters suggests strongly the powerful feature of the present theoretical framework, i.e. connecting solidification theories with industrial

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applications.

As mentioned above, a lot of parameters (including 𝜔roll) influence the determination of kissing

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point for Al-2 at.%Mg-1.5 at.%Zn alloy system, by which designing and choosing the apparatus parameters can be performed; see supplementary Fig. S5. Analogous to Section 3, the evolution of

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interface condition (𝑉, 𝑋𝑘il and 𝑋𝑘is or 𝑇i ), as characteristics of dendrite growth (Eq. (23)), can be deduced for any fixed processing parameter; see supplementary Fig. S6.

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4.2. Thermo-kinetic trade-off

Analogous to the strength-plasticity trade-off [72], the thermo-kinetic trade-off or the thermo-kinetic correlation, does exist commonly in, e.g., nucleation and growth in PTs [34-37,60-62,64], dislocation slip in PDs [73], and has been tested in free solidifications of

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undercooled Ni-18 at.%Cu-18 at.%Co alloy (Section 3.2; see also Fig. 9(a)). Is this thermo-kinetic trade-off also valid for the VTC processing? Analogous to Section 3.2, the relationship between ∆𝐺 and 𝑄eff controlling the typical VTC

processing is directly depicted in Fig. 9(b), where, with increasing 𝑉, a negative correlation between ∆𝐺 and 𝑄eff is observed obviously. Consequently, the decreased 𝑄eff with ∆𝑇 (< ∆𝑇(𝑉DL )) for both Fig. 9(a) and (b), is ascribed to the reduced difference between 𝑋𝑘il and 𝑋𝑘is and the declined 𝑇i from Eq. (25) related to the thermal and solutal diffusion; see also Fig. 6(b) and (c) and Fig. S6(a) and 24

S6(b). As for the undercooled solidification (Fig. 9(a)) with the negative TG in the liquid, both the declined 𝑇i and the suppressed solute partition come from the increased ∆𝑇, which promotes ∆𝐺 (black line), and simultaneously decreases 𝑄eff (red line); see also Section 3.2.2. Nevertheless, for the constrained solidification with the positive TG in the solid, the accelerated 𝑉 or the declined 𝑇i does not depend on ∆𝑇 but the cooling rate in the solid, which guarantees the VTC processing as an artificial controlling process and brings different behaviors to the thermo-kinetic trade-off. For

of

example, the plateau occurring in the 𝑉-∆𝐺 curve (Fig. 9(b)) originates physically from the upgraded 𝑇i arising from the diffusional concentration aggregation (Fig. S6(a) and (b)), but corresponds

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realistically to the continuously decreased 𝑄eff , thus forming the different trade-off relation from that shown in Fig. 9(a). As far as Eq. (18) highlights, the upgraded ∆𝐺 can be alleviated by artificially

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cooling, to weaken the diffusional concentration aggregation, and in turn, to dramatically decrease 𝑄eff . That is to say, a parameter couple of negligibly varied ∆𝐺 with continuously declined 𝑄eff is

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possible, but only in the constrained solidification with the positive TG in the solid. Up to now, the thermo-kinetic correlation has been validated for the VTC processing, which,

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different from free dendrite growth of undercooled melts, can be controlled by various artificial methods, i.e. adjusting the apparatus and the processing parameters.

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4.3. Connection between theoretical framework and processing parameters As compared to free dendrite growth by the negative TG, constrained dendrite growth by the positive TG alleviates the effect of recalescence [39], i.e. releasing of latent heat, particularly for sub-rapid solidification involved in the VTC, in combination with deformation initiated from the

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kissing point. That is to say, the as-solidified morphology formed immediately upon the kissing point can be basically determined solely by thermodynamics and kinetics of the dendrite growth, thus forming the correlations among ∆𝐺, 𝑄eff and as-solidified microstructure. Arising from the classical nucleation-growth theories, generally, different parameter couples by large ∆𝐺 (corresponding to high nucleation rate) and small 𝑄eff (corresponding to high growth rate), small ∆𝐺 and small 𝑄eff , large ∆𝐺 and large 𝑄eff , and small ∆𝐺 and high 𝑄eff tend to produce so-called fine dendrite bundles [74], coarse equiaxed dendrites [75], ultra-fine equiaxed dendrites [76] and directional 25

mono-crystal [77], respectively. Corresponding to the above different ∆𝐺-𝑄eff couples, such different as-solidified structures, further subjected to deformation and solid treatment, will lead to different resultant microstructure with different mechanical properties. Generally for the VTC, the kissing point (Fig. S1), corresponding to the contact of oppositely growing dendrites, is determined by combination of solidification and geometric conditions (Fig. 3), as reflected by 𝑉 (i.e. corresponding to 𝑉water) and the apparatus parameters (e.g. 𝑅roll, 𝐻melt ). On

of

this basis, the fixed set of apparatus parameters but with different alloy compositions, have been connected with the current model predictions (Fig. 10(a)), where for different alloy compositions, the

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evolution of 𝑉 (i.e. as reflected by 𝑉water) corresponds to different evolutions for the

parameter-triplets ∆𝐺-𝑄eff - 𝑉, which indicates that, the same kissing point reflected by the fixed set of

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apparatus parameters, can be satisfied by the different combinations of 𝑉 and alloy compositions, corresponding to different couples of ∆𝐺 and 𝑄eff . Furthermore, from the Fig. S6 and other works

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[39], the dendritic tip radius 𝑅 will decrease and the concentration will become more uniform with strengthening the coolant water flow. Therefore, fine microstructure can be achieved by designing the

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0 0 alloy following small 𝑄eff (corresponding to 𝑋Mg =1 at.%, 𝑋Zn =0.5 at.% in Fig. 10(a) at the same

∆G. Analogously, five different sets of apparatus parameters, in combination with the same alloy

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composition, have also been connected with the current model predictions (Fig. 10(b)), where, almost the same evolutions of ∆𝐺-𝑄eff couples with increasing 𝑉 are decided by the fixed composition (as indicated by the thermo-kinetic trade-off surface including five differently evolving parameter triplets; see yellow curved surface in Fig. 10(b)), and independent relations of 𝑉-∆𝐺 and 𝑉-𝑄eff are decided

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by different apparatus parameters; see correspondent coordinate system in Fig. 10(b). Clearly, different kissing points arising from different sets of apparatus parameters (only 𝑅roll considered here) can be satisfied by the same 𝑉 controlled by 𝑉water, but corresponding to different couples of ∆𝐺 and 𝑄eff , such as large ∆𝐺 and small 𝑄eff (corresponding to 𝑅roll = 0.215 in Fig. 10(b)), and small ∆𝐺 and high 𝑄eff (corresponding to 𝑅roll = 1 in Fig. 10(b)). Therefore, for a fixed alloy composition, different sets of apparatus parameters can be designed to further satisfy the artificial combinations of parameter-triplets ∆𝐺-𝑄eff - 𝑉, which corresponds to different kissing points. The 26

microstructure is closely related to deformation introduced by down-stream from the kissing points. Therefore, the processing optimization for VTC, from Figs. 8, 10(b), S5 and S4, can be performed by designing and choosing values of ∆𝐺 and 𝑄eff (corresponding to the apparatus parameters) to control the kissing points and further microstructure. It then follows that, the thermo-kinetic trade-off, like the strength-ductility trade-off, is absolutely held, but the trade-off behavior is relatively changing, i.e. the absolute value of ∆𝐺 and 𝑄eff varying

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along different trade-off relations can be modulated by tailoring the alloy composition and the apparatus and the processing parameters; see Fig. 10. This highlights the essence of the theoretical

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connection between the current theoretical framework and the VTC processing.

5. Conclusion

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Arising from the synergy of thermodynamics and kinetics, a more flexible thermo-kinetic

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modeling for the dendrite growth is herein developed for multi-component alloys, where, two inherent targets, i.e. the thermo-kinetic correlation and the theoretical connection between dendrite growth

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model and physically practical processing, have been realized. Accordingly, both the thermodynamic driving force and the effective kinetic energy barrier have been found to control quantitatively the dendrite growth, as reflected by so-called thermo-kinetic trade-off. Appling the present model into the

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VTC, a good prediction for the kissing point has been performed, particularly, the parameter-triplets, i.e. ∆𝐺-𝑄eff - 𝑉 has been proposed to tailor the alloy design and processing optimization for such industrial processing as the VTC. The thermo-kinetic correlation guarantees quantitative connections between the practical processing and the present theoretical framework, and thus opens a new

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beginning for connecting solidification theories with industrial applications.

Acknowledgements This work was supported financially by the National Key R&D Program of China (Nos. 2017YFB0703001 and 2017YFB0305100), the Natural Science Foundation of China (Nos. 51134011, 51431008 and 51671075), the Research Fund of the State Key Laboratory of Solidification Processing 27

(Nos. 2019-TZ-01 and 2019-BJ-02), the China Postdoctoral Science Foundation (No. 2018M643729 and 2019T120942), and the Natural Science Basic Research Plan in Shaanxi Province of China (No.

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2019JQ-091).

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[44] J. Svoboda, E. Gamsjäger, F.D. Fischer, Y. Liu, E. Kozeschnik, Acta Mater. 59 (2011) 4775-4786. [45] J. Svoboda, E. Gamsjäger, F.D. Fischer, Acta Mater. 54 (2006) 4575-4581. [46] M.A. Hartmann, R. Weinkamer, P. Fratzl, J. Svoboda, F.D. Fischer, Philos. Mag. 85 (2005) 1243-1260. [47] J. Svoboda, F.D. Fischer, P. Fratzl, A. Kroupa, Acta Mater. 50 (2002) 1369-1381. [48] J. Svoboda, F.D. Fischer, P. Fratzl, Acta Mater. 54 (2006) 3043-3053.

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[52] D.P. Bertsekas, Chapter 2 - The Method of Multipliers for Equality Constrained Problems, in: D.P. Bertsekas (Ed.), Constrained Optimization and Lagrange Multiplier Methods, Academic Press,

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[65] H. Liang, S.L. Chen, Y.A. Chang, Metall. Mater. Trans. A 28 (1997) 1725-1734. [66] D. Colognesi, M. Celli, A.J. Ramirez-Cuesta, M. Zoppi, Phys. Rev. B 76 (2007) 174304. [67] M. Vandyoussefi, A.L. Greer, Acta Mater. 50 (2002) 1693-1705. [68] M. Rappaz, W.J. Boettinger, Acta Mater. 47 (1999) 3205-3219. [69] Y.X. Shi, S. Sun, M.G. Guo, J.X. Zhang, Y. Liu, Foundry Technol. 38 (2017) 867-876. [70] J.P. Li, Z.L. Xu, C.G. Liu, Foundry 52 (2003) 783-785.

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Figure list:

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

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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.

33

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

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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

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inner-interface (∂ΩS/L), the bulk solid (S) and the bulk liquid (L) with solute fluxes (𝐽𝑘s,l,il,is),

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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

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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).

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Fig. 4. Evolution of the dendrite tip velocity (𝑉) with the bath undercooling (∆𝑇) of the undercooled Ni-18 at.%Cu-18 at.%Co alloy during free dendrite growth, where the full line denotes the predicted

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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

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between the two predicted results of PM and Wang at ∆𝑇 = 62.5 K.

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

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Fig. 6. Evolution of the interface behaviors at the dendrite tip with the bath undercooling (∆𝑇) of the undercooled Ni-18 at.%Cu-18 at.%Co alloy during free solidification predicted by the present model

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(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

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∆𝑇.

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concentrations 𝑋𝑘il , 𝑋𝑘is , and (d) solute partition coefficient K, evolving with the bath undercooling

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Fig. 7. Absolute value of the temperature gradient in the solid (|𝐺S |) and liquid (|𝐺L |) versus the

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coolant water flow (𝑉water ) during the overall VTC process of the Al-2 at.%Mg-1.5 at.%Zn alloy.

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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

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element calculations (dot line) reported by Stolbchenko et al [4].

Fig. 9. Evolution of the thermodynamic driving force ∆𝐺 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.

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Fig. 10. Evolution of the parameter-triplets (∆𝐺-𝑄eff -𝑉) reflected in the process of VTC for the

0 0 Al-Mg-Zn alloys: (a) different alloy compositions (𝑋Mg (at.%): 1, 2, 3 and 𝑋Zn (at.%): 0.5, 1.5, 2.5.) but

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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

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trade-off surface, existing five conditions with different roll radius and height of kissing points, i.e.,

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𝑅roll (mm): 215, 415, 615, 815, 1000 and 𝐻kiss (mm): 8.7, 19.4, 9.3, 39.4, 30.1.

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Table Table 1 Evolution of dendritic growth model arising from deviation between the classical theoretical hypothesis and the practical processing path Liquidus & solids slope

Solution approximation

Alloy component

Thermal transportation

Model parameters

Thermodynamics & kinetics

Localequilibrium

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

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Non-equilibrium level

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Correlation

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. 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, 𝑉DI (m/s)

19

Diffusion speed in the bulk liquid, VDL (m/s)

20

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Definition/Symbol

[26]

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[26]

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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. Value

Refs.

Outer roll radius of copper rolls, 𝑅roll (m)

0.215

[4]

Inner roll radius of copper roll, 𝑟roll (m)

0.200

[4]

Height of molten pool, 𝐻melt (m)

0.040 (0.025-0.075)

[4]

Roll gap, 𝑑gap (m)

0.003 (0.0015-0.0055)

[4]

Density of coolant water, 𝜌water (kg/m3)

1×103

Average flow of coolant water, 𝑉water (m3/s)

2×10-8

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Definition/Symbol

Heat capacity of coolant water, 𝐶water (J/(kg K))

Thermal conductivity in α-Al, 𝐾s (W/(m K))

PM

4.18×103

[69]

324

[70]

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Thermal conductivity in copper, 𝐾M (W/(m K))

[69]

300 (209-420)

[4]

78 (75-80)

[4]

Average heat resistance in the interface of roll/solid, 𝑅̅𝑖nterface (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, 𝑉DI (m/s)

1

[31]

Diffusion speed in the bulk liquid, 𝑉DL (m/s)

10

[31]

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Thermal conductivity in liquid melt, 𝐾l (W/(m K))

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

Refs.

Outer roll radius of copper rolls, 𝑅roll (m)

0.215

[4]

Inner roll radius of copper roll, 𝑟roll (m)

0.200

[4]

Height of molten pool, 𝐻melt (m)

0.040 (0.025-0.075)

[4]

Roll gap, 𝑑gap (m)

0.003 (0.0015-0.0055)

[4]

Density of coolant water, 𝜌water (kg/m3)

1×103

[69]

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Definition/Symbol

Average flow of coolant water, 𝑉water (m3/s)

3×10-8

PM

4.18×103

[69]

324

[70]

300 (209-420)

[4]

78 (75-80)

[4]

Average heat resistance in the interface of roll/solid, 𝑅̅interface (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, 𝑉DI (m/s)

0.001

[22]

Diffusion speed in the bulk liquid, 𝑉DL (m/s)

10

[31]

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Heat capacity of coolant water, 𝐶water (J/(kg K))

Thermal conductivity in α-Al, 𝐾S (W/(m K))

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Thermal conductivity in liquid melt, 𝐾l (W/(m K))

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Thermal conductivity in copper, 𝐾M (W/(m K))

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